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.

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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