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 ();
- 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 ();
- 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 ().
- 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: ();
- 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 fraction 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 = and 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: (). 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.
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?
- Is every square a rectangle?
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 , 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.