Look for and Express Regularity in Repeated Reasoning: Concepts to Procedures

It has been well said that the highest aim in education is analogous to the highest aim in mathematics, namely, not to obtain results but powers, not particular solutions, but the means by which endless solutions may be wrought. George Eliot, 1885

In the previous standard (Look for and use the structure of mathematics), the framers of CCSS-M and SMP placed an emphasis on students learning to observe and appreciate the structure of mathematics—paying attention to its unique properties and systems, big ideas, and organizing principles, appropriate to their age and grade level. The goal of this standard is to help students to see the nature and appreciate the power of mathematics.

The eighth standard: Look for and express regularity in repeated reasoning promotes the ability to recognize quantitative (numeric) or spatial (geometric) patterns, make connections, and use those patterns to arrive at an understanding of mathematics structures—the general methods, the reasons behind the efficiency and elegance of algorithms/procedures, work out effective strategies, properties of numbers and operations at their grade levels.

In the previous standard, students looked for and used the structures in mathematics to apply them to learn more mathematics and solve problems. In this standard, the aim is to develop those structures from concepts and even construct new ones—new to students or even new to mathematics at their grade level. This process provides skills for students to own, retain and communicate, efficiently, accurately, and with rigor, the understanding and mastery of mathematics content. It also means developing the awareness of their own learning and the nature of mathematics conceptualization—the patterns. But most importantly, the objective is to help students to create interest in mathematics and to continue it with accuracy, confidence, and enthusiasm beyond school.

Mastery of this standard allows students to complete the cycle of learning mathematics: try problems, recognize a pattern in the solution process, discern a pattern(s), see the mathematics structure in the pattern, generalize the terms of that structure, generate a conjecture, test the conjecture, and then arrive at a key concept, procedure, strategy, or relationship between different elements of mathematics.

What is a pattern?
A pattern is a relationship between many of pieces of data, that is on going, something that is repeating—a data or a relationship between data, and can be predicted either explicitly or implicitly. Can one have a pattern when there are one, two, or three pieces of data? One piece of data is an incident, two are a coincident, with three there is a possible beginning of a pattern. If there is definite relationship between these pieces of data, then one needs a fourth one to verify that pattern.

Observing patterns and regularity in processes is the key to learning and using the prerequisite skills for mathematics learning. Students experience and develop affinity for mathematics when they also develop the prerequisite skills for mathematics learning. These prerequisite skills include: ability to follow sequential directions, classification, organization, quantitative and spatial reasoning, visualization, deductive and inductive reasoning. In designing lessons, therefore, teachers must also include the development of these prerequisite skills in addition to introducing interesting and relevant content. However, they should be integrated rather than isolated exercises.

A central idea here is that mathematics is drawing general results (or at least good conjectures for the moment) by trying examples and looking for regularity in the results generated by these patterns. In this standard, we want students to experiment with examples and arrive at patterns and then generalize those patterns to arrive at workable rules and procedures.

When students have explored several problems related to a concept, a few inevitably say, “I see a pattern here, so do I have to do all this work?” This is where the teacher moves the learning into generalizing mode—using the pattern to arrive at a conjecture, testing it and then arrive at the procedure or rule. She can’t stop at the pattern level. Students need to test their pattern as well as to recognize it in diverse situations and construct counter examples or non-examples. For example, instead of teaching just the rules for combining integers, students should look at patterns of combining numbers to generalize and come up with the rule. In the number -3, the digit 3 is the numeral and -3 is the number. Thus, a number is a directed numeral.

Mathematically proficient students develop two opposing but intertwined abilities: persevering in solving problems and looking for generalizable, efficient methods (i.e., short-cuts). These students notice patterns in calculations, use those patterns to create shortcuts, and learn multiple algorithms for the same idea and choose the appropriate one for the context. This helps them to do more interesting mathematics with less effort, thus generating an inclination for mathematics.

On the other hand, low achieving students lack both these qualities. As a matter of fact, low achieving students work harder in mathematics classes as they persevere using inefficient and long-drawn out methods, for example, they will use counting for learning addition and subtraction facts rather than efficient strategies based on decomposition/recomposition. Partly, it is not their fault; they have been instructed to do so. Because of these laborious methods, they give up easily. For them mathematics becomes hard, uninteresting, and a collection of unrelated procedures because they are not given help to develop efficient strategies and practice in seeing patterns and mastering strategies.

Solving many problems develops students’ inductive reasoning, and then justifying and communicating their thinking develops deductive reasoning. Inductive and deductive thinking are essential components of the mathematical way of thinking. The SMP classrooms are expected, through these practices, to develop habits of higher order thinking skills. Synthesizing concrete models and examples into symbolic and abstract ideas, at the elementary school, are good examples of applications of inductive reasoning. In contrast, most middle school and high school concepts and procedures are based on deductive reasoning or analysis.

Habits of mathematical thinking (observing patterns, looking for structure and regularity, repeated reasoning, making conjectures, providing examples, and counter examples, etc.) enrich learners’ problem solving experiences. By looking for patterns, developing conjectures, making discoveries and inferences, generalizing results, monitoring the solving process, checking the reasonableness of answers, and devising new avenues to explore to make learning mathematics engaging, they develop the habit of persevering (increasing stamina) for problem solving and communicating what they know.

Mathematical practices such as constructing reasoning and critiquing the reasoning of others are examples of deductive reasoning; however, this particular standard is best developed when students have extensive experiences in inductive reasoning. Observation of regularity, patterns, structure, and symmetry develops inductive reasoning. Further, practice of formal mathematics logic and reasoning such as supporting arguments and formal proofs develops rigor and a deeper understanding of mathematics. Deeper understanding in mathematics is an integration of deductive and inductive reasoning: analysis and synthesis in the process of solving problems.

Being a student of mathematics means balancing both qualities. Sometimes one needs to persevere and to stick with a set of strategies that will definitely work, given time. Putting in the effort, even though it is hard, is necessary. But, the effort must be put in developing and using efficient methods and strategies. Getting stuck doing the same inefficient methods is not what being a mathematician is about. Mathematicians explore new approaches and look for efficient and elegant solutions. They want methods to be powerful and generalizable. They effectively use what they know and have done to become efficient, more powerful problem solvers. They look for and represent regularity in repeated reasoning in order to develop efficient shortcuts or notations. This standard urges us, as teachers, to prepare and remind students that mathematicians move from painstaking but reliable problem-solving methods to efficient procedures via looking for regularity in their repeated reasoning.

Look for and express regularity in repeated reasoning.
Regularity and repeated reasoning abound in mathematics. Many facts, diverse terms, concepts and processes reoccur in the development of mathematical ideas. In this standard, the key words are: “look for” and “express” regularity, which means the realization on the part of students that learning mathematics is looking for and seeking regularity, recurrence, and patterns, the role of regularities and patterns, and describing these with mathematical symbols and expressions. Experiencing regularity and repeated reasoning—the processes and procedures, that sometimes may be generalized and even created into shortcuts, may lead to developing a rule, creating a formula, or arriving at an efficient and elegant procedure.

For example, the idea of decomposition and recomposition of numbers recurs whenever we add natural numbers (58 + 64 = 50 + 8 + 60 + 4 = 50 + 60 + 8 + 6 = 50 + 60 + 8 + 2 + 4 = 50 + 60 + 10 + 2 = 50 + 50 + 10 + 10 + 2 = 100 + 20 + 2 = 122, 58 + 2 + 62 = 60 + 62 = 60 + 60 + 2 = 122 or 58 + 64 = 50 + 60 + 8 + 4 = 110 + 12 = 122) to rational numbers. Consider the example,
A recipe calls for 1cups of milk. Larissa has 3cups of milk. How much milk will be left for the cake recipe, she plans to make?

Using the decomposition/recomposition, one observes:
1⅞ +⅛ =2 and then 2 + 1⅛ = 3⅛.
Therefore, to go from 1to reach 3⅛, we took jumps of cap1and or cap2        

Both of these examples can be translated into a powerful method called Empty Number Line (ENL). Several other methods have emerged in the same way (e.g., the Bar Model, used in Singapore and other Asian countries).

The area of a rectangle is used in arriving multiplication procedures from two-digit by two-digit numbers to multiplication of complex numbers. This recurrence is the basis of making connections between concepts and procedures.

For example, consider the instructional sequence, in a problem in which students study the concept of rate and ratio in a mixture of juice (NCTM Practice Illustration, 2013).

For every 5 cups of grape juice, mix 2 cups peach juice. Students are asked to develop a general mathematical relationship between the amounts of juices (say x and y, representing the two juices) needed to create a particular amount of mixture with the same taste.
cap3

To graph this data, we begin with (0,0) and one finds that for each 1 unit one moves to the right, one moves up of a unit. Thus, when one goes 2 units to the right, one goes up 2· units. Similarly, when one goes 3 units to the right, one goes up 3· units and when one goes 4 units to the right, one goes up 4· units. Observing this regularity, one can generalize this to:
                        When one goes x units to the right, one goes up x· or (x) units.
Therefore, starting from (0, 0), to get to a point (x, y) on the graph, one goes x units to the right, then one goes up units. Therefore, one can relate the two variables, x and y by the expression: y = = (x)

While students may immediately notice some regularity, it is the process of expressing—from noticing a pattern in the observations about the data in the table—the pattern into statements like “if we increase the grape juice by 1 cup, we must increase the peach juice by of a cup to have the same” in language and ultimately to translating this statement into writing the equation y = (x) which constitutes the bulk of the mathematical work of the task. The formulation of CCSSM, at appropriate places, provides language to discuss this kind of expressive mathematical work—a result from the regularity of the data.

Mathematically proficient students notice if calculations are repeated. They look for general methods (by observing specific examples, they extend to generalized situations) and for shortcuts (from general cases, they arrive at efficient applications to specific cases). The observation of regularity and repeated reasoning is an important aspect of both deductive and inductive reasoning.

In the early grades, students notice repetitive actions in counting and computation, etc. For example, all addition and subtraction facts can be derived by using the strategies that employ the property of making ten and decomposition and recomposition of numbers (9 + 7 = 9 + 1 + 6 = 10 + 6 = 17, 9 + 7 = 6 + 3 + 7 = 6 + 10 = 16). Similarly, when children have multiple opportunities to add and subtract―ten and multiples of ten, they notice the pattern and gain a better understanding of place value. And then decomposition/recomposition is extended to multiplication in the form of distributive property of multiplication over addition or subtraction.

For example, an addition fact-practice exercise might result in seeing patterns and generalization: ask children to choose three consecutive whole numbers (e.g., 5, 6, and 7) and compare the double of the middle number to the sum of the two outer numbers. In this example, the two sums are 12 and 12. After they do this for several such triplets of numbers, they are likely to conjecture a pattern that allows them to state that when the two numbers are two apart, their sum is double of the middle number, 6 + 8 = 7 + 7, 19 + 21 = 20 + 20, etc. In later grades, they will realize that the average (mean) of three consecutive numbers is the middle number and later they will generalize the result to the triplet: (N -1), N, and (N+1) with the equation: (N-1) + (N+1) = N – 1 + 1 + N = N + N = 2N.

After several examples with addition, some students may ask what will happen when we ask the same question about multiplication about this triplet. Teacher asks children to compare the middle number times itself to the product of the two outer numbers in the triplet of the consecutive whole numbers. The two products are 36 and 35. After they do this for several triplets of whole numbers, they are likely to conjecture a pattern that allows them to multiply 9 × 11, 29 × 31, 99 × 101, etc. mentally because they expect it to be one less than 10 × 10, 30 × 30, 100 × 100 which they can do in their heads. Seeing this regularity is typically easy for fourth and fifth graders, but expressing it clearly and formally is harder. They do not have the language for communicating.

In later grades, students are able to understand the idea of naming the numbers. A simple non-algebraic “naming” scheme can be used to describe the pattern: the numbers are named “middle/inner” and “outer.” In this form, the triplet can be seen as “middle minus one”, middle, “middle + 1” or (middle – 1), middle, (middle + 1). Then they can attempt to state the result as: (middle – 1) × (middle + 1) = (middle)2 – 1. The step from this statement to standard algebra is just a matter of adopting algebraic conventions: naming numbers with a single letter m instead of a whole word like “middle,” and omitting the × sign: m2 – 1 = (m + 1)(m – 1). It is revisited in algebra as we teach this pattern in factoring. It is generalized into: x2 – 1 = (x + 1)(x – 1). Let’s take the first problem: if x = 10, then 10 × 10 = 100 and 9 × 11 = 99. If x is 10, then x2 – 1 = 99 and (x + 1)(x – 1) = 100 – 1= 99. It helps students to begin with number patterns before moving into using the concept of variables to describe the pattern.

Upper elementary students, for example, might notice that when dividing 25 by 3, after the initial step, they are repeating the same calculations and conclude they have a repeating decimal. In order to arrive at these general observations, the teacher needs to give examples of this kind of division to show that the repetition of digits is dependent on the number used for the divisor. For example, when the divisor’s prime factorization involves only the prime numbers 2 and 5, the resulting decimal will be terminating as 10 and 10n, for every positive integer n have a prime factorization as 2m5p, for some non-negative integers m and p. For example:
cap4

On the other hand, when prime factorization of the divisor involves some other prime numbers or does not have 2 and/or 5 as prime factors only, the division results in a non-terminating, repeating decimal. The repetition will occur before the number of digits is less than the divisor. For example, in the case of ⅔, the repetition will take place with less than three digits as the divisor is 3. In the case of these fractions cap5 the repetition will take place before in number of digits is 7 or less than 7 and 11 or less than 11 digits, respectively.

Middle school students might abstract the equation (y – 2)/(x – 1) = 3 by paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3. Let (x, y) be any point on the line passing through (1, 2) with slope 3, then applying the definition of slope formula using two points (x, y) and (1, 2), one has the equation:
cap6
This converts into the form: y 2 = 3(x − 1), or by simplifying the equation, we get the equation in the y = 3x −1 in the slope-intercept form. Therefore, in general, the equation of the line passing through the point (x1, y1) and slope m is: (y – y1) = m (x –x1).

Earlier we observed that (x – 1) (x + 1) = x2 – 1.   Similarly, noticing the regularity in the way terms cancel when expanding (x1)(x + 1) = x2 + x – x – 1 = x2 − 1, (x – 1)(x2 + x + 1) = x3 − 1, and (x – 1)(x3 + x2 + x + 1) = x4− 1, etc., we can generalize it to:
                                       (a – b) (a2 + ab + b2) = a3 – b3
and by symmetry, we have:
                                       (a + b) (a2 – ab + b2) = a3 + b3.

However, if we continued the repetition and the pattern leads to the general formula for the sum of a geometric series:
Let S = 1 + x + x2 + x3 +… + xn-1 + xn.
Then, multiplying both sides by (x − 1)(assuming, of course, if (x − 1) ≠ 0, or x ≠1, as we can only divide or multiply by a non-zero number), we have
(x − 1)S = (x − 1)(1 + x + x2 + x3 + … + xn)
(x − 1)S = xn− 1
S = (xn− 1)/(x− 1).
Therefore, we have:
(x − 1)(1 + x + x2 + x3 + … + xn) = xn− 1
And, when |x|<1, xn approaches 0 for n is very large, then the above relation gives us the sum for a geometric series:
1 + x + x2 + x3 +… + xn + …
S = 1/(x−1).

Regularity also means looking for similarity and symmetry. The development of divisibility rules is a good example of regularity, symmetry, and repeated reasoning. For example, when we consider the divisibility test for 2, we look for the one’s place being even as the value of digits in other places (10’s, 100’s, etc.) are already divisible of 2 and will not leave any remainder when divided by 2 as its value is a multiple of 10n for every natural number n which in turn is a multiple of 2. Thus, a six digit number: 135,798 is divisible by 2 as 135,798 = 100,000 + 30,000 + 5,000 + 700 + 90 + 8, each one of the numbers is divisible by 2, therefore, their sum is also divisible by 2. Therefore, a number is divisible by 2 if the digit in the one’s place is an even number.

Similarly, when we consider the divisibility test for 4, we look for the number formed by the ten’s and one’s place digits being divisible by 4 as all the other digits in the number will not yield any remainder as their value is a multiple of 10n, for every natural n greater than 1 is a multiple of 4 (i.e., 102 =100, 103 = 1,000, etc.). Therefore, a number is divisible by 4 if the number formed by the ten’s and one’s digits is divisible by 4. Example: In the number 333,324, the ten’s and one’s digit form the number 24, that is divisible by 4, so 333,324 is divisible.

Now, we can develop a test for divisibility by 8 by following this pattern, e.g., 111112 is divisible by 2 (look at one’s place); 111132 is divisible by 4 (look at 32—the number formed by the ten’s and one’s place); and 111128 is divisible by 8 (look at 128—the number formed by hundred’s, ten’s, and one’s place). Thus, for the divisibility by 8, we look at the number formed by the 100’s, 10’s, and 1’s digits as all other digits in the number will have values as multiples of 10n for every natural number n > 2, and when divided by 8 will not leave any remainder. Thus, a number is divisible by 8 if the number formed by the hundred’s, ten’s and one digit is divisible by 8. Or simply, if a number is divisible by 2 and then the new quotient is again divisible by 2, then the given number is also divisible by 4 and if the number is divisible by 2 and then by 2 and then again by 2, then the number is divisible by 8. For example, 1448 is divisible by 2, 4, and 8, so is 111,448.

For elementary and middle school children above results are achieved by observing the nature of many numbers and actually performing the division or the teacher playing a game of seeing the pattern in the divisibility and then arriving at the tests. 

Many middle school and almost all high school students should be given experiences in deriving these results. In general, the number N = an an-1 an-2 ……. a0 (In the expanded form it is N =10n an +10n-1 an-1 + ……+ 102 a2 +101 a1 + 100 a0 ) is divisible by 2, if a0 is an even digit and an, an-1, an-2, ….., a1 can be any digit. N is divisible by 4 if the number the number a1a0 (which in the expanded form is 101 a1 + 100 a0) is divisible by 4. Similarly, N is divisible by 8 if the number a2 a1 a0 (which in the expanded form is 102 a2 +101 a1 + 100 a0) is divisible by 8 as 10n an, 10n-1 an-1, 10n-2 an-2, …103 a3 are all divisible by 2, 4, 8 for any n ⫺ 3.

The divisibility by 5 and 10 is quite clear: the one’s place should have 0 or 5 in the case of divisibility by 5 and 0 in the case of divisibility by 10 as all other places are multiples of 10n, where n ⫺ 1.

Let us consider the divisibility by 3. We can see a pattern in the numbers that are divisible by 3. For example, in the numbers 33; 24; 234; 111; 232323 the sum of the digits in these numbers is a multiple of 3 and we observe that the numbers are divisible by 3. Once again, the teacher gives the examples of numbers that are divisible and not divisible by 3. And create two columns. One for the numbers divisible by 3 and the other with those not divisible by 3. By observing these numbers and share their observations. The teacher provides examples, counter examples, and non-examples to test their conjectures. Students arrive at a pattern. Thus, by observation they arrive at a conjecture (a conjecture is an observed pattern) for the divisibility of 3: A number is divisible by 3 when the sum of its digits is a multiple of 3.

In a general case, we can use mathematical reasoning to prove this statement. The proof for high school students is a very good example of understanding place value, laws of exponents, and the concept of division. Let us consider a six digit number: N = abcdef where a, b, c, d, e, f are digits in the number and can assume values between 0 and 9, except a ≠ 0, otherwise it will not be a six digit number (we could easily extend it to a more general number and any number of digits, N = an an-1 an-2 ……. a0 =10n an +10n-1 an-1 + ……+ 102 a2 +101 a1 + 100 a0, where the coefficients are digits ).
N = abcdef = a×105 + b×104 + c×103 + d×102 + e×101 + f×1
= a×100000 + b×10000 + c×1000 + d×100 + e×10 + f×1
= a×(99999+1) + b×(9999+1) + c×(999+1) + d×(99+1) + e×10 + f×1
Now, if we divide the number by 3, the numbers 99,999; 9,999; 999; 99; and 9 will not have any remainder. When divided by 3 each one of the expressions (99999+1), (9999+1), (999+1), (99+1), and (9+ 1), and 1 will have a remainder of 1. Thus, the expressions a×(99999+1), b×(9999+1), c×10(999+1), d×(99+1), e×10, f×1when divided by 3 will have remainders a, b, c, d, e, and f respectively. Therefore, the right hand side of the expansion of abcdef will have only a + b + c + d + e + f as the remainder. Thus, when the number N= abcdef is divided by 3, then the remainder is a + b + c + d + e + f. Therefore, the number N = abcdef is divisible by 3 if the sum of the digits a + b + c + d + e + f is divisible by 3. Thus, the divisibility test for 3 is proven. Similarly, the test for divisibility by 9 can be developed using the same reasoning: A number is divisible by 9 if the sum of its digits is a multiple of 9.

The number 6 is a product of 2 and 3. The test for 2 and 3 combined works for 6. Thus, a number is divisible by 6 if it is divisible by 2 and 3. This test works because the numbers 2 and 3 have an interesting relationship. 6 = 2×3 and 2 and 3 are relatively prime (two numbers are called relatively prime when their greatest common factor is 1).

This suggests that this divisibility test can be generalized to any whole number. For example, a number is divisible by 24, if it is divisible by 8 and 3; since, 24 = 8×3 and 8 and 3 are relatively prime number. For example, 144 is divisible by 24 because 144 is also divisible by 8 and 3. However, a number may not be divisible by 24 if the number is divisible by 12 and 2 as 12 and 2 are not relatively prime numbers (2 is their greatest common factor). For example, 36 is not divisible by 24 although 36 is divisible by12 and 2 and 24 = 12×2. The condition of two factors being relatively prime is important. Thus, in general, a number n is divisible by another number c, if n is divisible by a and b, where c = a×b, and a and b are relatively prime. Thus, for most purposes—learning operations on fractions, simplifying radical expressions, etc., the divisibility test for up to ten are adequate. However, there is not a “good” test for the divisibility by 7. There is a procedural test for the divisibility by 7 developed in the 7th century.

A great deal of research shows that one can predict which students will do well in higher mathematics are those who are well-versed and fluent in long division and operations on fractions. This is particularly so in a course on algebra. And the four major prerequisite skills for understanding and mastering the concept of fractions and operations on them are:

  • Multiplication tables
  • Divisibility rules
  • Short-division, and
  • Prime factorization.

These are the tools of success in mastering fractions and then algebraic operations. These skills are best derived by patterns and then practiced using these patterns.

To make the observation of regularity, pattern, repeated reasoning, and structure possible, both teachers and students need to pose a series of questions such as:

  • Is there a process—operation, property, condition that I am repeating?
  • Can I use this repeated idea to create a pattern and then to develop a conjecture?
  • Can I make a generalization of the conjecture?
  • Can I express this abstractly based on this repeated reasoning?
  • Does this abstraction result into a procedure, method, or strategy?
  • Are there exceptions to this generalization/abstraction?
  • Can I make this method, procedure more efficient and elegant (this is like editing a piece of writing)

Teachers can ask questions such as:

  • You claim that you observed a pattern, how do you know it is a pattern?
  • Can you describe your pattern, conjecture, or method?
  • Explain why your method makes sense?
  • Can you describe your method in formal mathematics terms and reasoning?
  • How would you explain why it works?
  • Have you checked whether the answer seem reasonable?
  • What have you learned about ….?
  • How would this work with other numbers or situations? Does it work all the time? How do you know?
  • What do you notice when …?

Mathematics is about constructing and solving problems, and interesting problems must be the focus of students’ mathematical experiences; they should be the content of daily mathematics lessons. Problems should be accessible to children, yet they should be modestly challenging and at times creatively frustrating. These kinds of problems should be the focus when students and their teachers are engaged in sharing their thinking process. These thinking processes— having ideas, not having ideas, seeing relationships, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work are the back-bone of the standards of mathematics instruction.

Standards of mathematics practice—techniques and methods, when routinely practiced, will implement the content advocated in the CCSSM. Arising naturally out of practice, content will not be in isolated from; it will be an organic outgrowth of the problem background. When a concept and procedure are learned, they should be practiced till they convert into skills and tools for doing meaningful mathematics. Doing mathematics is different than knowing mathematics. However, practice is a good reinforcement tool if students have learned the content first.

The Standards of Mathematics Practices (SMP) aim to develop mathematics content, the mathematical way of thinking, and mastery of the content with deeper understanding. As students work on solving problems, they are guided to observe regularity and symmetry, discover relationships, and make conjectures. They are asked to evaluate the reasonableness of their intermediate results and arrive at mathematical concepts. They articulate these results and then prove or disprove their results using formal reasoning and their prior knowledge of related concepts. The success of this evaluation process is dependent on several skills that can be developed with the following activities:

  • Encourage students to share multiple methods for solving the same problem and then request that students use someone else’s method to solve a similar problem.
  • After sharing and naming multiple methods for a problem, tweak the parameters and conditions of the problem and ask students “Which method would you use to solve the problem if it were like this?”
  • Playfully and explicitly add an element of time to students’ problem solving, once they have had experience developing methods for a particularly problem structure. For example, if students have painstakingly drawn pictures and calculated the sum of the interior angles of rectangles, rhombuses, pentagons, and hexagons, and shared their different methods for doing so, ask them to think about ways they could get more efficient. Then ask them to make a bid for how many polygons they could find interior angle sums for in 5 minutes. Pass out the requested number of puzzles to each team, and start the clock. While you’ll have to pay attention to group dynamics and support students to be good team members even under pressure, you’ll also hear good ideas about mathematical shortcuts that didn’t come up when students didn’t have any reason to need more efficient methods.

Another purpose of this standard is to make sense of the formulas we use by finding a pattern or relationship in numbers generated during an exploration of the topic. It is to understand that there are purposeful connections between procedures and concepts or between the manipulation of tools and a procedure. For example, students can explore the area of a triangle by building rectangles, drawing the diagonal, and realizing that the area of a triangle is half of the area of a rectangle formed with the same base and same height—thus the formula is developed and then relate to the parallelograms of the same base and height.

It is important to design lessons where patterns are revealed and lessons that encourage students to make generalizations. Some of these tasks involve students working with prior knowledge in a non-routine way.

The learning that takes place through these lessons is incredible. Using the best-designed lessons is not the only necessary component in developing this practice in students, so be sure to facilitate conversations that include comments and questions such as:

  • Notice repeated calculations and reasoning
  • What generalizations can you make?
  • Look for general methods and shortcuts
  • What mathematical consistencies do you notice?
  • How would you prove that …?
  • What would happen if …?
  • What predictions or generalizations can this pattern support?
  • How would this strategy work with another number? Does it work all the time? How do you know?
  • Maintaining over sight of process while looking at details evaluate reasonableness of results
  • Finding new structures/methods, generalizing
  • Can you find a shortcut to solve the problem? How would your shortcut make the problem easier?

 

 

 

 

 

 

 

Look for and Express Regularity in Repeated Reasoning: Concepts to Procedures

Look For and Make Use of Structure: Understand the Nature of Mathematics

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. G.H. Hardy

The Common Core State Standards for Mathematics (CCSS-M) include both content standards and standards for mathematical practice (SMP). The content standards define “what students should understand and be able to do.” The standards for mathematical practice describe “varieties of expertise that mathematics educators…should seek to develop in their students.” The “what” part encourages students to amass a body of content whereas the “why” part develops students’ mathematical way of thinking. These practices help them become better learners of mathematics and problem solvers. The why part in teaching adds value to both student learning and formative assessment by the teacher. It informs the teacher and the students. Unless we give students opportunities to work on tasks that target the standards for mathematical content and require students to explain their reasoning with models, diagrams, equations, or oral and written explanations of the structure of the mathematics of the task, we might find ourselves with a limited or false sense of student understanding.

Looking for patterns in information and making use of the structure in mathematics ideas is a fundamental process of mathematics learning and an important component of the mathematical way of thinking. We must present students with tasks that address the content standards with rigor, using as many of the standards of mathematics practices as possible. It is crucial for students to look for and make use of a structure because this practice requires them to reason about the underlying mathematical structure and unity of mathematics ideas.

Children naturally seek and use structure. They learn their native language by observing patterns and then extending them. If the extension works, they become bold and create language expressions. Mathematics has far more consistent structure than our language, but too often it is taught in ways that don’t make that structure easily apparent. If, for example, students’ first encounter with the addition of same-denominator fractions drew on their well-established spoken structure for adding the counts of things—three books plus four books make seven books, three hundred plus four hundred make five hundred, and three globs plus four globs make seven globs, no matter what a glob might be—then they would already be sure that three ninths plus four ninths makes seven ninths. Developing the linguistic structure first is important so that we add or subtract only if the two “things” we are adding are of the same type or have some common property or common characteristic. Instead, children often first encounter the addition of fractions in writing, as 3/9 + 4/9, and they therefore invoke a different pattern they’ve learned—add everything in sight—resulting in the incorrect and nonsensical 7/18.

Structure defines a language and form defines ideas. Mathematics is a language and collection of wonderful ideas. First we need to acquire the structure and form of the language and then we can play with it. Creativity is part of all learning and it takes place in organized chaos. In the chaos of problems, in the midst of new information, students need to see the pattern—organization and structure of the problems in order to understand them and enter into the solution process.

“Look for” “patterns,” and “structure” are key phrases in the seventh standard of math practice. It calls for going beyond the given information in the problem and shifting the perspective to discern relationships between pieces of information either explicitly or implicitly or predict the structure in the information. The skills involved in the process are:

  • Look for relationships (explicit and implicit) between the information given in the problem and also what is hidden
  • Look for pattern/structure in the problem or concept under discussion
  • Step back for an overview/shift perspective
  • See something as a whole or as combination of parts
  • Using familiar/known structures to see something in a different way

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, using Cuisenaire rods might notice that the 3-rod and 7-rod joined together are the same length as the 7-rod and 3-rod joined, therefore, infer that three and seven more is the same amount as seven and three more and that, with a few more example, results in the generalization: commutative property of addition (3 + 7 = 7 + 3; light green + black = black + light green; lg + bk = bk + lg).

Children may sort a collection of shapes according to their different attributes, for example, of how many sides the shapes have and arrive at a definition of that category of objects: A triangle is a three straight-sided closed figure or three sided polygon; a quadrilateral is a four straight-sided closed figure or four-sided polygon; a decagon is a ten straight-sided closed figure or ten-sided polygon; and a n-gon is a n straight-sided closed figure or n-sided polygon. The concepts are clearer and learned better when the structure is emphasized. The examples below, one from geometry and one from arithmetic make the structure of mathematics evident.

Example 1: Structure of Classification
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Example 2: Decomposition/Recomposition and Distributive Property
Students see that a rectangle made using 7 brown Cuisenaire rods (8-rods) defines the multiplication fact 7 × 8 (area model) and then they create two rectangles from it: 7 by 5 and 7 by 3 of known facts 7 × 5 = 35 and 7 × 3 = 21, therefore they know 7 × 8 equals the well remembered 7 × 5 + 7 × 3, which when several such examples are used is generalized into the distributive property of multiplication over addition. In fact, the standard procedure/algorithm of multiplying two multi-digit numbers is derived from the distributive property of multiplication over addition.
Capture2

The properties of operations (addition and multiplication) such as associative, commutative, and distributive properties and properties of equality (=), such as reflexive, symmetric, transitive, and equivalence are indispensable in achieving fluency with computations and algebraic manipulation. Without them many students waste time in simple computations such as:
Compare the two expressions
2×3×2×5×2 and (b) 3×2×5×3×2×2
What is the relationship between the two expressions?

Students invariably compute both of them rather than just noticing that both of them have a common factors: 2, 2, 2, 3, and 5 and the second expression has an extra factor of 3; therefore, the second expression is 3 times the first product. And during the computation they will compute from left to right rather than recognizing that two of the factors are 2 and 5; therefore, it is easier to multiply them first and then the others. This shows they are not looking for the inherent structure of mathematics (commutative and associative properties and the pattern of multiplying by 5 and then multiplying by 10).

Let us consider a third grade problem to see the importance of this structure.
Robert has 48 candies with 6 pieces of candy in each bag. How many pieces of candy does Robert have? Jessica has 16 bags of candy with 3 pieces of candy in each bag. How many pieces of candy does Jessica have?

Compare Robert and Jessica’s bags of candy. Who has more and how do you know? Make a diagram or write an equation that explains how you know who has the most and why?

Some students claim Robert and Jessica have the same amount of candy even though one of them has 8 bags and the other has 16 bags. Explain with a diagram or an equation why you agree or disagree.

This simple task asks students to work with several content standards:

  • Interpret products of whole numbers. Students wrestle with the meaning of the factors 8 × 6 and 16 × 3 in a multiplication problem.
  • Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities. Students compare how much candy Robert and Jessica have.

This simple task also asks students to use a variety of mathematical practices:

  • Students must make sense of the problem and persevere as they attempt to determine the way to represent both students’ amounts of candy with a diagram and equations.
  • Students must reason abstractly and quantitatively because the task is a contextual situation and students are required to write an equation to represent each student’s candies. When students re-contextualize the algorithms in the context of the situation and explain the meaning of the expressions, they will demonstrate if they can work quantitatively.
  • Construct viable arguments and critique the reasoning of others. Students are likely to reason that Jessica and Robert have the same amount of candy because both have 12 pieces of candy, thus constructing a viable argument.
  • Model with Mathematics. Students’ equations, diagrams of the bags of candies and their written explanation will let us know if they have a means of modeling with mathematics.

However, by asking the last part of the question, they are engaged in examining the structure of multiplication. Without the final prompt, we may fail to find out if students really understand the reason why both expressions equal 12 candies. The focus on 2 bags versus 4 bags draws students’ attention to the number of equal groups in relationship to the number of items in each of the groups. Ideally, students will explain that the two students’ amounts are equivalent because although Jessica has more bags, he has fewer candies in each whereas Robert has fewer bags but each bag has more candies so, in the end, the total number of candies are the same.

The distributive property derived using the area models of multiplication is then applied to multiplication of binomials and relates to factoring of trinomials. Let us consider the following example:
Capture3
With a few more examples, students see the pattern and (x + 2)(x + 3) = x2+ x(2 + 3) +2Ÿ3 = x2 + 5x + 6 can be generalized into (x + a) (x + b) = x2+ x(a + b) + ab.

Every problem should be examined, even after it has been solved, from different perspectives to get depth, to make connections, and to gain efficiency in the solution process. The reexamination is to step back for an overview and shift perspective.

Mathematically proficient students look for the overall and inherent structures and patterns in mathematics. For example, they can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects (or a compound figure as one geometric entity and at the same time composed of several other figures with specific properties). They also observe the nature of numbers in complex expressions. For example, they can see 5 – 3(xy)2 as 5 minus a positive number times a perfect square (which in the case of a real number is also positive) and use that to realize the value of the expression cannot be more than 5 for any real numbers x and y, and later they realize that in the expression f(z) = 5 – 3(xy)2, f(z) has a maximum value of 5 for all real values of x and y.

Mathematically efficient students observe that patterns and repeated reasoning can help them solve more complex problems. For young students this might be recognizing fact families, that arithmetic operations have their inverse operations, properties of operations (commutative, associative, and distributive properties), of numbers (prime, even, divisibility, etc.), and as students get older, they can break apart problems and numbers into familiar relationships. For example, in a Kindergarten class I asked them to calculate the expression: 5 + 1 + 7 + 5 + 8 + 2 + 3 + 9. The students who knew how to make 10 and the commutative property right away were able to give the answer: 40.

There are specific ways mathematics ideas are learned by students. They learn them through

  1. Using concrete models,
  2. Applying analogies,
  3. Seeking and extending patterns, and
  4. Applying formal logic and reasoning.

In the other standards, there was an emphasis on using concrete models, analogies in solving problems, and logic and reasoning. However, students learn quickly that finding a pattern simplifies the logic and makes access to the solution approach and solving the problem easier. Recognition of patterns helps students realize effective conceptualization. Larger/complex concepts can be seen as the generalization and extensions of simpler and primary concepts. As a result, the bigger task becomes simpler, and the custom command can be used to solve the “small problem.”

Knowledge is cumulative. What a student is capable of learning depends upon what she already knows. We can see more patterns when we have more learning in store.

To observe and seek structure, we may assume that many or most students are properly equipped to make sense of new information and are able to observe patterns. We need to constantly help students to learn and amass more skills and more knowledge. Students need to be guided down the path of their learning—to seeing patterns and structures. Teachers should remain central to the activity of imparting knowledge to students.  Learning is most effective and enjoyable when it is carefully sequenced and scaffolded by the teacher and when students see structure and patterns in the information, they make connections.

Structure is everywhere in mathematics, and when we understand the structure embedded in a concept like addition/subtraction or multiplication/ division, we have a deeper understanding of the concept.

There are deep ideas that nourish the different branches of mathematics. One can think of specific mathematical structures of and patterns in (a) Number systems, (b) Algorithms, (c) Shapes and relationships, (d) Data, and (e) Relationships, such as rate of change or ratio. These structures describe and give rise to attributes such as: (a) Linear or Non-linear, (b) Periodic or Random, (c) Symmetric or Chaotic, (d) Continuous or Discontinuous/ discrete, (e) Exact or Approximate, and (f) Maximum vs Minimum.

The skills of using models, applying analogous thinking and conditions, and taking advantage of pattern analysis are as important in mathematics learning as the use of logic and reasoning. The most important of these is pattern analysis.  Therefore, a central idea of mathematics is the study of patterns—to discern, extend, create, apply in order to make conjectures based on these patterns and then to arrive at general results, principles, and procedures. By using formal logic one converts the conjectures arrived by discerning patterns, if possible, into well-founded results—concepts, definitions, and procedures/algorithms. Seeing and revealing hidden patterns are what mathematicians do best. That applies to both developing and accomplished mathematicians.

We need to help our students see and reveal patterns in the given information. The guide to this growth is not just calculations and formulas but an open ended search for pattern. Pattern analysis involves four sub-skills: (i) discerning and recognizing patterns to understand the problem, (ii) extending patterns to enter into the solution process, (iii) creating new patterns to verify the validity of the solution process and generalize the approach, and (iv) applying patterns to discover and develop procedures.

Mathematically proficient student look for patterns and structure in mathematical situations. Students who are proficient at making use of structure are able to identify and create viable and efficient ways of looking at and using patterns. Students may then extend the pattern and derive the mathematical expression or equation related to that pattern. Once a pattern is understood, students can explain the process to their peers.

 As students look for and make use of structure, some of the questions they may ask include

  • Is there a pattern in the problem situation?
  • Can I state the pattern or structure as a rule?
  • Is this a rule that holds true every time?
  • If there the rule does not work every time, can I adapt it to work every time?
  • When does it not work?

Mathematically proficient students use structure to make connections and deepen conceptual understanding. The understanding of connections also enables students to shift perspectives and see the pattern from a different view.

To increase the likelihood of student pattern recognition, teachers can incorporate work on pattern analysis throughout the year. For example, a morning mathematics meeting framework, such as “calendar time” in K-2 grade classes is an excellent time to incorporate extra math pattern/structure exposure.

In the early grades, students can recognize, identify and apply pattern knowledge through calendar work, weather graphing and exploring daily numbers (e.g., date, number of days in school, and the number of the day) to incorporate extra math/structure exposure. For example, one of the activities that I have found to be useful in developing patterns and increasing children’s numbersense is “skip counting.” Skip counting should be done in each grade, even if it is just for a few minutes. For example, in Kindergarten: counting backward and forward by 1, 2, and 10 starting from any number; First grade: by 1, 2, 5, and 10; Second grade: 1, 2, 5, 10, and 100; Third grade: by 1, 2, 5, 10, 100, 1000, and unit fraction (e.g., 1/3, 1/5, etc.); Fourth grade: by proper fraction (e.g., 2/7, 3/10, etc.); Fifth grade through eighth grade: by mixed fraction (e.g., 1⅜,Capture4 , etc.).  Here is an example of counting activity in a second grade classroom.

Procedures

  1. Oral only
  • Announce the counting number and beginning number
  • Ask who has the next one, several times till all children had a chance
  • Once the pattern is established, continue around the room asking each student
  1. Oral and written
  • Announce counting number and beginning number
  • Record on the board to show the pattern, both vertically and horizontally
  • The number of entries in the columns changes everyday. For example, in the above example, the number of entries in each column is 4. Next day, there may be 8 entries in each column.
  • Ask students to write the next 5 numbers on their papers

Example:
149     169  189
154     174  194  ___  ___          ___   ___   ___
159     179  199
164     184  ___                                 ___

As children are writing their numbers, work with the student(s) who are having difficulty in counting (e.g., 199 + 5 = ?).

Teacher: What should you add to 199 to make 200?
Student: 200.
Teacher: Great! What more is left from 5 to be added?
Student: 4.
Teacher: What is 4 more than 200?
Student: 204.
Teacher: Now use the same strategy for the next number,

  1. Discussion

Once children have generated the five numbers, ask them to check each other’s work and then the teacher records them in the columns. After the entries have been recorded, identify an empty place in the column 4 row 2 (see the empty place 1), discuss with your partner to find what number will fit in this place. When you have the number raise your hand, both persons from the team hands must be raise their hands for me to recognize the team. One hand will not be recognized. Write all the responses on the side of the board and ask them to justify their answers. If the number is wrong, locate it in the right place with a comment: it belongs in this place. Do not disclose the pattern. If a team finds the correct answer and in their explanation discloses the pattern that is fine. Now identify another place (see column 5 row 2). Continue for several examples. A variety of questions can be formed, for example,

  • Ask students to fill in the places you have identified on the board.
  • Ask students to calculate the difference between 2 of the identified numbers. For example, what is the difference between 184 and 169? Etc.

The same process can be repeated in different grades with different numbers. For example, the counting will be by mixed fraction (say 1 starting from (say 7). The same kinds of questions are asked.

Mathematics is learned by doing and it should always include discovering patterns, developing and crafting conjectures, and providing correct and incorrect, rough and elegant, emerging, elegant, and meaningful explanations to understand the inherent structure of mathematics. Patterns abound in information generated in any field. Mathematics is no longer just the language of physics and engineering but now an essential tool for banking, manufacturing, social science, and medicine. Active mathematicians seek pattern wherever they arrive. For example, when mathematicians look at the numbers on any random page of a newspaper they discovered a pattern, the first digit of numbers, 1 occurs more often (about 30% more), 2 comes next (about 18% more) and so on. This phenomenon (Banford’s Rule) occurs often wherever random data is involved.

Mathematicians train their eyes to see patterns. “I see” has always had two distinct meanings for them: to perceive with the eye and to understand with the mind. Learning to see and recognize patterns and symmetry trains the mathematical eye. This “seeing” is the outcome of actions, such as:

        (a) Experimenting,

        (b) Discovering quantitative and spatial data,

        (c) Representing in language, pictures, diagrams, words, symbols,

        (d) Visualizing,

        (e) Classifying and organizing,

        (f) Identifying and defining,

        (f) Modeling,

        (g) Computing/approximating/estimating,

        (i) Generating conjectures,

        (i) Verifying, and

        (j) Proving.

These actions and processes result in: (a) generations of and representations in symbols, such as number symbols, variables, infinity, etc., (b) development of relationships—terms and expressions, (c) arriving at formulas, (c) representations through forms, models, figures, diagrams, and (e) structures, processes and algorithms.

The processes create structures in mathematics; however, some of these may have dichotomies. For example, numbers can be discrete or continuous; figures can be regular and irregular; relationships can be equations or inequalities; forms can be linear or non-linear; formulas can be explicit (closed) or recursive; etc. Students should be able to understand and apply dichotomies such as, (a) Discrete vs. continuous vs. chaos, (b) Finite vs. infinite, (c) recursive vs. explicit, (d) Algorithm vs. existential, (d) Stochastic vs. deterministic, (e) Exact vs. approximate. A sound education in mathematics requires student encounters with all of these.

At the elementary level, these relationships and structures are evident in numbers and their operations, in geometry, and in daily activities. The structure of mathematics gives the power to the learner. A procedure arrived at in some specific situations is applicable to infinitely many situations. Consider the normal distribution curve (Gauss Curve, Bell Curve), which was arrived at in a few specific situations, but it is now applicable to a myriad of situations—where there is randomness, there is the Bell curve and the results of normal distribution are applicable from IQ scores to population distribution, agriculture production to manufacturing widgets, and dispensing of coffee from the coffee machine to the distribution of the cars by age on the highway. Such structures abound in mathematics.

As an example of structure, consider the concept of place value: when children are asked to make a number, say 127, they are being asked to decompose 127 into hundreds, tens and ones in all the possible ways they can find. There are two kinds of structures: canonical decomposition: 127 = 100 + 20 + 7; or non-canonical decompositions: 6 tens and 67 ones; 12 tens and 7 ones; 1 hundred, 2 tens, and 7 ones; 1 hundred and 27 ones; etc.  In order to be efficient or complete in arithmetic operations, students need to use exchanges—of a ten for a group of ten ones or a hundred for ten tens—systematically, etc. The canonical decomposition and recomposition is at the heart of our base-ten number system. That is, there is a structure of systematic exchanges, which students must look for and make use of whether it is addition or subtraction procedure.  We can say that this task invites students to engage through both its “look for” and “make use of” structure. Children should explore the activity before systematic exchanges are suggested by the task or by the teacher. Unless they experiment first, the practice won’t be fully engaged in. Once a child can read a 3-digit number, she can read any number. Similarly, to read a multi-digit number (say, 9876543210), we explore the patterns to understand the structure of place value. First, we group them in 3 digits as 3-digit numbers, starting from right to left:

109,876,543,210.

Then, we name these groups; the second group (543) is called the thousand’s group. The “commas” in the number represent the name of the group and their names are called when you reach that comma. Since 543 is a three digit number it will be read as five-hundred-fort-three and it is the thousand’s group, it will be read as: five-hundred-fort-three thousand. Here, since the digit 5 in 543 is in the hundreds place, the value of digit 5 in 543 is 500 and value of digit 5 in 543,210 will be read as: five-hundred thousand. The next group is one (mil) group (llion) away from thousand so will be called group of millions. 876 being a 3-dgit number will be read as eight-hundred-seventy six. Therefore, the 876,543,210 will be read as: eight-hundred-seventy-six million… The next group is two (bi) groups (llion) away from thousand so will be called group of billions. 109 is a 3-didit number so will be read as one-hundred nine. Therefore, the 109, 876,543,210 will be read as: one-hundred billion, eight-hundred-seventy-six million… The place value of the digit 1 in 109 is one-hundred and the place value of 1 in 109,876,543,210 will be 100 billion. The next group will be trillion as it is 3 (tri) groups (llion) away from thousand, etc. The next group is quadrillion (quadri-four, llion-group), pentillion, sextillion, septillion, octillion, nonilllion, decillion, and so on. There is a pattern that defines the structure. This helps children to remember the place value.

To be useful, mathematical concepts, like power,  must be understood in their components, including their strength, reach and limits. It is when we understand the primary components of a concepts and their integrated relationships—the bonds of structure between them, that we understand the power and limitations of the methods and procedures derived from the concepts. For example, the power of the ‘long division procedure’ is really understood when we understand the structure of place value, estimation, and the four arithmetic operations that build the structure of the procedure. The long division procedure is such a powerful arithmetic achievement that depriving children of the economy, the fluidity, and the power of this procedure is to deprive them of many other related ideas, for example, the division of polynomials by binomials or the structure and behavior of the curves described by these polynomials.

Mathematically proficient students discern patterns, generalize ideas from specific situations, and then apply general mathematical rules to specific situations. Young students, for example, might notice and understand that our number system is based on the decimal system and certain numbers play an important role: 1 (iterative nature of natural numbers—1 more than a number is the next number), 10 (after a group of ten numbers we move to the next decade), and the teens numbers (10 + 1 = 11, 10 + 2 = 12, 10 + 3 = 13, etc. Therefore, it is important that children understand and master the structure of the number 10 (9 + 1, 8 + 2, 7 + 3, 6 + 4, 5 + 5), 1 more than a number, and the teens numbers. If the students know: (a) 1 more or one less than a number, (b) Making 10, (c) Teens numbers, and (d) commutative property, they can easily learn all the other addition and subtraction facts. All of the addition and subtraction strategies (number + 9; doubles, near doubles, missing double, near ten, etc.)[1] are extensions of understanding the making ten (structure) and decomposing/recomposing number patterns. For example, 8 + 7 = 8 + (2 + 5) = 10 + 5 = 15, or 8 + 7 = (5 + 3) + 7 = 15, or 8 + 7 = (3 + 5) + (5 + 2) = 3 + 10 + 2 = 10 + 3 + 2 = 10 + (3 + 2) = 10 + 5. And, then they can extend it to the procedure of addition: 28 + 37 = 20 + 8 + 30 + 7 = 20 + 30 + 8 + 7 = 50 + 15 = 65. This regrouping is a part of the fundamental structure of arithmetic. Almost all of the arithmetic procedures are derived from this structure of number 10.

Similarly, in the process of developing a proof for a theorem in geometry, they recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.

How to Help Students Look For, Use, and Reinforce Structures in Mathematics
This is mostly a learned behavior. In order that students look for and use structures of mathematics in learning new concepts and problem solving, teachers should point out the structures and patterns by asking enabling questions. Teachers can instill and then observe the same in students’ work.

  • What observations do you make about this data, this figure, these numbers, etc.?
  • Can you transform this expression into another form?
  • What do you notice when you simplify, transform…the …?
  • What different components, ideas, and concepts make this expression, equation, figure, problem, etc?
  • What parts of the problem might you estimate or simplify first?
  • What property of numbers, quadrilaterals, graph, … can you apply?
  • How do you know if something is a pattern?
  • What patterns do you find in…?
  • Are the conditions of a pattern satisfied here?
  • What are the characteristics of the pattern in this problem?
  • What ideas that we have learned before were useful in solving this problem?
  • What are some other problems that are similar to this one?
  • How does this relate to …?
  • In what ways does this problem connect to other mathematical concepts?
  • Can we generalize the results from this problem/situation?
  • Can you make a conjecture from this situation?
  • Does everyone agree with this conjecture?
  • If you do not agree with the conjecture, can you give a counter example?

The most important mathematics activity that this standard asks students to demonstrate is to see relationships between mathematical objects and the relations they can form between different elements. They should know what a conjecture is (e.g., an observed pattern). They should learn to express conjectures linguistically and symbolically: How to make them. How to justify them. How to disprove by providing a counter example. What a counterexample is.

  • Can you describe the pattern using your own words?
  • Can you generalize a pattern?
  • Can you describe this pattern using mathematical symbols as an expression, inequality, or a formula?
  • Can you give an example for which this conjecture does not hold good?

Making conjectures, giving examples and counter examples demonstrate the mathematical way of thinking—the understanding and knowledge of structure. For example, ask students to observe the patterns in the following numbers and articulate their conjectures:

 Capture5

In the first group of numbers, the “first difference” becomes constant. It means that the next number is 3 more than the previous one.  In other words, the numbers can be described as:

1st     2nd                                                   100th term               nth term

1, 1 + 3(1), 1 + 3(2),  1 + 3(3), …, …, 1 + 3(100 – 1), …,        1 + 3(n-1) + …

If we call the value of the term as y and the term as x, then the relationship between y (nth term) and x (1 + 3(n -1) is expressed as: y = 1 + 3(x – 1) or y = 3x – 2. In other words, one level of difference gives rise to a linear relationship, where the rate of growth is 3.

In the second group of numbers, the second difference becomes constant. The intuition based on the first set of numbers seems to indicate the relationship may be a second degree (quadratic) expression. It means if call the value of term is denoted by y ad the terms by x, then it suggests:

y = ax2 + bx + c

Assuming that this is the case, when we substitute x = 1, x = 2, x = 3, e get

     9 = a×12 + b×1 + c or 1 = a + b + c

          12 = a×22+ b×2 + c or 4 = 4a + 2b + c

          17 = a×32+ b×3+ c or   9 = 9a + 3b +c

When we solve these 3 simultaneous linear equations in a, b, and c, we get a = 1, b = 0, and c = 8. The sequence defines a relationship; the sth-term y can be defined in terms of the position of the term x as y = x2+8. Thus, a second level difference defines a quadratic relationship. 

A pattern begins to develop since in the next group of numbers, we find that the third level difference is constant. We can, therefore, conjecture that this defines a cubic relationship, and when we actually calculate, it is a cubic relationship. The fourth group, therefore, is a quartic relationship.

Procedural mathematics, particularly arithmetic, can be taught with or without attention to pattern. The CCSS-M acknowledges that students do need to know arithmetic facts at an automatized level, but random-order fact drills rely on memory alone, whereas patterned practice can develop a sense for structure as well. Learning to add 8 to number—not just to single digit numbers—by thinking of it as adding 10 and subtracting 2 can develop just as fast recall of the facts as random-order practice but it also allows students to generalize it and therefore be able to add 18 or 28 to any number mentally. The structure is a general one, not just a set of isolated memorized facts, so students can use it to add 19 or 39, or 21 or 41, to any number, too. With a bit of adjustment, they can use the same thinking to subtract mentally. This is, of course, exactly the way we hope students will mentally perform 350 – 99 (1 more than 99 is 100, so 350 – 100 = 250, therefore, 350 – 99= 251).

In elementary school, attention to structure also includes the ability to defer evaluation for certain kinds of tasks. For example, when presented with 29 + 15 ☐ 29 + 14 and asked to fill in <, =, or > to compare the two expressions, first and second graders are often drawn—and may even be explicitly told—to perform the calculations first. But this is a situation in which we want the students’ attention on the structure, ✪ + 15 ☐ ✪ + 14 or even u + 15 ☐ u + 14, rather than on the arithmetic. This can then be generalized later in middle and high school to:

For what value of x, the x2 + 2x + 9 > x2 + 2x + 7 is true?

This same skill of deferring evaluation—putting off calculation until one sees the overall structure—helps students notice that they don’t have to find the common denominator for 1¾ – ⅓ + 3 + ¼ – ⅔ but can simply rearrange the terms to make such a trivial computation that they can do it in their heads. Similarly, to find the value of the expression
Capture6
they do not have to make any calculations, if they know that Capture blog 6.02is equal to 1, if a = b and b ≠0. Those who do not see the structure, will go through the whole calculation as a large number of high school students actually did on a state test. They found the sum of the numerator and denominator by finding the common denominator and then performed division and multiplication.

When students begin to solve algebraic equations, the same idea will help them notice that 3(5x – 4) + 2 = 20 can be treated as “something plus 2 equals 20” and conclude, using common sense and not just “rules,” that 3(5x – 4) = 18. And 3 times something is 18 means that 5x – 4 = 6. From such reasoning, they can learn to derive rules that make sense.

Similar thinking is present in higher mathematics. For example, use a graphing tool to graph each of the following equations. Describe what shape you see in your graph and what is changing on the graph each time.

a.  x2 + y2 = 1,    b.  x2 + y2 = 4,    c.  x2 + y2 = 9
d. 
(x + 2)2 + y2 = 4,    e. x2 + (y + 3)2 = 4,    f. x2 + (y – 3)2 = 4

How might this help you to describe the graph of x2 + y2 = 16 and x2 + (y – 6)2 = 4 ? And what is the relationship between the two curves? In what ways are they similar and different?

The structure means some properties, some relationships, some characteristics are present in a particular problem, in a particular concept, and then they are present in a global situation—they are present in a class of problems. In pattern analysis, we are dealing with inductive reasoning. In pattern analysis, we move from particular situations to a general situation whereas in looking for structure we move from general to specific. Looking for and making sense of structure means we are using deductive reasoning. In most of elementary mathematics, we use less of deductive reasoning, but more of inductive reasoning. Mathematics is all about patterns and structure—inductive and deductive reasoning, so it isn’t something that should be taught as a single standard but rather as a practice that we use when thinking mathematically.

These types of conversations need careful facilitation. We shape students’ thinking by the way in which questions are posed during instruction as well as by how questions are written on assessments. I end with a list I like to incorporate as I support students in becoming more proficient with this mathematical practice:

  • Why does this strategy work, and can a solution be found using this strategy?
  • What pattern do you find in ___?
  • What are other problems that are similar to this one?
  • How is ____ related to ____?
  • Why is this important to the problem?
  • What do you know about ____ that you can apply to this situation?
  • In what ways does this problem connect to other mathematical concepts?
  • How can you use what you know to explain why this works?
  • What patterns do you see?
  • Is there a structure? How can you describe the structure?

Some educators argue that we should give students as many tasks as possible that draw on or require using many mathematical practices when determining the solution path for a task that targets a mathematical concept. Some argue that by continually giving students high-level tasks, they will eventually engage with all the math practices. I argue that we must design strategically to engage students in all the mathematics practices regularly.

[1] See the list of Strategies for Mastering Addition Facts and how to teach them in How to Master Arithmetic Facts Easily and Effectively (Sharma, 2008). List is available free from the Center.

Look For and Make Use of Structure: Understand the Nature of Mathematics

Attend to Precision: The Foundation of Mathematical Thinking

The sixth of the Standard of Mathematics Practice (SMP) in Common Core State Standards (CCSS-M) is: Attend to Precision. The key word in this standard is the verb “attend.” The primary focus is attention to precision of communication of mathematics—in thinking, in speech, in written symbols, in usage of reasoning, in applying it in problem solving, and in specifying the nature and units of quantities in numerical answers and in graphs and diagrams. With experience, the concepts should become more precise, and the vocabulary with which students name the concepts, accordingly, should carry more precise meanings.

The word “precision” calls to mind accuracy and correctness—accuracy of thought, speech and action. While accuracy in calculation is a part, clarity in communication is the main intent of this standard. The habit of striving for clarity, simplicity, and precision in both speech and writing is of great value in any discipline and field of study. In casual communication, we use context and people’s reasonable expectations to derive and clarify meanings so that we don’t burden our communication with too many details that the reader/listener can surmise anyway. But in mathematics (thinking, communicating, and writing), we base each new idea/concept logically on earlier ones; to do so “safely,” we must not leave room for ambiguity and misconceptions.

Students can start work with mathematics ideas without a precise definition. With experience, the concepts should become more precise, and the vocabulary with which we name the concepts can, accordingly, should carry more precise meanings. But we should strive for clarity and precision constantly. Striving for precision is also a way to refine understanding. By forcing an insight into precise language (natural language or mathematical symbols), we come to understand it better and then communicate it effectively. For example, new learners often trip over the order relationships of negative numbers until they find a way to reconcile their new learning (–12 is less than –6) with prior knowledge: 12 is bigger than 6, and –12 is twice –6, both of which pull for a intuitive feeling that –12 is the “bigger” number. Having ways to express the two kinds of “bigness” and the sign defining the direction helps distinguish them. Learners could acquire technical vocabulary, like magnitude or absolute value, or could just refer to the greater distance from 0, but being precise about what is “bigger” about –12 helps clarify thinking about what is not bigger. With such a vocabulary, one can express the relationship between the two numbers more precisely.

The standard applies equally to teachers and students and by extension to textbooks, modes and purpose of assessments, and expectations of performance. To achieve this, teachers need to be attentive to precision in their teaching and insist on its presence in students’ work. They should demonstrate, demand and expect precision in all aspects of students’ interactions relating to mathematics with them and with other students. Teachers must attend to what students pay attention to and demonstrate precision in their work, during the learning process and problem solving. This is not possible unless teachers also attend to the same standards of precision in their teaching.

Teachers, while developing students’ capacity to “attend to precision,” should focus on clarity and accuracy of process and outcomes of mathematics learning and in problem solving from the beginning of schooling and each academic year. For example, teachers can engage their students in a “mathematics language talk” to describe their mathematics activity. The emphasis on precision can begin in Kindergarten where they talk about number and number relationships and continues all the way to high school where they furnish mathematics reasoning for their selection and use of formulas and results.

Attention to precision is an overarching way of thinking mathematically and is essential to teaching, learning, and communicating in all areas of mathematical content across the grades.

For the development of precision, teachers should probe students to defend whether their requirements for a definition are adequate as an application to the problem in question, or whether there are some flaws in their group’s thinking that they need to modify, refine and correct. Just like in the writing process, one goes through the editing process, students should come to realize that in mathematics also one requires editing of expressions to make them appealing, understandable and precise.

However, communication is hard; precise and clear communication takes years to develop and often eludes even highly educated adults. With elementary school children, it is generally less reasonable to expect them to “state the meaning of the symbols they choose” in any formal way than to expect them to demonstrate their understanding of appropriate terms through unambiguous and correct use.

The expectations according to the standard are that mathematically proficient students

  • communicate their understanding precisely to others using proper mathematical terms and language: “A whole number is called prime when it has exactly two factors, namely 1 and itself” rather than “A number is called prime if it can be divided by 1 and itself.
  • use clear and precise definitions in discussion with others and in their own reasoning: e.g. “A rectangle is a four straight-sided closed figure with right angles only” rather than “A four-sided figure with two long sides and two short sides.”
  • state the meaning of the symbols they choose, use the comparison signs ( =, >, etc.) consistently and appropriately, for example, the names of > and < are not greater and smaller than respectively, but depend on how we read them: x > 7 is read as: x is greater than 7 or 7 is less than x; 2x + 7 = -5 + 3x is bidirectional (2x + 7 => -5 + 3x and 2x + 7 <= -5 + 3x).
  • are careful about the meaning of the units (e.g., “measure of an angle is the amount of rotation from the initial side to the terminal side” rather than “measure of an angle is the area inside the angle or the distance from one side to the other”), identifying and specifying the appropriate units of measure in computations, and clearly labeling diagrams (e.g., identify axes to clarify the correspondence with quantities and variables in the problem, vertices in a geometrical figure are upper case letters and lengths are lower case letters, and the side opposite to the <A in ΔABC is denoted by “a”, etc.).
  • calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context (e.g., the answer for the problem: “Calculate the area A of a circle with radius 2 cm” is A = 4π sq cm not A = 12.56 sq cm; if x2 = 16, then x = ± 4, not x = 4, whereas √16 = 4, etc.).
  • know and state the conditions under which a particular expression, formula, or procedure works or does not work.

Beginning with the elementary grades, this means that students learn and give carefully formulated explanations to each other and to the teacher (at Kindergarten level it may mean that the child explains her answer for 8 + 1 = 9 as “I know adding by 1 means it is the next number. I know 9 is next number after 8” or can show it concretely as “Look here is the 8-rod add the 1-rod and I get the 9-rod.” By the time they reach high school, they have learned to examine claims—their own and others’ in mathematical conversations, make explicit use of definitions, formulas, and results, and proper and adequate reasoning. At the high school level the explanations are rooted in any or more of these:

  • demonstrating it concretely,
  • showing by creating and extending a pattern,
  • application of analogous situation, or
  • logical reasoning—proving it using either deductive or inductive reasoning or using an already proved result.

What Does the “Attention to Precision” Look Like?
Effective mathematics teachers who use precision and efficiency in their teaching and encourage precision in their classrooms produce mathematically proficient students. Mathematically proficient students understand the role of precision in mathematics discourse and learning. They understand that mathematics is a precise, efficient, and universal language and activity. Precision in mathematics refers to:

Language

  • Appropriate vocabulary (proper terms, expressions, definitions), syntax (proper use of order of words), and accurate translation from words to mathematical symbols and from mathematical symbols to words.
  • Knowledge of the difference between a pattern, definition, proof, example, counter example, non-example, lemma, analogy, etc. at the appropriate grade level.
  • Reading and knowing the meaning of instructions: compute or calculate (4 × 5, √16, etc., not solve), simplify (an expression, not solve), evaluate (find the value, not solve), prove (logically, not an example), solve (an equation, problem, etc.),
  • Know the difference between actions such as: sketch, draw, construct, display, etc.
  • Precise language (clear definitions, appropriate mathematical vocabulary, specified units of measure, etc.).

Teacher instruction about vocabulary must be clear and correct and must help children to understand the role of vocabulary in clear communication: sometimes formal terms and words distinguish meanings that common vocabulary does not, and in those cases, they aid precision; but there are also times when formal terms/words camouflage the meaning. Therefore, while teachers and curriculum should never be sloppy in communication, we should choose our level of precision appropriately. The goal of precision in communication is clarity of communication and achieving understanding.

A teacher can use familiar vocabulary to help specify which object(s) are being discussed—which number or symbol, which feature of a geometric object—using specific attributes, if necessary, to clarify meaning. Actions such as teaching writing numerals to Kindergarten by “song and dance” is a good starting point, but ultimately the teacher should use the proper directional symbols, e.g.,

  • To write number “4” the teacher first should point out the difference between the written four (4) and printed four (4). Then she needs to show the direction of writing (start from the top come down and then go to the right and then pick up the pencil and start at the same level to the right of the first starting point and come down crossing the line).
  • When discussing a diagram, pointing at a rectangle from far away and saying, “No, no, that line, the long one, there,” is less clear than saying “The vertical line on the right side of the rectangle.”
  • Compare “If you add three numbers and you get even, then all the numbers are even or one of them is even” with “If you add exactly three whole numbers and the sum is even, then either all three of the numbers must be even or exactly one of them must be even.”
  • Compare giving an instruction or reading a problem as “when multiply 3 over 4 by 2 over 3, we multiply the two top numbers over multiply two bottom numbers” to “find the product of or multiply three-fourth by two-third, the product of numerators is divided by the product of denominators.”

Elementary school children (and, to a lesser extent, even adults) almost never learn new words effectively from definitions. Virtually all of their vocabulary is acquired from use in context. Children build their own “working definitions” based on their initial experiences. With experience and guidance, the concepts should become more precise, and the vocabulary with which children name the concepts will carry more precise meanings. Formal definitions generally come last. Children’s use of language varies with development but typically does not adhere to “clear definition” as much as to holistic images. If the teacher and curriculum serve as the “native speakers” of clear Mathematics, young students, who are the best language learners around, can learn the language from them.

Quantities
Accuracy (know the difference between exact, estimate, approximation and their appropriateness in context) and appropriate level of precision in use of numbers (level and degree of estimation, significant digits, significant powers, units of measurement), correct classification and location of number on the number line (e.g., to locate ⅞, one divides the unit segment into halves and then each half into fourths, and then each fourth into eighths and then locates ⅞ rather than arbitrarily divide the unit segment into eight parts), correct relationships between numbers (e.g., √(140) is between 11 = √(121) and 12 = √(144), because, we have 121 < 140 < 144, therefore, √(121) < √(140) < √(144), but √(140) much closer to 12 as 140 is much closer to 144 than 121), selection of appropriate range and window on graphing calculator, tool selection (when to use what tools–paper-pencil, concrete models, diagrams, abstract, or calculator), and appropriate meaning of numbers in the outcome of operations (what role do the quotient and remainder play in the outcome from the long division algorithm, etc.). Precise numbers (calculate accurately and efficiently; given a context, round to an appropriate degree of precision)

Teachers should use written symbols correctly. In particular, the equal sign (=) is used only between complete expressions and signals the equality of those two expressions. To explain one way to add 42 + 36, we sometimes see it written (incorrectly) this way: 40 + 30 = 70 + 2 = 72 + 6 = 78. This is a correct sequence of calculator buttons for this process but not a correct written mathematics expression: 40 + 30 is not equal to 70 + 4; only the last equals sign is correctly used. We need the = sign to have a single, specific meaning. Also, the equal sign should not be misused to mean “corresponds to”: writing “4 boys = 8 legs” is incorrect.

Models
Appropriate choice of concepts and models in the problem solving approach: choice of strategy in addition/subtraction (8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, 8 + 6 = 4 + 4 + 6 = 10 + 4 = 14, 8 + 6 = 2 + 6 + 6 = 2 + 12, 8 + 6 = 8 + 8 – 2 = 16 – 2 = 14, or 8 + 6 = 7 + 1 + 6 = 7 + 7 = 14 rather than “counting up” 6 from 8 or 8 from 6), appropriate multiplication/division model (the only models of multiplication work for fraction multiplication are “groups of” or “area of a rectangle” not “repeated addition” and the “array” models), which exponential rule, which rule of factoring, which rule for differentiation, what parent function to relate to, what formula to use, etc.

Reasoning, Symbols, and Writing Mathematics
Appropriate and efficient use of definitions, reasons, methods of proof, and order of reasoning in solving problems and explanations. For example, children should know the reasons for using the “order of operations” or that the solutions of equations have domains and range. Precise usage of symbols and writing:

  • Choose correct symbols and operators to represent a problem (knowns and unknowns; constants and variables),
  • State the meaning of the symbols and operations chosen appropriate to the grade level (multiplication: 4×5, 45, 4(5), (4)5,(4)(5), a(b), (a)b, (a)(b), ab),
  • Label axes, shapes, figures, diagrams, to clarify the correspondence with quantities in a problem, location of numbers,
  • Show enough appropriate steps to communicate how the answer was derived,
  • Organize the work so that a reader can follow the steps (know how to use paper in an organized and systematic form—left to right, top to bottom),
  • Clearly explain, in writing, how to solve a specific problem,
  • Use clear definitions in discussion with others and in reasoning
  • Specify units of measure and dimensions,
  • Calculate accurately and efficiently.

At the elementary level, even the simplest of things such as: the proper way of forming numbers and mathematical symbols, writing the problems solving steps in a sequence: ([3(4 + 8) – (4 ÷ 2)] = [3(12) – (2)] =[36 − 2]= 34 rather than 4 + 8 = 12 × 3 = 36 −2 = 34). Similarly, clarity in reading numbers and mathematical symbols needs to be  emphasized from the beginning (e.g., ¾ is read as “3 parts out of 4 equal parts” rather than “3 out of 4,” “3 divided by 4” rather than “3 over 4.”

It is difficult to change inappropriate and incorrect habits later on. For example, when elementary grade teachers do not emphasize the importance of aligning multi-digit numbers in their appropriate place values, this creates problems for children later. New symbols and operations are introduced at each grade level, so it is important for the teacher to introduce them correctly and then expect precision in their execution.

Similarly, when middle and high school students are not instructed to write fractions properly, it creates problems. The following high school lesson illustrates the point. The problem on the board was:

27.1

To solve the equation, in order to eliminate fractions in the equation, the student suggested we multiply the whole equation by the common denominator of all the fractions in the equation (a correct and efficient method). When I asked for the common denominator, the student said: 9x because the denominators are 3x, 9 and 3. The error is purely because of lack of precision in writing fractions in the equations.

Precision often means including units when specifying numerical quantities. But not always. The purpose of precision is never to create work, only to create clarity. Sometimes a number is clear by itself, other times a unit is needed, sometimes a whole sentence is required: the situation determines the need. For the same reason, label graphs and diagrams sufficiently to make their meaning and the meanings of their parts clear.

Exposure and consistent questions from the teacher such as the following help students to be accurate, precise and efficient:

  • Is this the right way of writing the expression (number, symbol, etc.)?
  • Does the diagram you have drawn show the elements asked for or given in the problem?
  • Is this the right unit for the quantities/numbers given in the problem?
  • What mathematical terms apply in this situation?
  • Is the term you used the right one in this situation?
  • How do you know your solution is reasonable and accurate?
  • Explain how you might show that your solution answers the problem?
  • How are you showing the meaning of the quantities given in the problem (e.g., problem says: “the length of the rectangle is 3 more than twice the width)? Does your rectangle demonstrate the right dimensions? Your rectangle looks like a square.
  • What symbols or mathematical notations are important in this problem?
  • What mathematical language, definitions, known results, properties, can you use to explain ….?
  • Can you read this number (symbol, expression, formula, etc.) more efficiently?
  • Is ___ reading (saying, writing, drawing, etc.) correctly? If not, can you state it correctly and more efficiently?
  • How could you test your solution to see if it answers the problem?
  • Of all the solutions and strategies presented in the classroom, which ones are exact/correct?
  • Which one of the strategies is efficient (can achieve the goal more effectively)?
  • What would be a more efficient strategy?
  • Which one is the most elegant (can be generalized and applied to more complex problems) strategy? Etc.

The number and quality of questions in a classroom bring the attention of students to appropriate and precise conversation. In a fourth grade geometry lesson, I had the following exchange with the students: 

Sharma: Look at this rectangle (I was holding one of the 10 by 10 by 1 rectangular solids in my hand) while touching the 10 by 10 face, I asked: What are the dimensions of this rectangle?

A student raised his hand and said: “That is not a rectangle. It is a square.”

I said: “yes, it is a square. Can you also call it a rectangle? Is it also rectangle?”

“No!” He declared emphatically.

I asked the class: “How many of you believe that it is not a rectangle?” Almost every hand went up.

When I asked them what the definition of a rectangle was, almost all of them said: “A rectangle has two long sides and two shorter sides.” I drew a quadrilateral with 2 long sides and 2 short sides that did not like a rectangle.

Another student said: “The sides are parallel.” I drew a parallelogram.

The student said: “No! That is not what I mean. Let me show you what I mean.” He drew a correct rectangle.

One student said: “A rectangle has four right angles and 2 longer sides and 2 shorter sides. Like this.” He drew a correct rectangle.

We had a nice discussion and came to the conclusion that a rectangle is: A straight-sided closed figure with four right angles. I also emphasized the meaning of the word “rectangle.”  It is made up of two words “recta” and “angle.” The word “recta” means right.  Therefore, a rectangle has only right angles. With this discussion and the precise definition, they were able to accept and see the face of the object I was showing as a rectangle.

This episode, in one form or the other, is repeated in many classes, from urban to rural classrooms, in many elementary schools. The same misconception is present even in many classrooms in many middle and high schools students. This is an example of lack of precision in teaching and, therefore, lack of precision in student understanding and expression.

There are many examples of such misconceptions. For instance, children often misunderstand the meaning of the equal sign. The equal sign means is “the same as,” “equal in value” “equal in some specified characteristic—length, area, quantity, volume, or weight,” but most primary students believe the equal sign tells you that the answer is coming up to the right of the equal sign. When children only see examples of number sentences with an operation to the left side of the equal sign and the answer on the right, this misconception is formed and generalized. Teachers should, therefore, emphasize the true meaning of the equal sign. From the very beginning—Kindergarten children should be shown that the equal sign “=” is a two-way implication. For example, Kindergarteners should be shown and know the simple facts as: 2 + 8 = 10, 8 + 2 = 10 & 10 = 2 + 8, 8 + 2 = 10 and first graders need to see equations written in multiple ways, for example 5 + 7 = 12, 7 + 5 = 12, 12 = 5 + 7, 12 = 7 + 5, and 5 + 7 = 2 + 10, 5 + 7 = ☐+10, ☐ + 2 = 9 + ☐. Although most above average and many average children are able to realize this level of understanding of the concept of equal or equal sign, there are many average and children with learning disabilities who have difficulty in reaching that level of understanding. This level of precision in understanding can be achieved by using Cuisenaire rods, the Invicta math balance for teaching arithmetic facts, and proper and appropriate language usage and questioning by teachers.

If students are taught using imprecise language, they will necessarily learn imprecise language and concepts, because language is the basis of mathematics learning. Later, they will not only resist when asked to use precise language in mathematics, but they will also have difficulty applying the concepts. A sequence of ideas begins to take place in students’ mind when we ask questions and emphasize language.
Questions instigate language.
Language instigates models.
Models instigate thinking.
Thinking instigates understanding.
Understanding produces conceptual schemas.
Conceptual schemas produce competent performance.
Competent performance produces long lasting self-esteem.
Self-esteem produces willingness to inquire and learn.

With proper language and conceptual models a great deal can be achieved. It is not too late to instill precision even at the high school level; however, if it is not emphasized at the elementary and middle school levels, it is much more difficult to do so. This does not mean we give up; it only means we redouble our effort and find better ways of doing it, such as using concrete models, patterns, and analogies when we are introducing new mathematics concepts and procedures.

As students progress into the higher grades, their ability to attend to precision will expand to be more explicit and complex if we constantly use proper language and symbols.

As students develop mathematical language, they learn to use algebraic notation to express what they already know and to translate among words, symbols, and diagrams. Possibly the most profound idea is giving names to objects. When we give numbers names, not just values, then we can talk about general cases and not just specific ones.

Correct use of mathematical terms, symbols, and conventions can always achieve mathematical precision but can also produce speech and writing that is opaque, especially to learners, often to teachers, and sometimes even to mathematicians. Good mathematical thinking, therefore, requires being correct, but with the right simplicity of language and lack of ambiguity to maintain both correctness and clarity for the intended audience. If we are particular about this in the first few grades, it becomes much easier to attend to precision in later grades.

Attend to Precision: The Foundation of Mathematical Thinking

Use Appropriate Tools Strategically – Part II

Teachers’ Role in Tool Building and Using
Teachers play a critical role in the development of the strategic use of tools. First, they make a diversity of tools available to students. From the beginning of the year, students should know where the math tools are in the room and how they will be used throughout the class. From the beginning, a teacher should declare: “Just like many, you play a sport of your choice and get better at it with practice. Similarly, each one of you should become an expert in a mathematics tool or strategy and its usage in mathematics concepts and procedures.” Then the teacher should present varieties of situations or problems where that tool is applicable and effective (e.g., different types of rules or protractors). Then she works on the strategic use of the tools. The first step is modeling their appropriate use.  One can begin by using phrases like “I bet a ruler would help me divide this into even pieces” or “I wonder if using a graph paper would help me organize my work” or “I think I could use a calculator to double check my accuracy on this one.” The key factor in getting students to use mathematical tools efficiently is exposure—multiple and varied exposures.

Students need to see how teachers make decisions about using tools so they know what appropriate use is. Using a calculator to solve 50 + 50 is not appropriate—but it is appropriate to check a complicated computation. A teacher should not admonish students who choose tools inappropriately. Instead, she should ask them to share their reasoning for using a particular tool in a specific way. They should also make explicit how a particular tool or approach will connect to an abstract mathematics idea.

As the year continues and new tools are introduced, students will be able to apply their current knowledge of mathematical tools to the new ones. It is important to plan tasks that will require multiple learning tools.

In order for students to be proficient they need to start using the tools independent of the teacher. They will then pick tools based on the needs of the problem and plan. They will also be able to visualize the results after using the tool.

Introductory Part of the Lesson (Teacher Directed: Didactic and Socratic Roles)
For Teachers: Decisions

  • What are the goals of this lesson?
  • What language, concept, procedures, and skills do I want students to develop?
  • What activities and tools are best suited for this purpose?
  • What tools and methods my students are already familiar with?

Middle Part of the Lesson (Socratic and Coaching Roles)

  • Have the students acquired the concept/procedure using this tool, what tools can further expand, enhance, or deepen these ideas?
  • Can they apply these tools in problem solving with my support and in collaboration with their colleagues?

Last Part of the Lesson (Coaching and Supporting Roles)

  • What problems can I assign students that will provide opportunities of applying these and previous tools in solving them?
  • Can they create/construct problems that can be solved by these tools?
  • Discussion to establish the efficiency of tools and develop proficiency and competence in the use of tools and integrate these tools with earlier tools.

Teachers must recognize that tools do not produce understanding, problem solving, and solutions. These come when teachers ask questions and make connections between the tool and the concept and when students do the same. Providing students with protractors does not ensure that they will measure the angles and find angle sums of triangles with accuracy. Similarly, a graphing calculator doesn’t consider user error or misconception when graphing a linear equation. The teacher should therefore bring to students’ attention the strength and limitations of the tool and its usage.

User error (including a broken ruler) can occur with a basic ruler or a calculator. If we have the concept and understanding, we adjust. We try the tool again, maybe a little differently.

Initially it is the teacher’s questions that help students in the tools’ usage, but then the teacher needs to transfer their usage to students and the questioning process to facilitate this. The questions we want students to ask when selecting and using tools include:

  • Do I need a tool in this situation?
  • Which tool will work for this situation?
  • Is this the right tool?
  • How does it work?
  • What tool is the best to use in this situation?
  • Do the results align with what I was expecting?
  • Do the results make sense?

Most importantly, when children select, use, make decisions, compare the usage of tools, they think and develop metacognitive processes. As a result, their cognitive ability and potential increase. They become better learners. This development is centered on teachers facilitating the process by asking questions and creating cognitive dissonance in students’ minds. Thus, we encourage the thinking behind the tool as well as the procedure for using the tool. We should require our students to predict what their findings might be prior to using the tool and then require them to reflect on the results and if the answers make sense.

For Students: Problem Solving
The standard says:
[S]tudents consider the available tools…. Proficient students…make sound decisions about when each of these tools might be helpful.
[They]… use technological tools to explore and deepen their understanding.

These phrases focus on the student. The goals for students are: Use available tools recognizing the strengths and limitations of each. Use estimation and other mathematical knowledge to detect possible errors. Identify relevant external mathematical resources to pose and solve problems. Use technological tools to deepen understanding of mathematics.

Students learn through the questions teachers pose to help them think and sort through the ideas that are forming. Helping students to generate questions results in their seeking and using tools.

  • What information do I have?
  • What is stated in the problem?
  • How will I represent the information in the problem?
  • What tool(s) can help me visualize and represent this information to understand its nature?
  • What do I know that is not stated in the problem?
  • What approach should I consider trying first?
  • What other mathematical tools could I use to visualize and represent situations and conditions in the problem?
  • What is the expected range of the answer for this problem?
  • Should I make an estimate for the answer?
  • What estimate can I make for the solution? Should I change the numbers for that purpose?
  • What will be the unit of my estimate?
  • In this situation, would it be helpful to use…a graph, number line, ruler, diagram, calculator, or a manipulative?
  • Why was it helpful to use…?
  • What can using a ____ show us that ____ may not?
  • In what situations might more information be helpful …?

The following problem illustrates students’ reasoning:
Three-fourth of the yard was converted into a vegetable garden. Two-third of the garden is used for gardening herbs. What fraction of the garden is herbs?

Student One: I am going to assume that the garden is a rectangular shape. Let me represent the rectangle as my whole. The herb garden is going to be ⅔ of. Approach One: I can see my problem as coloring first ¾ of the garden representing the garden and then I color ⅔ of the ¾ in another color. That will represent the herb garden as: ⅔ of ¾ = ¼+¼=½.
blog26.1

That is a correct approach to get the answer; however, it does not help us to arrive at the procedure.

Student Two: I am going to assume that the garden is a rectangular shape. Let me represent it as a 1 by 1 rectangle. The herb garden is going to be a ¾ × ⅔ or ⅔ × ¾ rectangle. By definition ¾ × ⅔ is the area of the rectangle with the dimensions of ¾ and ⅔. Therefore, the vertical side of the rectangle is divided into four equal parts and the horizontal side into three equal parts and the ¾ × ⅔ rectangle is formed.
blog26.2

As the diagram suggests, this area is 2 by 3 parts out of 3 by 4 parts and that is represented as (2×3) out of (3×4) or blog26.3. In other words, we have
blog26.4

The representational tool “area as multiplication” is more efficient and strategic compared to the “groups of” tool for multiplication. Although at the whole number level, the four models of multiplication: repeated addition, groups of, an array, and the area of rectangle are equally good as an introduction, the most efficient is the area model.

When students are sufficiently familiar with the tools, the teacher should pose problems that are tool specific and help students to sharpen their tools by practicing them daily till they are proficient, a better tool is available, or a better way of using that tool is possible.

I suggest teachers devote about 20% to 25% of time in every lesson on tool building (achieving proficiency in the use of physical tools and fluency in the use of thinking tools). For example, arriving at a particular definition relating to a mathematical idea is part of conceptual development. But, mastery in using definitions, concepts, and procedures is tool building. As students get older, they need to add more versatile tools or create new tools by combining tools to their mathematical toolbox.

The teacher starts with definitions, develops conceptual understanding and arrives at the procedure, but the children should not remain at the definition and procedural level; they should solve problems—solving problems builds tools and their efficient use builds mathematics stamina. Proficiency in the use of tools and concepts are built when we use them regularly and apply them in different settings. Some concepts become tools and then these tools are used to learn new concepts and new tools. This iterative process continues in learning mathematics. When we use tools, we learn new mathematics. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a whole number and concluded that it must be a different kind of number.

 

 

Versatile tools (exact, efficient, elegant) build mental schemas/models that last. What makes a tool like the Cuisenaire rods, Base Ten blocks, Invicta balance, Empty Number line, Bar Model or the area model truly powerful is that it is not just a special-purpose trick or temporary crutch for a particular type of problem but is faithful to the mathematics and is applicable to many domains and concepts. Because these tools help students make sense of mathematics, they last. And that is also why the CCSS and SMP mandates them.

Mathematically proficient students are able to identify relevant external mathematical resources such as people, books, digital content on a website and use them to pose or solve problems. They use technological tools to explore and deepen their understanding of concepts and make connections between concepts and procedures.

Technology Tools
Hands-on tools are useful; however, as we prepare students for the world of work and the power of technology, students need to understand the range of use, strengths, and limitations of these tools. Today, technological computations tools (e.g., calculator) are common outside the classroom, so the classroom needs to reflect this reality. With the different technologies—calculators, smart phones, tablets, and laptops, the question of when and how to use technology becomes even more important.

Technology can allow greater opportunities to visualize, explore, predict, and compare mathematical ideas. A parallel practice for teachers, therefore, is to augment the use of appropriate technological tools for mathematics instruction at the appropriate time. The use depends on teachers deciding first the mathematical goals of instruction and then which tools may be most effective in accomplishing them.

The question of when to use technology tools is a question about the most productive use of valuable classroom time. The answer to that question may lie in the reasons we teach mathematics today:

  • to understand numbers and patterns found in nature (number concept, numbersense, numeracy)
  • to acquire math tools, know when and how to use those tools
  • to make fast and accurate predictions and check the reasonableness of answers
  • to grow and maintain mental power
  • to identify unknowns in a situation, represent and deal with them
  • to think logically and clearly when solving problems
  • to feel comfortable with quantitative and spatial thinking demands in a technological world

Using a calculator as a tool should be a strategic decision. Calculators should be used with caution in elementary school (that means very carefully). Calculators should be available as computational tools, particularly when many or cumbersome computations are needed to solve problems. However, when the focus of the lesson is on developing computational skills or algorithms, the calculator should not be used. It should be a tool that provides access, simplifies the task, or confirms accuracy. It doesn’t make sense for a fifth grader to use a calculator for 8 + 13. However, it may make sense for a first grader to confirm the sum that way. But, for 18.1759 + 27.19427, use the calculator.

In Elementary School, the following conditions are satisfied then and only then I will allow the use of calculators: (a) student knows the arithmetic facts, (b) good in estimation, (c) understands the concept/procedure for which the calculator is being used (e.g., in how many 4 are there in .04, the student knows which one is the divisor and the dividend). Teachers should encourage students in all contexts to estimate first and then if necessary to use calculators. Students who know “about” how much an inch is can tell when they use a ruler if their answer is reasonable. Students who understand how to round and estimate when multiplying money know if what they plug into a calculator makes sense.

Students should have the opportunities to discuss when they might use tools (e.g., concrete, technology, paper/pencil, mental math), and they should know when and where tool use is appropriate. Tool use is not an ideological decision: it is neither dedication to “times tables,” “long division,” and “repetition and memorization” nor is it allegiance to “fuzzy math” reform—preaching concept over content, exposure over mastery, insight over “right.” Tool use is the judicious decision of both and is the integration of the linguistic, conceptual, and procedural components. This can happen if we:

  • allow students to spend less time on tedious calculations and more time first on language development, conceptual understanding and solving problems
  • help students develop better number sense (number concept, arithmetic facts, and place value)
  • allow students to study mathematical concepts they could not attempt if they had to perform the related calculations themselves and use the tool to develop the concept by seeing the patterns
  • allow students who would normally be turned off by math because of frustration or boredom to increase their mathematical understanding and help them to acquire fluency of arithmetic facts by seeing number relationships
  • simplify tasks while helping students determine the best methods for solving problems
  • make students more confident about their math abilities, once a problem is solved using a calculator, make the number manageable and help them solve without the calculator.

While few educators deny the usefulness of calculators at the high school level, we need to rethink their use in the lower grades. Inappropriate use of calculators prevents students from seeing the underlying structure and beauty in mathematics, inhibits them from seeing mathematical relationships, and gives them a false sense of confidence about their math ability. Students who do not do long division, who quickly pull out their calculator to find the answer, do not understand the underlying principle of division. For example, when I asked eleventh grade students: What is the largest remainder you could expect when you divide a six-digit number by 11? most took out their calculators and started calculating by choosing random numbers. Many gave unreasonable answers.

If properly used, technology is an important means to achieve the goals of instruction. However, specific examples of technology use do not start appearing in the content standards (CCSS-M) until the middle school grades, and most appear at the high school level.

There are elements of mathematics at all levels (not just elementary) where some basic facts simply must be memorized (of course, after understanding and using them in problems). For example, to succeed in algebra, one should have mastered operations on fractions and integers. And success in mastering fractions is dependent on the mastery of:

  • understanding, fluency, and applicability of multiplication tables,
  • divisibility tests,
  • prime factorization, and
  • short division.

Similarly, at each level new skills and relationships emerge (e.g., laws of exponents, etc.), which need to be automatized (once again first understanding and then using them in problem solving situations) and without rote.

Many teachers, not just at third, fourth, and fifth grade levels, but at the high school and college levels, report that students do not know their basic facts or the concept of fractions. This is due to the ineffective use of tools. That is, students are overly dependent on concrete models, counting, number line, calculators, graphic organizers, multiplication charts, etc.

Others disagree, however, claiming that calculators help younger students grasp the underlying principles of mathematics by allowing them to spend time on those principles rather than on the rote computations necessary to solve them.

To build a beautiful piece of furniture, a carpenter does not use only one tool but recognizes the relationship between the object to be made and the tools needed. It is up to individual teachers, of course, to find the right way to achieve the many and complex goals of mathematics learning. And most teachers strive to do just that.

Examples of Tool Usage

Common Core Curriculum Standards (CCSS-M, 2010) call for specific situations and context to use specific tools. For example, the following geometric standard calls for the use of classic geometric tools:
Make formal geometric constructions with a variety of tools and methods (compass     and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle, bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

This high school standard on “Interpreting Categorical and Quantitative Data” suggests the use of several appropriate tools. Here’s an example:
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

The high school standards for number and quantity include this paragraph: Calculators, spreadsheets, and computer algebra systems can provide ways for students to become better acquainted with these new number systems and their notation. They can be used to generate data for numerical experiments, to help understand the workings of matrix, vector, and complex number algebra, and to experiment with non-integer components.

In high school algebra, the standards suggest uses for spreadsheets of computer algebra systems: A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave. Here’s an example:
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Kindergarten through Second Grade The focus of these three years is to master Additive Reasoning and its applications and identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures. To achieve these goals, the tools may include counters, Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, and concrete geometric shapes (e.g., pattern blocks, 3-d solids). Students should also have experiences with educational technologies such as virtual manipulatives and mathematical games and toys that support conceptual understanding, but calculator is not advisable. During classroom instruction, students should have access to various mathematical tools as well as paper (for applying alternative methods of adding and subtracting—concrete models, Empty Number Line, Decomposition/recomposition, transforming a problem by translating, by place value methods, and standard procedure), and determine which tools are the most appropriate to use. For example, find the difference: 93 – 46.

Here are the methods in order of efficiency.
Concrete Tools
Cuisenaire rods, BaseTen blocks

Pictorial/Representational Tools
Hundreds’ Chart, Empty Number Line (small-big, big-small, small-big-small jumps)

Abstract/Symbolic/Procedural Tools
Decomposition/ recomposition methods:
26.12

The standard procedure:
26.13

The fundamental concept in these grades is decomposition/recomposition. Decomposing and recomposing numbers should be done with manipulatives and models until it becomes something students can do mentally. Then we should go to the standard algorithms and mnemonic devices.

Third and Fourth Grade The focus of these two years is to master Multiplicative Reasoning and its applications; identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures; and introducing the concept of fractions. To achieve these goals, the tools may include Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, concrete geometric shapes (e.g., pattern blocks, 3-d solids), and factions strips. By the end of fourth grade, mathematically proficient students have mastered numeracy skills (ability to execute the four whole number operations correctly, consistently, fluently, with understanding in the standard form), use available tools (including estimation) when solving problems and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter and area. Or they may use graph paper or a number line to represent and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative size of units within a system and express measurements given in larger units in terms of smaller units.

Fifth and Sixth Grades The focus of these two years is on Proportional Reasoning, introduction to integers and equations, and their applications to quantitative and geometrical situations. Mastery of the concept of and operations on fractions in different forms—parts to whole, comparison of quantities (ratio, rate, etc.), comparison of a quantity with a standard (decimals and percents), comparison of comparisons (proportions), concept of integers and its applications; identifying, recognizing, drawing, and using 2 and 3-dimensional shapes and figures and their relationships. To achieve these goals, the tools may include Cuisenaire rods, Invicta Balance, place value (base ten) blocks, hundreds number boards, number lines, fraction and decimal strips, geoboard, concrete geometric shapes (e.g., pattern blocks, 3-d solids), and factions strips. Technological tools—calculators, Apps, Geometric Sketch pad, etc. Mathematically proficient students consider the available tools (including estimation) when solving problems and decide when certain tools might be helpful. For instance, they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions. They use graph paper to accurately create graphs and solve problems or make predictions from real world data.

At any level, what is critical is that students are given opportunities to use each tool and to learn when its use is appropriate. For example, is it better to use a tape measure or a ruler to measure the length of a room? Why? In what situation would you use a protractor? Why would pattern blocks be a tool for helping students understand the need for a common denominator when adding or subtracting fractions? Is this the only tool students should experience with this specific content? Questions such as these will help teachers determine how tools foster mathematics learning most effectively.

Use Appropriate Tools Strategically – Part II

Use Appropriate Tools Strategically: Right Tools for the Right Job – Part I

One of the Standard of Mathematics Practice (SMP 5, CCSSI 2010, p. 7) calls for selecting appropriate tools and using them strategically. The two words “appropriate” and “strategically” apply to students as well as teachers. What does appropriate and strategic mean in the use of a tool? The answer depends on our interpretation of tools, our expectations for using them, and their role in gaining mathematical maturity for our students.

Simply, a tool is anything that aids in accomplishing a task—learning a concept/procedure. It is appropriate if it makes the concept transparent and provides the learner access to the concept. A tool is an appropriate tool in the context of what it is for and who is using it and for what purpose. Appropriateness of a tool, thus, is a function of the concept, the user, and the standard of mastery expected. A tool is appropriate if it helps the student learn the concept at the expected level.

Without strategic use, any tool, including an appropriate one may be ineffective and may not produce optimal results. However, we need to have a common definition of “using a tool strategically.” If the tool produces optimal results—develops language, concepts, and procedures with rigor and efficiently, the tool is being used strategically.

The number line is sometimes regarded just as a visual aid for children—as a physical tool. It is, in fact, a sophisticated image used even by mathematicians; it is a thinking tool. For young children, it helps develop early mental images of addition and subtraction that connect arithmetic with measurement, mental arithmetic, and standard algorithms. Rulers are just number lines built to specifications. In Kindergarten and first grade, it is the starting of solving a problem like 9 – 5 = ?

blog 25.1

This number line image shows “the distance from 5 to 9.” It gives visual and conceptual richness to the problem and in extension the flexibility of thought. For example, if children are given the problem:

My team scored 9 points on Monday and 5 points on Tuesday. Then,

  • How many more points did they score on Monday than Tuesday?
  • How many fewer points did they score on Tuesday than Monday?
  • What was the difference between the scores on Monday and Tuesday?
  • How many more points should they have scored on Tuesday so that their score was the same as on Monday?
  • How many less points should they have scored so that their score would have been as on Tuesday?
  • How many extra points did they score on Monday if the goal of the game was to score only five points?

The number line can answer all of the questions raised in diverse contexts. Children who see subtraction that way can use this model to see the problems with larger quantities and different numbers. For example, let us consider the problem: 63 –27 as “the distance between 28 and 63.” To do so without crossing out digits and borrowing and following a rule, they may only barely understand.

blog25.2

But this number line easily explains the procedure and extends to mental calculations and applications in real life situations. In fact, it leads them to forming mental models of subtraction and helps achieve fluency in problem solving—both addition and subtraction. The number line model also extends naturally to decimals, fractions, integers, and elapsed time. For example, when students are asked to solve the problem:
If the temperature in the morning was -20 and reached 50 at noon time, what was the change in the temperature?

Many students answer it as 30, showing that they do not have the conceptual understanding and visual image in their mind for the problem. However, with the use of the number line, they can see that the distance from -2 to 5 is the number we must add to -2 to get 5: From -2 to 0 is 2 units and from 0 to 5 as 5 units, therefore, the total distance from -2 to 5 as 7, and they can generalize to solve a problem like: 42 – (-36), which can also be seen as distance from -36 to 42, using the Empty Number Line as the sum of distances from -36 to 0 (=36) and then from 0 to 42 as (=42) or 36 + 42.

Number line, on one hand, unifies arithmetic, making sense of what is otherwise often seen as a collection of independent and hard-to-remember rules and, on the other hand, it is generalizable and one can leap into algebra. The number line remains useful as students study data, graphing, and algebra: two number lines, at right angles to each other, label the addresses of points on the coordinate plane.

To find the difference 231 – 197 by counting on the number line by tens or ones is an inefficient use of the number line. But treating the problems as an addition problem and using the number line as Empty Number Line is effective and efficient as it improves numbersense and mental math.

blog25.3

A tool by itself is neither appropriate nor strategic. It is its use that determines whether it is appropriate and strategic.

Tools are meant to help teachers and students make sense of mathematics and its role in the world around us. They are to make teaching efficient and to support accurate, rigorous, and proficient learning. It is, therefore, our responsibility to know how to select, understand, and use the tools strategically to develop our students’ proficiencies in learning and their competence in mathematics. Ultimately, the strategic use of tools is when teachers are able to transfer the control of the use of tools to students and they use them strategically.

For students it means that they acquire the facility to use appropriate tools strategically in learning and solving problems in mathematics. It is one of the important skills of mathematically proficient students.

Teachers’ Role in Using Tools Strategically
An important element of the strategic use of tools depends on the goals of instruction. A teacher first considers mathematical goals of her instruction and then decides which tools may be most effective in accomplishing them. It means to select tools to get the concept across the students and then to use them with optimal results in learning and achievement.

Appropriateness of a tool means that the concept becomes transparent to the students and they can see the congruence of representations through the tools and the abstract/symbolic form. To achieve this, the teacher asks:

  • Does it show the concept exactly—is the representation transparent?
  • Is it efficient to demonstrate the concept or procedure?
  • Is it easy to work with, to manipulate?
  • Does the student see it efficiently and clearly?
  • Can this tool be used to extrapolate, generalize, and abstract the concept from the current manifestation?
  • Is it available to them?
  • Will the student be able to use this tool easily and effectively?

Effective, appropriate, and strategic use of tools is important at all grade levels, but the types of tools and how they are used can differ. Golf players know when to use which “iron.” They constantly practice their usage of the tools. The same is true for a “budding” mathematician and a real mathematician alike. In developing students’ capacity to “use appropriate tools strategically,” teachers make clear to students why the use of tools will aid their problem solving processes.

Proficient students are sufficiently familiar with tools appropriate for their grade to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

Using Appropriate Tools Strategically
Mathematically proficient students consider the available tools when solving a problem and they use them strategically. The framers of CCSS-M seem to refer to two kinds of tools: physical and thinking tools. In the case of physical tools, one is looking for proficiency, and in the case of thinking tools, one wants fluency.

The physical tools (commercially prepared or constructed by teachers and students) might include pencil and paper, concrete manipulative models (sundry counting objects, fingers, TenFrames, Cuisenaire rods, Algebra tiles, Base-Ten blocks, Invicta Balance, fraction strips, games and toys, straight edge, rulers, diagrams, two-way tables, graphs, graphic organizers, protractor, compass, calculator, spreadsheet, computer algebra systems, statistical package, or dynamic geometry software, geometry sketch pad, geogebra, iPad apps, Smart Phone Apps, Graphing Calculator, Algebra Computer System, Statistics Package, Spread Sheet, etc.).

Manipulatives are objects that appeal to several senses and that can be touched, moved about, rearranged, and otherwise handled by children. Using manipulatives in the early grades is one way of making mathematics learning more meaningful to students as they are used to make abstract ideas more concrete and transparent. Modeling with manipulatives is the first step in creating an environment where students can begin to understand abstract mathematical concepts in a variety of contexts and ways. For example, an elementary teacher might have students select different color tiles to show repetition in a patterning task. A middle or high school teacher might have established norms for accessing tools during the students’ group learning and problem solving processes to make things and see geometrical relationships from them.

However, a manipulative does not by itself carry the intended meaning and does not guarantee that mathematical understanding will result from use. It is the expertise of the teacher in the use of manipulatives and the amount of time and experiences students are given to interact with the manipulatives that lead to increased achievement.

Counters of many kinds, Base-10 blocks, Cuisenaire rods, Pattern Blocks, measuring tapes, spoons or cups, and other physical devices are all, if used strategically, of great potential value in the elementary school classroom. They are the “obvious” tools. But, physical tools should satisfy the following properties: they should (a) be exact and transparent, (b) be efficient, and (c) be elegant. The physical tools serve three purposes:

(a) generate the language of that mathematical idea,
(b) help develop the conceptual schema of the idea, and
(c) derive the procedure related to the idea.

Concepts must be developed and reinforced by the tool. The use of the tool itself should support reasoning rather than mere procedure. Reasoning develops understanding. And understanding develops mental math and strategies. The idea of understanding holds true for other tools and transfers to paper/pencil as well. With understanding, physical tools develop into thinking tools. For example, the practice of making ten by the help of Cuisenaire rods develops the mental math strategies suing making ten, for example, 8 + 6 = 8 + 2 + 6 or 4 + 4 + 6; 17 – 9 = 7 + 10 – 9 = 7 + 1, etc.

Understanding helps students realize accuracy and proficiency. Consider 9.1888 + 11.1020. If I use a calculator, I should know that my sum will be in the neighborhood of 20. I need to reconsider if my calculator result is dramatically different. This transcends grade level. For example, if I determine that my slope is negative and my line rises from left to right, then something is not right.

In other words, students must derive and understand outcomes of operations with and without a calculator but also reinforce this understanding while using the calculator. If the Sin of an angle comes out to be more than 1, then, there is something wrong. Similarly, using a protractor in measuring an angle, it is more than just “lining it up the right way.” Understanding enables them to be proficient in diverse situations and even with diverse protractors.

When we have developed the language, concept, and procedure, using physical tools, students should convert them into thinking tools and then practice the procedure and the skills related to that idea. The physical tool should always be converted into thinking tools.

The thinking tools refer to vocabulary, written or mental strategies (decomposition/recomposition, properties of operations, etc.), conceptual schemas (e.g., area model of multiplication), approaches (e.g., prime factorization for LCM, etc.), skills (e.g., facts, translation from native language to math symbols, etc.), and procedures (standard or alternative). The mathematical thinking tools deal with intellectual and cognitive skills.

Cognitive/Learning Skills
A major outcome of using concrete materials as tools for mathematics is the development of prerequisite skills to anchor mathematics ideas.

  • Following sequential directions: every procedure and task analysis is dependent on this skill,
  • Pattern analysis: mathematics is the study of patterns in quantity and space; recognizing, identifying, extending, creating, and applying are integral part of tool usage,
  • Spatial orientation/space organization: observing and identifying spatial orientation, organization, and relationships is essential in tool usage,
  • Visualization: holding and manipulating information are essential for mathematics, particularly for mental math and planning problem solving and selecting tools. Tools that have patterns, color, shape, and size (e.g., visual cluster cards, Cuisenaire rods, etc.) develop visualization and therefore enhance working memory.
  • Estimating: along with number concept, numbersense, the key skill implicated in dyscalculia is estimation; using appropriate concrete tools (non-counting materials) help develop estimation,
  • Deductive and inductive reasoning: The development of formal/ abstract/logical reasoning begins when children use concrete tools effectively,
  • Collecting/classifying/organizing: These are developmental concepts; children begin at concrete level and then are transitioned to abstract/formal levels (e.g., collecting data—look up information on Internet, in a book, in one’s notes, and read teacher comments on home work and tests, etc.),
  • Metacognition: Learning about one’s learning—what works and does not work.

Mathematical Skills
The purpose of many physical tools is to acquire abstract/formal tools to prepare students for college and careers. This is achieved when they have these tools:

  • linguistic: read the problem (e.g. focus on instructions), know the vocabulary, rewrite the problem in one’s own words, underline and understand the key words, recall and define the key terms, translate terms from English language to mathematical language and symbols, ask questions, etc.;
  • conceptual: describe what the problem means, identify what mathematical concept is involved, what the unknowns are, what the knowns are, draw diagrams/figure/curve, make tables, create relationships between knowns and unknowns, write mathematical expressions, equations/inequalities, see patterns, solve a special case, recall an analogous situation or problem, consult a related solved problem, generalize, etc.;
  • arithmetic: know decomposition/recomposition of numbers, master arithmetic facts, understand place value, describe the relationship between the quantities, estimate the outcome, create an empty number line, make a bar model, make a concrete model, draw a picture, create or use a graphic organizer, etc.;
  • algebraic (identify the variables, write a formula, equation, or inequality, construct a table, chart, graph, or diagram, sketch the function, identify the parent function, create a prime factor tree or successive prime division chart, use a graphic organizer, etc.),
  • geometric tools (draw a figure or diagram, classify data or information, look for spatial relationships, etc.);
  • probabilistic and statistical (draw a Venn diagram, make a graph, create a tree-diagram, make lists, make a model, consult result tables, guess and check, etc.)

Mathematically proficient students gain entry to the problem situation and the solution process by using appropriate physical tools, manipulative materials—such as Cuisenaire and BaseTen blocks, for example, at the elementary school level, fraction strips, fraction bars, algebra tiles at higher grades or thinking tools (writing relationships between knowns and unknowns) to model a problem. For example, mathematically proficient high school students analyze graphs of functions and solutions and their behaviors with a graphing calculator and realize that technology can enable them to visualize the results of different assumptions on the conditions of the problem, explore their consequences, compare predictions with data, and the role of assumptions and constraints on the solution process.

Thinking tools also develop the ability to make sound decisions about when each of these tools might be helpful and gain the insight from their optimal use and also their limitations. This certainly requires that students gain sufficient competence with the tools to recognize the differential power and efficiency they offer.

It also requires that their learning include opportunities to decide for themselves which tool serves them best and why. In order for students not to become dependent on a particular tool and strategy and to develop flexibility of thought, it is important that the curriculum and teaching include the kinds of problems that involve the use of different tools. Students use tools efficiently and deepen their understanding by using different tools to solve the same problem. For example, from time to time, a particular tool is used until students develop a competency that would allow them to make sound decisions about which tool to use. The proficiency in the use of a tool is developed when we use it frequently, discuss its use from different perspectives, and apply it in several problems. In many situations, paper and pencil are inefficient and using them is not strategic. We must therefore develop the notion that mental computations are possible, reliable, and often more efficient. However, students should have skills to detect possible errors by strategically using estimation and other mathematical knowledge.

Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful (e.g., the flat piece in the BaseTen blocks kit represents 10×10 =102 at the third grade level, 1×1 = 12 at fifth grade level, and a×a = a2 at seventh grade level).

As we explore the connections between different types of concepts (e.g. numbers relationships) to use them flexibly, we need to explore the similar interactions between different types of tools to be able to use them flexibly and strategically.  For example, using BaseTen blocks for place value or for addition and subtraction operations encourages children to count, but combining BaseTen blocks and Cuisenaire rods precludes that possibility. Similarly, learning how to solve linear equations can follow the sequence for the strategic use of several tools:

  • Invicta balance to derive and learn the properties of equality,
  • Cuisenaire rods, BaseTen blocks, and Algebra tiles to learn arithmetic and algebraic manipulations and then to arrive at the procedures, and properties of operations,
  • Paper and pencil to record these activities and procedures, then practice these operations formally,
  • Graphing tool to see the behavior of the equations, functions, and solutions,
  • Using computer algebra system (CAS) to take more complex equations and see their relationships and behaviors. To have proficiency in the strategic use of tools, the role of questions and classroom discussions is critical. The teacher can ask questions to help students to identify, select and use tools effectively.

To have proficiency in the strategic use of tools, the role of questions and classroom discussions is critical. The teacher can ask questions to help students to identify, select and use tools effectively.

Use Appropriate Tools Strategically: Right Tools for the Right Job – Part I

Model with Mathematics: Real World to Mathematics and Back

No problem can withstand the assault of sustained thinking. Voltaire

The Standards for Mathematical Practice (SMP in CCSS-M) describe mathematically productive ways of thinking that support both learning and applications by modeling mathematics in the classroom. Providing these experiences has to be an intentional decision on the part of the teacher. In other words, students learn mathematics concepts and procedures using models, on one hand, and use mathematics to model real problems, on the other. This activity is the basis of scientific and many social science innovations. Students need to experience this aspect of mathematics in the classroom from the very beginning.

Models and practical applications of mathematics have three distinct roles in mathematics learning. The first purpose of modeling and applications of mathematics is to motivate students to learn, engage, and see the relevance of mathematics. Here students learn mathematics by using concrete and representational models. This requires choosing the right manipulative, instrument, model or pedagogical tool to learn a mathematics concept, procedure or mathematical language.

In the second case, students apply mathematics—when they have learned a concept, skill, or procedure, to solving real life problems. The second case brings the appropriate mathematical knowledge and methods to match the demands of the problem. The third aspect deals with generating new mathematics or a model to solve a problem where one direct mathematics idea is not available. Throughout history, this twin process of modeling to learn new mathematics and solving novel problems by developing/ discovering models has solved real problems and generated new mathematics ideas—concepts, procedures. This is the interplay of pure and applied mathematics.

The facility of modeling mathematics is an example of the mathematical way of thinking and demonstration of competence in mathematics.

The modeling process, as application, spans all grade levels and applies mathematics that students know up to that grade level to solve “real” and “meaningful” problems. A simple example of modeling is the application of fractions to solve problems relating to rates of increase and decrease in various situations.

Deep Mathematical Understanding and Flexibility
A great divide often exists between students’ conceptual understanding, their procedural skills, and their ability to apply what they know. An even larger divide is that students may have conceptual and procedural knowledge but they have difficulty in applying mathematics ideas and realizing the power and relevance of mathematics.

The belief that applying mathematics is complex and complicated for many students and is separate from learning the concept and skills often leads many teachers to stop short of this most important step of teaching problem solving as part of each lesson. However, application of mathematics should not be a separate activity. While students who learn mathematics in a traditional fashion perform well on customary, standardized assessments, they tend to do poorly on tasks that require them to apply the math concepts to real problems. Students who learn mathematics through a modeling lens are better able to perform on both traditional and non-routine assessments.

Students too often view what happens in the math classroom as removed from and irrelevant to the real world. When a task can tap into a student’s innate sense of wonder about the world around him or her, that student becomes engaged in the problem-solving process. But when we can pique the interest of students through problems that have a basis in reality, we encourage them to question, investigate, and problem solve. Modeling bridges this gap and allows students to understand that to resolve many of the situations around them involve and require mathematics. When students engage in rich modeling tasks, they develop powerful conceptual tools that increase their depth of understanding of mathematical concepts and improve their abilities and interest in mathematics.

The concept of mathematical modeling, as a mathematics practice, has an important place in implementing the Common Core State Standards for Mathematics (CCSS-M). This practice emphasizes a student’s ability to realize the power of mathematics by applying mathematical tools to solve problems. Mathematical modeling demonstrates the power of mathematics for learners. Throughout their schools, students should use mathematical models to represent and understand mathematical relationships.

Levels of Knowing and Modeling
At each stage of mathematics learning (intuitive, concrete, pictorial/representational, abstract/symbolic, applications, and communication) and in mastering its components (linguistic, conceptual, and procedural) problem solving plays an important role. At the intuitive and concrete levels, a real life problem not only acts as a “hook” for students to see the role of mathematics as an important set of tools but also gets them interested in that concept.

At abstract/symbolic and the applications levels, applying the concepts, procedures, and skills shows how those elements are used and integrated, so students learn the strength and limitation of a particular mathematical tool.

When students have acquired a set of concepts and procedures and face a real life problem, they try to model the problem in mathematical form and solve it. This takes several forms: word problems, problem solving, and modeling. Because these are not isolated activities, problem solving, modeling, and application must be embedded throughout students’ learning of mathematics.  To make sense of developments in the natural, physical, and even social sciences and to solve the related problems involves looking for and developing mathematical models.

By incorporating mathematical modeling in their classrooms, teachers can motivate more students to enter STEM fields and to solve real life problems in social sciences and humanities. Integrating computers and calculation tools with mathematics methods, many of the social science problems are amenable to mathematical modeling.

Problem Solving: Model for Introduction to Mathematics Concept
Real life examples can introduce mathematics concepts and bring the real world into the mathematics classroom.  A real world scenario motivates students to see mathematics as relevant to their lives and increases the desire to learn that mathematics idea. In this situation, a teacher moves students, explicitly, from real-world scenarios to the mathematics in those scenarios.

For example, an elementary school teacher might pose a scenario of candy boxes with an equal number of candies in each box and represents it as repeated addition and then relates and extends “the repeated addition of a number,” “groups of objects,” or the tile pattern in the yard to see the “area of a rectangle” into the concept of multiplication.

An upper elementary grade teacher poses a scenario of candy boxes with a number of candies with different flavors in each box to help students identify ratios and proportions of flavors and ingredients.

A middle school teacher might represent a comparison of different DVD rental plans using a table, asking the students whether or not the table helps directly compare the plans or whether elements of the comparison are omitted.

A high school teacher shows several kinds of receptors (parabolic dishes) and poses a set of questions to instigate a discussion why parabolic receptors are optimal shapes to receive the sound, radio, and micro-waves. This discussion instigates the study of parabolas, in particular, and quadratic equations in general. Similarly, the discussion of waves of different kinds might instigate a discussion of Sinusoidal curves in an algebra, trigonometry or pre-calculus class.

A statistics teacher brings in a big bag of MMs to the class and asks:  “Without counting all of the MMs, how do we determine the number of MMs of different colors, as close as possible to their distribution in the bag?” Students might say: “It is easier to count them, why go through all that?” The teacher responds by posing the problem: “Yes, you can count the MMs, but how do we determine the population of fish in the pond or the number of particular species of animal in the wild as we cannot directly count them?” In this process, she shows the power of sampling method in real life and therefore the reasons to learn it.

The role of these problems is to motivate students to learn mathematics and show the power of learning the tools of mathematics. To achieve this goal, the problems have to be of sufficient interest and diversity. They should show the relevance to the topic being studied. The mathematics in them should be transparent.  Finally, they should be accessible to children.

Problem Solving: Applications of Mathematics
The first application of a mathematics concept, procedure or a skill is in the form of word problems. Word problems, while more demanding than pure computation problems, are typically presented in the context of a specific mathematics content area or skill, and are solved with a particular method or algorithm; therefore, students do not apply much mathematical reasoning. Word problems can serve as one example of problem solving; however, typical word problems in mathematics classrooms are “concocted.”  Often, they have no resemblance to realty, so to call them as applications is stretching the meaning of the word “problem solving.” However, if teachers routinely mix different types of problems, involving several mathematics concepts, they can help solicit mathematical reasoning.

Problem solving, on the other hand, is when students need to decide what mathematics, concept, skill, or procedure is involved for solving the problem. Problem solving is more advanced than word problems because it requires students to (a) translate native language to mathematics expressions, relationships (equations, inequalities, formulas, etc.), (b) interpret what mathematics skills, concepts, and procedures are needed to solve the problem, (c) make assumptions and approximations to simplify a complicated situation, and realize that these may need revision later, (d) determine how to find the answer, and (e) to make sense of the solutions in terms of the conditions of the problem and the solution sought.

Problem solving takes place when students have acquired a certain set of mathematics concepts, procedures and skills and the teacher presents problems that they can solve by using these newly learned skills. The role of the teacher is to identify and present these problems to students. These are focused application problems where the math content and the skills needed to solve problems have a close match between the problem and skill set, but they are not recipe oriented.

There are three kinds of applications of mathematics: (a) intra-mathematical, (b) interdisciplinary, and (c) extra curricular.

In intra-mathematical applications, a student learns a new mathematical skill, concept, and procedure and can apply this to solve problems in other parts of mathematics. To be successful in this context, teachers must be cognizant of the connections that can be made in different parts of the mathematics curriculum at that grade level and even higher grades. The role of modeling in mathematics, in this context, is making connections between different branches of mathematics and discovering new relationships about mathematics concepts. Students who engage in modeling in the math classroom have increased mathematical autonomy and flexibility in the ways they use mathematics.

For example, in early grades, students may learn the property of commutative property by using Cuisenaire rods:
2 + 5 = 5 + 2
models with math 1

In this case, the model is used to learn a mathematics concept. On the other hand, they may use Cuisenaire rods to solve an addition problem at first grade (I spent $9 on Monday and $7 on Tuesday.  How many dollars did I spend?) by constructing and then writing an addition equation to describe the situation.
model with math 2

In the middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. On the other hand, a student might use paper folding to see the division of fractions or visual cluster cards to learn the operation on integers.

In inter-disciplinary applications, a student learns a new concept, mathematical skill, or a procedure and can apply it to another discipline.  For example, a student learns the concept and operations of fractions and now can apply this knowledge in the “shop class.”  A student learned the concept of transformations in geometry and now can create a collage by using tessellations in art class.  The student just learned how to solve linear equations, so now she can use this skill to solve a problem in chemistry class. A school-based project integrating learning from several disciplines is a good example of this type of application. To achieve this objective of inter-disciplinary applications, teachers should be aware of the interconnections of the mathematical concepts and the use of mathematics in other disciplines of students’ curriculum.

In extra-curricular applications, a student learns a mathematics concept, procedure, and a skill and applies these to problems in everyday situations outside of the curriculum.  Here the teacher finds problems from the real world to connect with mathematics skills.

The goal of mathematical modeling, at this stage, is for students to pose their own questions about the world and to use mathematics to answer those questions. Quite naturally, most students want to know there is some utility in what they’re learning, that a lesson is not just an isolated lesson with no future use. In each section, in each module, they should be able to see what they are learning as relevant to their own lives and their own careers.

Discovering Mathematics: Modeling as Content Category
Throughout history, individuals have generated mathematics knowledge to solve practical problems. On the other hand, some mathematicians focus on mathematics for the sake of mathematics. Many others are interested in mathematics for its power, its tools, its approach to problem solving and modeling problems. The mathematical tools available at any given time are the means for innovation, inventions, determining the standards of living at that time. For example, in the twentieth century, most science, engineering and technology problems were tackled by the tools of calculus, but with the advent of calculators and computers, it is possible to extrapolate the data and find solutions using discrete methods. In such problems, there is need to integrate the mathematical tools that are based on continuous models (functions, calculus, etc.) and discrete models (finite difference methods, probability, statistics, etc.). In this scenario, students use their mathematics skills to discover new mathematical tools and skills.

Modeling as conceptual content category means using mathematics models to generate and learn new mathematics concepts. It is more than just using a concrete material, pictorial representation to learn a mathematics concept. For example, the study of transformations (both rigid and dynamic) to geometric and algebraic objects gives rise to the study of geometrical concepts, understanding of curves, functions, and conic sections. Similarly, in statistics and probability, we create, model, or simulate an idea to study it.

Modeling with Mathematics
When students themselves find or encounter real world problems and want to solve them, they are modeling mathematics at the highest level. One distinct difference between typical problem solving and mathematical modeling is that modeling frequently involves interpretation or analysis of an essentially nonmathematical scenario. This content conceptual category reflects a modeling cycle involving a series of operations.  Students must:

  1. identify a problem,
  2. study the scenario that gave rise to the problem to determine what the important factors or variables are, interpret these mathematically, identify variables in the situation and selecting those that represent essential features,
  3. observe the nature of the data—looking for regularities (e.g., if the data is increasing at a constant rate, it may be modeled by a linear system; if the change is constant at the second level of iteration, it can be modeled by a quadratic function; if each entry in the data is a constant multiple of the previous entry, it is modeled by a geometric/exponential function, etc.),
  4. develop and formulate a tentative mathematical model by selecting arithmetical, geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables,
  5. use the model to analyze the problem situation mathematically, draw conclusions, and assess them for reasonableness of the solution,
  6. analyze and perform operations on these relationships obtained by the modeling process to draw conclusions,
  7. test the solution to determine whether it makes sense in the context of the problem situation,
  8. interpret the results of the mathematics in terms of the original situation, and
  9. if the solution makes sense and they have a mathematical model for this type of problem, validate the conclusions by comparing them with the situation, then either improve the model or express the model formally in mathematical terms – if it is acceptable, and report the conclusions and the reasoning behind them.

However, if the problem is not adequately solved, the learning from this trial is incorporated to improve the model.

The iterative process and interpretation of the solution are hallmarks of the modeling process. A vital part of modeling is interpreting the solution and comparing it to reality.

By high school, students might use pictorial, numerical, algebraic, geometrical, trigonometric, functional, probabilistic, statistical, and computational methods to solve real life—social and physical science problems.  Choices of functions, diagrams, assumptions, range of constraints, and approximations are present throughout the modeling cycle.

Why Modeling?
There are common misconceptions about mathematical modeling. Many teachers view mathematical modeling as a process of showing the students how to approach or solve a problem. The first two types of applications described above are not true examples of mathematical modeling. They are uses of models to learn mathematics. The key feature of mathematical modeling, as defined by the NCTM and the CCSSM, is that students seek or encounter a problem, to solve. The teacher is the facilitator and guide in the process, but the modeling is done primarily by the students under her guidance: students select a problem, select the mathematics, integrate the skills and concepts, and then explain what they have done.

Mathematical modeling is not just a type of word problem or problem solving—it is mathematics being practiced; it is applications of mathematical ways of thinking. Modeling represents a shift from learning math to doing math. Modeling can be differentiated from word problems as it does not usually call for the use of one method or algorithm in order to solve the problem. Standard word problems or even problem solving in school mathematics curricula do not model realistic problem situations for problem solving, whereas modeling presents students with realistic problem-solving experiences requiring strategizing, using prior knowledge, and testing and revising solutions in real contexts.

The inclusion of modeling in the math classroom increases student engagement, depth of understanding, and provides opportunities for investigation, contribution, and success for all learners. Students involved in problem solving and inquiry-based activities such as modeling develop a positive disposition toward mathematics.

There is a place in the classroom for each type of problem, but it is possible to take a typical word problem and adapt it in such a way that it increases the depth of knowledge required to solve it. Open-ended modeling problems allow students to use mathematical tools and prior knowledge including measurement, proportions, map reading, scale drawings, and geography to make decisions and justify those decisions in a real-life problem. Students may use multiple methods to come to a conclusion and multiple representations to demonstrate their understanding. They must explain their solutions and use writing in mathematics to explain their reasoning.

Mathematical modeling is an effective practice for all students even for those who have a history of poor performance on traditional mathematics tasks. By incorporating modeling tasks into the classroom, we recognize all students as important contributors to the decision- making and investigation of the problem at hand. Because a good modeling task will be based in real-life experience, all students have the ability to make contributions based on their prior knowledge. Such tasks also emphasize and require a broader range of mathematical abilities than algorithmic exercises, and therefore allow a broader range of students to emerge as being capable. Students who may have a history of poor performance in math when their abilities and understanding are assessed solely on narrowly defined tasks and assessments can demonstrate significant ability and potential when given the opportunity to problem solve in a real-life modeling context. It is therefore important for teachers to emphasize to students that any (mathematically valid) solution for which they can make a strong argument is “correct.” This flexibility in thinking, and departure from the idea of only one correct solution, encourages students.

When students understand that they each have a unique contribution and valid voice in problem solving, they are more likely to become involved. Even students who have struggled in the past will contribute and share their thinking, and will be less likely to rely on the work of more successful students. Because modeling tasks can utilize a broad range of mathematical abilities, a broader range of students can emerge as capable mathematical learners.

Not all modeling problems have several feasible solutions, but the process always presents opportunities for different approaches and diversity of thinking, resulting in greater chances for success for students with diverse backgrounds and experiences. As these students gain confidence in their ability to contribute to the problem solving, they begin to develop a sense of mathematical autonomy.

Mathematical Empowerment
Our job as teachers is to present the tools, show students when and how they are used, and then provide a context in which they can choose the appropriate tools for a given problem. When we provide a larger toolkit, students begin to approach problems in a variety of (perfectly valid) ways and gain confidence in their abilities. The more opportunities are given to students to make mathematical decisions, the more they are encouraged to use their tools to explore and reason about mathematical problems. They grow increasingly confident in offering their ideas and methods and are better able to take the initiative when presented with a novel task.

Students should also see the teacher as a problem solver. When a teacher participates in problem solving as a senior learner and persists in solving a problem with students and explore different ways of solving the problem, students learn from that cognitively and affectively. When they see the teacher excited about learning, interested in students’ ideas and thinking, and willing to explore new ways of teaching and learning, even unmotivated learners convert into students who are excited about learning and proud to consider themselves mathematical learners. When teachers are open to finding different ways of solving a problem, they are more likely to create mathematically confident students.

Success and empowerment are the key factors to student motivation. The student who has tasted success is more persistent in solving problems and demonstrates meta-cognition in learning. Successful learners are curious and desire greater challenge. As they build confidence in their own thinking, their intrinsic motivation increases. They are viewed, and view themselves, as developing math experts. They have the confidence to make mathematical decisions, to approach a problem from one direction and, if necessary, change direction and try another. They will use the mathematical tools in their toolkit and decide when and how they might be helpful in a given problem. Therefore, a vital part of our work as math teachers is to encourage our students to become originators of ideas rather than merely recipients of content.

The introduction of modeling into the math classroom across all grade levels will increase student understanding, interest, and appreciation for the power of mathematics tools. Whether engaged in shorter, more focused modeling problems or more extensive, multi-day projects, students gain confidence in their proficiency as mathematical learners and make connections between mathematics concepts and real-life applications. All students, regardless of background or history with mathematics, have the opportunity to contribute and learn through experiences with mathematical modeling.

Characteristics of Effective Problem Solvers
First, to be a problem solver, one has to be mathematically proficient. Mathematically proficient problem solvers can and do apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Students who regularly experience modeling with mathematics as a problem solving tool acquire unique characteristics and think of themselves as mathematicians.

They do not shy away from selecting complex problems and they:

  • simplify a complex problem and identify important quantities to look at relationships and they can represent this problem mathematically,
  • ask what mathematics do I know to describe this situation either with an equation or a diagram and interpret the results of a mathematical situation,
  • look for the mathematics learned to apply to another problem and try to solve the problem by changing the parameters of in the problem.

They regularly ask:

  • What model (quantitative, geometrical, algebraic, statistical, probabilistic, or mixed) could be constructed to represent the problem?
  • What are the ways to represent the information in the problem (e.g., create a diagram, graph, table, equations, etc.)?
  • What tools and approaches are appropriate to the problem at hand?
  • How to select and decide which argument makes sense and is reasonable in the context of the problem?
  • How to justify the appropriateness of the solution, explain why it makes sense, and how to convince the group of the reasonableness of the solution?
  • How to make sure that the results make sense?
  • How to improve/revise the model?
  • How do I incorporate the comments and concerns of others in the approach?
  • What is the best way of presenting the solution to others?
  • What further extensions, generalizations, investigations might be interesting or necessary?

In order to support such learners, the classroom must be a place that encourages choice and provides positive feedback regarding competence.  Teachers in such classrooms:

  • Assure all students that they are capable and competent, and their ideas are worth sharing with others and encourage student collaboration,
  • Presents problems that encourage student initiative and provide the opportunity for a variety of approaches and representation,
  • Make available appropriate manipulatives and instructional materials for exploration,
  • Practice and integrates the three roles: didactic, Socratic, and coaching,
  • Spend less time talking and more time listening to student questions and reasoning,
  • Ask more questions, give measured and focused feedback without curtailing creativity and initiatives, seek suggestions for improving solutions, encourage alternative solution approaches.

In such classrooms, students are more likely to dive into a problem and less likely to ask a question like “What do we do now?” As students succeed, they will be more likely to identify a problem, discuss methods for approaching the problem, and begin investigating and discussing different possible methods of solving the problem. When students are motivated from within—when they are excited about participating in their own learning for learning’s sake rather than because of pressure or external rewards—they become empowered learners.

Example 1  In the early grades, students have concrete, pictorial and arithmetic models available. This might be as simple as writing an addition equation to describe a situation. For example:
My team scored 91 points on Monday and 37 points on Tuesday.
How many more points did they make on Monday than Tuesday?
What is the difference in points on the two days?

This problem could also ask:
How many fewer points did my team score on Tuesday than Monday?
How many more points should my team have made on Tuesday to have the same score as on Monday?

A second grade class worked out the problem as:
One child said: ‘I subtracted 1 from 91 to make it 90 and then subtracted 30 from 90 to get 60 and then subtracted the remaining 6 from 60 to get 54. I know 6 and 4 are pairs to make 10. I took 37 away in all. My answer is 54.’

David was all excited. He said: ‘I have a better method. I subtracted 40 from 91 to get 51 and then added 3 to 51 to make sure that I actually subtracted 37. I subtracted and also added.’

Another child said: ‘I first added 50 to 37 and then 3 and then 1 to get the same answer 54.’ He said: ‘I know 7 and 3 are pairs to make 10.’ He showed the work on the empty number line as follows:

Another child said: ‘I added 50 to 37 to get to 87, but then I just added 4 directly to 87 to 91 as I know that 7 plus 4 is 11, so 87 plus 4 is 91.

Another child said: ‘I did it a little differently. I added 3 to 37 to make it 40. And then I added 1 to 40 to make it 41. And then I added 50 to 41 to get 91. I also got 54.’

Another said: ‘I added 4 to make 41 and then added 50 to get 91.’

Another said: ‘I added 3 to get 40 and then 50 to get 90 and then added 1 to get 91. And the answer is the same 54.’

Then students wrote their equations:  91 − 37 = 54 or 37 + 54 = 91.  They concluded that the team made 54 more points on Monday than Tuesday.

All of these examples show that the students are able to apply their understanding of (a) number concept (as demonstrated in the decomposition/ recomposition of numbers), (b) addition and subtraction facts (as seen in number sense), and (c) place value. The discussion and the recording of different number relationships demonstrate all the standards of mathematics practice.

Example 2  In the middle grades, students have pictorial, arithmetical, algebraic and geometrical methods available. For example, a student might apply proportional reasoning to plan a school event or analyze a problem in the community:
Our school survey revealed that 7 out of 8 students have access to iPads for their homework. If there are 128 students in our grade, I wonder how many students have access to iPads. Is that more than the number of students with iPads in the eighth grade where, according to our English teacher, 91 students have access to iPads?

Students in a seventh grade class demonstrated modeling by constructing the following table and then corresponding equation.

Some students in the class approached the problem as:
# of Students with iPads: ___ ___ ___ ___ ___ ___ ___ (7 equal line segments)
Total # of Students:  ___ ___ ___ ___ ___ ___ ___ ___ (8 equal line segments)
# of Students with iPads in my grade: ___ ___ ___ ___ ___ ___ ___
# of Students in my Grade: ___ ___ ___ ___ ___ ___ ___ ___ = 128
The goal now is to find: “What does one line segment represent?”
Now, 8 (___) = 128,
# of students represented by ___ = 128 ÷ 8 = 16
# of Students in my grade with iPads = 7 × 16 = 112.
Thus, the number of students with iPads in my class = 112. Then they extended it to the 8th grade to compare the corresponding numbers.

Example 3 Observing social and political world events, a group of researchers wanted to know how things go viral on the Internet. In other words, the team wanted to understand how political and social movements, ideas, or products could catch on or fail to do so.

Since two phenomena—disease and social movements use the same word “spread,” the team borrowed from mathematical models used in epidemiology. In biology and medicine, scientists have studied the spread of viruses, disease, and epidemics by using mathematical models. This suggests a possible model for the spread of social and political phenomena.

In medicine, there are susceptibles, infected, and impacted (dead, cured, quarantined); similarly, the researchers reasoned that they could construct a new model to examine the spread of ideas. The team showed that while an individual’s resistance to the spread of a “contagion” might be high, when bombarded by that contagion from many directions, such as happens through Facebook or Twitter, transmission occurs, i.e. you view the activity or participate in it as well. That synergy leads to explosive transmission and we say that something has gone “viral.” This is not only a wonderful example of the use of mathematical modeling to explain a real-world phenomenon but also an example of the generalizability of mathematics and mathematical models. The same mathematics and the same types of mathematical models that can be used to study, for example, the spread of Ebola can be used to study the spread of ideas.

Example 4 Modeling with mathematics means that the students not only understand the concept and procedures but also see that particular method(s) may have limitations and that the context of the problem calls for the applicability and efficiency of model and the method. For example, after students have learned to recognize and use the linear system, they explore real life situations and learn that some situations are modeled by linear systems only under certain conditions. For students to become proficient in modeling, they should have experiences that relate to modeling. Here is an example of a problem (from a textbook commonly used in high schools) a teacher presented to his students to understand modeling:

When a cake is first removed from the oven, its temperature is 370°F. After 3 hours, its temperature is approximately 70° F, the temperature of the kitchen.

  • Does this situation represent a linear system? Why do you think so? If so, represent it as a linear system. If not, why not?
  • Use the information above to write two ordered pairs (x, y), where x represents the time (in hours) since the cake was removed from the oven and y represents the temperature (in degrees Fahrenheit) of the cake at that time.
  • If it is a linear system, write the linear relationship between x and y, in any of the following forms, with general values (two point form; a point and slope form; slope-intercept form; standard form)
  • Find the slope of the line through the two points identified in step 2.
  • Write the linear equation in slope-intercept form or point-slope form.
  • Use the equation from step 5 to estimate the temperature of the cake after 1 hour, after 2-hours, and after 4 hours.

The problem, as given in the book, was straightforward; however, the questions above are reformulated to make sure that the students not only understand the problem but also have a deeper understanding and make connections between different concepts and relate the problem to a realistic situation. After this, the teacher made this problem even more rigorous by asking a series of further questions:

  • Why do you think the information given to you in the problem satisfies the conditions of a linear relationship?
  • Under what conditions can this be modeled by a linear relationship?
  • You know from geometry that two points determine a line, is that condition satisfied here?
  • What does a linear relationship look like in general?
  • What minimum conditions do you need to be able to find the linear relationship in this situation?
  • What is unknown in the linear function you just gave?
  • What is unknown in this equation?
  • When you look at your ordered pairs, will the slope be positive or negative?
  • What will be the orientation of the line?
  • How will you find the slope of this line?
  • What is the formula for slope?
  • Can you find the slope geometrically?
  • You said: “The formula for slope is .”
  • What do m, (y2−y1) and (x2−x1) represent in the formula?
  • What do y2, y1, x2, x1 represent?
  • Will the formula…give the same slope for your line?
  • Why do you think so?
  • Can you prove that the two formulas represent the same slope?
  • Please draw a rough sketch of the line.
  • Based on this sketch, what can you predict about the temperature in the future?

Then the teacher asked his students to solve the problem. Students calculated the slope by considering two points (two ordered pairs): (0, 370) and (3, 70). As the teacher was walking around in the room looking at their work, he asked students:

  • What does -100 mean here?
  • What will be the temperature in five hours? 10 hours?

At this point there was a great deal of discussion amongst students and they began to question whether it was really a linear model. Students came to the conclusion that it was a linear model only till the temperature of the cake reached room temperature, and after that it was not a linear model. The teacher introduced several examples of non-linear and mixed models. Students even brought the idea of a step function.

This is an example of teaching with rigor, making connections, and how mathematics is used to model real-world prolems. The teacher focused only on one problem during the lesson, but students understood the concept at a deeper level rather than solving several problems just applying a procedure.

The requirements of rigor—understanding, fluency, and ability to apply, are parallel to our expectations in reading. A child is a good reader when he or she (a) has acquired fluency in reading (displays speed in decoding, chunking, blending of sounds using efficient strategies indicating phonemic awareness, and word attack), (b) shows comprehension (understands the context, intent, and nuances of meaning in the material read), and (c) is able to use it in real life with confidence (pragmatics—able to read a diversity of materials from different genres and reads for interest and purpose). Mastery in any of these elements alone is not enough because reading is the integration of these skills.

Similarly, rigor in mathematics means a student demonstrates intra- and inter-conceptual understanding, fluency in performing computational procedures and their interrelationships, knowledge of the appropriateness of a particular mathematical conceptual and procedural tool, and ability to apply mathematics concepts and procedures in solving meaningful, real-life problems. Finally, it is demonstrated in their ability to communicate this understanding. To achieve the same level of mastery as in reading, mathematics educators need to balance these elements in expectations, instruction, and assessments.

The writers of the CCSS-M were careful to balance conceptual understanding, procedural skill and fluency, and application at each grade level.

 

Model with Mathematics: Real World to Mathematics and Back

Construct Viable Arguments and Critique the Reasoning of Others: How Do You Know? Prove it!

The third of the Standards of Mathematics Practice (SMP) is mathematicians’ key occupation: construct viable arguments and critique the reasoning of others in a mathematical discourse. They discover, invent, and develop mathematics knowledge by constantly engaging in this process.

Nature of Mathematics Knowing
Mathematics is learned and generated by observing concrete situations and models, identifying and extending patterns, using analogous situations, and applying formal logic and reasoning to new and old situations. Developing formal reasoning provides a stronger base for learning and the development of mathematical ideas. In two previous Standards of Mathematics Practice (SMP), the emphasis was on understanding the problem—the language and concepts involved in the problem and then taking the specific concept to a general situation.

Developing reasoning, supporting one’s argument, critiquing another’s approach should not be reserved for high school geometry or advanced calculus; they should be part of all mathematics learning from Kindergarten on. Kindergarteners and first graders should be as familiar as high achieving high school students with the appropriate language (vocabulary, syntax, and mathematics sentence structure) and the development and practice of reasoning and logic (deductive and inductive; direct and indirect) such as: “prove it” “how did you know?” “how did you find out?” “defend your answer” “how can you be sure?” They should know answers to these questions and many others such as: “What definition or result did you use in this approach?” “What is wrong with this answer?” “Do you agree with …?” “why do you agree with … reasoning?” “What conclusions can you make from this?” “Is this a correct inference?” “Do you agree with that person’s reasoning?” “Why?” “Why not?” Development of and insistence on providing reasoning for their statements is not to make mathematics difficult; it is to understand mathematics better, deeper, and with understanding. Such mathematical thinking offers students the choice whether they want to be generators of mathematics knowledge or its users.

The origin of reasoning is intuition. When children’s intuitive answers are encouraged, they feel confident and are ready for formal reasoning. Mathematics is about removing obstacles to intuition and keeping simple things simple. Doing good mathematics is the interplay between intuition and reasoning—making things simple.

Viable Arguments and Critique of Others’ Reasoning
Mathematically proficient students understand and use stated assumptions, definitions, derived formulas, proven theorems, and established results in constructing arguments in the process of forming equations, relationships, and representations.

They can give examples for terms and definitions. They make conjectures and build a logical progression of statements to explore the truth of their conjectures and ideas.

They are able to analyze situations by breaking them into cases and recognize and use counter examples. They justify their conclusions, communicate them to others using mathematical language, and respond to questions and the arguments of others using appropriate reasoning.

Mnemonic Devices and Mathematical Reasoning
One hallmark of mathematical understanding is the ability to justify, in ways appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, where a mathematical rule comes from, and how and when that can be applied.

There is real difference between students who can give the sum 8 + 6 as 14 by counting up or by rote memorization and the difference 17 – 9 by counting down or rote memorization and those who can find the sum by using strategies: decomposition/ recomposition of numbers, making ten, and knowing teens numbers. They see sum 8 + 6 as the outcome of strategies: 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14, or 4 + 4 + 6 = 4 + 10 = 14, or 2 + 6 + 6 = 2 + 12 = 14, or 8 + 8 – 2 = 16 – 2 = 14, or 7 + 1 + 6 = 7 + 7 = 14. They develop mastery (understanding, fluency and applicability) and develop efficient procedures.

There is a world of difference between a student who can summon a mnemonic device (DMSB = Does My Sister Bite or Dead Mice Smell Bad or Does McDonald Sell Burgers) to conduct the long division procedure: divide, multiply, subtract, and bring down and the student who knows why particular digits in the quotient are in a certain place or what will be the probable size (estimate in the correct order of magnitude) of the quotient before and when he completes the division procedure. Learning and applying procedures by just memorizing mnemonic is not mathematics.

Similarly, using the mnemonic device PEMDAS (= Please Excuse My Dear Aunt Sally to implement the order of operations: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction) is purely procedural and shows a lack of understanding. It is important to know the reasons behind this order of operations (for details see the post on Order of Operations).

  • Addition and subtraction are one-dimensional operations (linear—for example joining two Cuisenaire rods or skip counting on a number line); they are at the same and the lowest level of operations. If both operations appear in the same expression, they are executed in order of appearance, first come first serve (Capture4);
  • Multiplication and division are two-dimensional operations (as represented by an array or the area of a rectangle), therefore, are at a higher level than addition and subtraction, they must be performed before addition and subtraction and if they both appear in an expression should be treated as first come first serve (Capture5);
  • If all the four operations: addition, subtraction, multiplication, and division appear in a mathematical expression, the order should be: Two-dimensional operations first and then the one-dimensional operation in order of their appearance (Capture7).
  • Exponential expressions are multi-dimensional (depending on the size of the exponent, e.g., a 10-cube = 103 is a 3-dimensional expression with an exponent of 3 and a base of 10; therefore, exponentiation operation is more important than multiplication (and division) and definitely higher than addition and subtraction, therefore, must be performed before all of them. Therefore, the order of operations so far is: (Capture9);
  • Grouping operations are expressions included in groups such as brackets, braces, parentheses either transparent and/or hidden (compound expressions in the numerator and denominator of a fraction, function and radical operations are hidden operations. They may involve some or all of the above operations in multiple forms, therefore, are or higher preference than all of the above operations. In transparent grouping operations, the order is parentheses, braces, and brackets. The hidden grouping operations are performed in the context. Inside a grouping operation, the same order as in the above operations is kept.

Hidden operations such as: fraction and radical operations need to be brought to students’ attention. For example, the fraction expression has hidden groupings as the numerator and denominator involve extra operations, even though there is no transparent grouping operation. In order to simplify the fractionCapture16 read it as: (3 + 5) ÷ (3 -1). Therefore, before we simplify the fraction (performing the division operation), we simplify the hidden operations in the numerator and the denominator. Similarly, function and radical operations are hidden: e.g., f(a) = 3a where a = Capture1and x=2).

Therefore, the grouping operations (parentheses, braces, and brackets in this order) are performed first. The hidden grouping is contextual. Then exponential operations need to be performed. After that multiplication and division are in order of their appearance. The last operations to be performed are addition and subtraction in order of their appearance. Therefore, the grouping operations should be performed before all of the other operations. The order of operations, therefore, should be: (Capture2). Here, G represents grouping operations—transparent, hidden, and both.

The mnemonic devices are important for remembering the sequence of activities in a multiple step procedure or operation; however, the use of acronyms and memory reminders should be only after students have understood the concepts and procedures and the reason for a particular order of operations. They do not take the place of conceptual understanding and derivation of procedures.

Similarly, there is a difference between a high school student who uses the mnemonic (FOIL) to expand a product such as (a + b)(x + y)= ab +ay +bx +by and a student who can explain where the mnemonic comes from (application of the distributive property of multiplication over addition, applied twice: (a + b)(x + y)= (a + b)x +(a + b)y = ax + bx + ay +by. The student who can explain the rule understands the mathematics and can use the mnemonic device productively as he may succeed at a less familiar task such as expanding (a + b + c)(x + y +z) or (2x + 3y)(-2x2+6xy -5y2).

Another practice that does not develop mathematical reasoning in students is the emphasis and introduction of procedures before the appropriate conceptual schemas are developed. It is important to develop the language containers and the conceptual understanding before a procedure is introduced. Fluency of a procedure or skill without conceptual strategies robs students of applying mathematics with understanding and reasoning. It is, therefore, important to assess them both for understanding before students are asked to apply them. Both conceptual and procedural understanding can be assessed by teachers by using mathematical tasks of sufficient richness and constantly asking the question: how do you know it?

The student should first have the conceptual understanding and then use it to acquire the procedure and only then should mnemonic devices be introduced to remember and automatize the steps.

When mnemonic devices and algorithms/procedures are introduced before conceptual understanding and the development of language containers (vocabulary, terminology, language expressions), students do not show interest in conceptual understanding and apply these without knowing the reasons behind them.

When students are given mnemonic devices before they understand the concept and procedure and the related reasoning, it may be difficult for them to apply the concept, defend their work and reasoning, and communicate their results and understanding. The classrooms where use of these mnemonic devices as a proxy for mathematics is paramount, real interest and passion for mathematics are absent and difficult to achieve.

Deductive and Inductive Reasoning
Many in the general public and non-mathematicians and even some teachers have the misconception that mathematics is a collection of sequential procedures, and the only justification for their actions is the sequence of steps and best case the use of deductive logic. It is true that the foundations of mathematics including arithmetic are established by formal deductive logic. However, in learning school mathematics and even in some higher mathematics, there is an interplay of deductive and inductive logic. In inductive logic, one moves from many specific examples to a pattern, that helps develop conjectures and then we arrive at a general principle—theorem, formula, and procedure. It is a right hemispheric activity—looking for patterns.

Deductive logic, on the other hand, starts from the general principle—formula, theorem, definition, etc. and proceeds to its application to specific situation. It is a left-hemispheric activity—engaging in sequential reasoning. Mathematics reasoning is the interaction of these opposite but complementary activities; it is similar to the corpus callosum integrating the flow of these activities from one side of the brain to the other. In that sense mathematics reasoning is a whole brain activity—integration of thinking originating from kinesthetic to linguistic to spatial orientation/spatial organization to inter- and intra-personal to logico-mathematical intelligence. The integration of inductive and deductive reasoning spans from seeing a concept geometrically/spatially to following step-by-step procedures using sequential procedural logic.

Mathematically proficient students are able to reason deductively and inductively about data, concept, and procedures, making plausible arguments that take into account the context from which the data arose and understanding the nature and quality of the concepts and procedures.

Applying Mathematical Reasoning as Communication
Just like any communication, mathematics communication has two parts: expressing one’s ideas succinctly with reason and understanding others’ ideas correctly. It is defending one’s ideas but also understanding others’ ideas and identifying strengths and finding fallacies in arguments from both sides. It is not enough to defend one’s argument, it is equally important to:
(a) understand and identify others’ reasoning and its validity and effectiveness
(b) recognize the fallacies in one’s own and other’s reasoning and arguments, and
(c) correct the fallacies in one’s own and others’ arguments and approaches in a mathematics context, e.g., problem solving.

This means students are able to compare the soundness, effectiveness, and efficiency of the two (or many) plausible arguments and approaches, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is and how to fix it.

Mathematical reasoning is developmental and contextual. Children are capable of developing reasoning according to their age and mathematics concepts. For example, elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. The concrete materials they use to show their reasoning should be (a) effective, (b) efficient, and (c) elegant. Such concrete arguments should make sense and be correct even though they may not be generalizable or made formal until later grades. For example, even Kindergarten students can easily see the commutative property of addition using Cuisenaire rods and then generalized to numbers and even variables.

Capture3

Later, students learn to determine domains to which an argument/reasoning based on language, diagrams or formal logic applies. At the same time, students, in all grades, can observe, listen or read the arguments of others, decide whether they make sense, and ask questions and add to clarify or improve the arguments.

Developing Mathematics Reasoning
Mathematical reasoning develops when we provide students experiences that help them acquire the component skills of such reasoning. Teachers’ questions aid the development of mathematical reasoning:
(a) What strategy can be used to find the sum 9 + 7?

  • I can count 7 after 9.
  • Can you give more efficient strategy?
  • I can use blue and black Cuisenaire rods.
  • Can you give a strategy without concrete materials?
  • I can use Empty Number Line.
  • Can you give any of the addition strategies?
  • Making ten: (9 + 1 + 6 = 10 + 6 = 16)
  • Making ten: (6 + 3 + 7 = 6 + 10 = 16)
  • Using doubles: (2 + 7 + 7 = 2 + 14 = 16)
  • Using doubles: (9 + 9 – 2 = 18 – 2 = 16)
  • Using missing double: ( 8 + 1 + 7 = 8 + 8 = 16)

(b) What strategy can be used to find the difference 17 – 9?

  • Using teens number: 17 – 9 = 7 + 10 – 9 = 7 + 1 = 8
  • Using making ten: 17 – 9 =10 + 7 – 7 – 2 = 10 – 2 = 8
  • Using doubles’ strategy 17 – 9 =18 – 1 – 9 = 18 – 10 = 8
  • What to add to 9 to get to 17: 9 + 1 + 7 = 17 = 9 + 8 = 17, so 17 – 9 = 8.

(c) What is the nature of the figure formed by joining the consecutive mid points of a quadrilateral?

  • To get a sense of the outcome of this construction, I will first consider a special case of quadrilateral: a square or a rectangle.
  • What does the constructed figure look like in such a special case?
  • What if the quadrilateral is concave? Is this assumption correct? What is your answer in this case? Why?
  • What if it is convex quadrilateral? What is your answer in this case?
  • Is it true in both cases?
  • Is it true for any quadrilateral?
  • Can you prove it by geometrical approach?
  • Can you prove it by algebraic approach?

(d) How many prime numbers are even?
What is the definition of prime numbers?
How many factors does an even number have?

  • 2 has 2 factors, namely, 1 and 2
  • 4 has 3 factors, namely, 1, 2, and 4.
  • 6 has 4 factors, namely, 1, 2, 3, and 6.
  • All even numbers, except 2 have more than 3 factors.

What conjecture can you form?

For upper grades:
Can you predict the nature of any even number?
Can you prove that ___ is the only even prime number?
Is a square number a prime number?
Why is a square number not a prime number?
If n is a prime number, what can you say about n + 1?

(e) Is the product of two irrational numbers always an irrational number?

  • What is the definition of an irrational number?
  • Is every number an irrational number?
  • Why? Can you prove it?
  • If not, why?

Can you give a counter example to justify your answer?

(f) Will the range of the data change if every piece of data is increased by 5 points?
David says: It will increase by 5. Is he right?
Why? Can you prove it?
Melanie says: It will not change. Is she right?
If not, why? Can you prove it?
Can you give a counter example to justify your answer?

(g) What other central tendencies are affected by such a change? Why? Explain.
One of your classmates just stated: Such a change will not change the median of the data, is this true? Why?

When students are given opportunities to make conjectures and build a logical progression of statements to explore the truth of their conjectures, they learn the role of reasoning and constructing arguments. The teacher should constantly ask questions such as: “How did you get it?” “What did you do to get this?” “Can you explain your work?” When teachers ask children to explain their approach to finding solutions and the reasons for selecting the particular approach, children develop the ability to communicate their understanding of concepts and procedures and the ability to trust their thinking. Some questions are applicable to all grade levels:

  • What mathematical evidence would support your assumption/ approach/strategy/solution?
  • How can we be sure of that ….?
  • How could you prove that …?
  • Will it work if …?

However, some questions should be at grade level. For example, at the high school level the questions can be more content specific.

Question: Your classmate claims that the quadratic equation: 2x2 + x + 5 = 0, has no real solutions. This can be followed by questions such as:

  • What is a solution to an equation?
  • What is a real solution?
  • Do you agree with this claim?
  • Why? Why not?
  • What information in the equation assures you that it does not have any real solutions?
  • How did you determine that this does not have a real solution?
  • Can you change the constants in this equation so that it will have two real solutions?
  • Only one real solution.

Teachers should analyze general situations by breaking them into special cases and ask students to recognize, use and supply examples, counter examples, and non-examples. This can be exemplified by questions such as:

  • What were you considering when …?
  • Why isn’t every fraction a rational number?
  • Is every rational number a fraction?
  • Is every fraction a ratio?
  • Is every ratio a fraction?

To help children how to learn to justify their conclusions, communicate them to others, and respond to the arguments of others, teachers can ask questions such as:

  • How did you decide to try that strategy?
  • Do you agree with David’s statement? “Between two rational numbers, there is always a rational number.”
  • Why do you agree?
  • How will you find it?
  • Why don’t you agree?
  • Do you have a counter example?
  • Is this statement true for all real numbers?
  • Why?
  • Is every square a rectangle?
  • Why?

Analogies and Metaphors as Aids to Mathematical Reasoning
Students’ reading comprehension is improved when their thinking involves the understanding of analogy, metaphor, and simile. Similarly, the use of analogies and metaphors is an example of reasoning in mathematics, particularly in the initial stages of learning a concept. Students need to learn to reason by using analogies and reason inductively about data, making plausible arguments that take into account the context from which the data arose. For that teachers need to follow with questions such as:

  • What is different and what is same about this problem and the other you solved before?
  • Did you try the method of the previous problem?
  • Did it work?
  • If it did not work, how did you know it did not work?
  • Why did it not work?
  • Could it work with some changes in your approach? Why or why not? What changes would you make?
  • How did you decide to test whether your approach worked?

One of the important aspects of thinking children need to develop is to know the conditions under which a particular definition, formula, or procedure applies and the parameters of its limitations. To develop this ability, teachers could ask questions about the content under discussion. For example:

  • Is 3,468 divisible by 4?
  • Yes or no?
  • Why? Justify your answer without actually dividing the number by 4.
  • Is this number divisible by 12? Why? Justify your answer without actually dividing the number by 4.
  • In a fraction Capture blog 6.02, if a = b, and b ≠ 0, then the fraction is equal to 1.

Do you agree wit this statement? If so, can you prove it? If not, can you give or construct a counter example to this situation?

Students need focused training and support in comparing the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument, explain what it is. For this a teacher may articulate questions that focus on:

  • How to differentiate between inefficient and efficient lines of reasoning?
  • How to focus and listen to the arguments of others and ask questions to determine if the reasoning and the direction of the argument make sense?

Finally, teachers should ask clarifying questions or suggest ideas to improve/revise student arguments.

All skills, from cognitive to affective to psychomotoric, can be improved by efficient and constant practice. In classrooms where expectations of high levels of rigor are standard, students develop proper mathematics reasoning and are keen to identify others’ reasoning and critique it.

 

Construct Viable Arguments and Critique the Reasoning of Others: How Do You Know? Prove it!