“students should see and expect that mathematics makes sense.”
Reasoning and Proof should be fundamental in teaching mathematics from prekindergarten through 12th grade. But, what is it? And is it taught throughout grade levels?
What is Reasoning and Proof? And do teachers use it?
It is my understanding, that Reasoning in mathematics is understanding that everything in math should make sense; and, Proof is the reasoning that math follows specific assumptions and rules that students can see and show through evidence. Throughout their school career (K – 12), mathematics teachers should have students make conjectures (an informed guess), ask questions, work with problems, and be able to explain their reasoning with evidence. Within my elementary school experience, the use of reasoning and proof when teaching math is extremely limited. I have experienced teachers showing students how to work a specific type of math problem, giving students practice, and then assessing students. I have heard many teachers tell their students to go back and check their work, but have not heard many teachers ask their students if the problem worked, why or why not, and show me how you know.
Important Main Ideas with Reasoning and Proof
Throughout my readings on Reasoning and Proof, the authors stressed the importance of presenting students with challenging problems and encouraging them to ask questions of themselves and their peers. My understanding is that the communication process that follows leads to deeper understanding. By asking “why” teachers ignite students’ natural curiosity. Students must reach beyond the mechanics of a problem and begin to think for themselves. But teachers must ask the “right” questions such as “Will this work with different numbers?” “What do you notice?” “Do you see a pattern?” “Is it always true?” “Is there another way to show this?”
To establish a life-long habit of questioning and explaining math concepts, teachers must encourage students to investigate using concrete materials such manipulatives and calculators at younger grades and mathematical representations and symbols in the older grades. In lower elementary grades, students learn that even and odd numbers alternate and that “odd” numbers have one left over. They can use counting blocks or drawings to represent this information. Likewise in upper high school grades, students can use a scatter plot to represent what happens to a correlation coefficient under linear transformation of the variables.
Students should be able to explain their proofs. Elementary students are not expected to come up with elaborate or formal “proofs;” but, we should teach them to come up with convincing arguments to justify their ideas and solutions. In high school mathematics however, students should be able to “present mathematical arguments in written forms that would be acceptable to professional mathematicians.” If students can explain and justify their reasoning behind the answers then their understanding is much greater than if they can simply answer a problem correctly.
Further, as teachers, it is our responsibility to give students specific and appropriate feedback as they use reasoning and proof to help them develop this critical thinking process. And, as stated in Yopp’s article, we must identify and value arguments that explain why a statement is true and not just give examples and counterexamples. He noted that logical necessity should not be taught in elementary mathematics, but teachers who understand logical necessity can better help students hone their reasoning skills and learn to develop formal reasoning.
As pre-service teachers, we are cautioned to develop a reasoning and proof “habit of mind” as we learn to teach mathematics. Meaning that we should consistently incorporate this type of questioning and explanation into our daily math lessons. Teaching students to multiply is important, but teaching them to understand why multiplication works and how to logically examine this operation is even more important.
As future mathematics teachers, our goal should be to have students consistently question, investigate, hypothesize, investigate further, argue using logic, and then show evidence based on mathematically accepted concepts and truths.
How can Reasoning and Proof be Emphasized in Math Tasks?
Example 1: Divisibility Rules – With this example, a group of pre-service teachers are studying divisibility rules and conjecture that if a whole number is divisible by two and divisible by three then it must be divisible by six. This is indeed correct, but another student further “surmises that if a number is divisible by two distinct numbers, then the number is also divisible by the product of those numbers.” The student supports this conjecture by using examples instead of using variables to test to make a more general case. This statement is not true and shows the “limitations in generalizing from inductive reasoning and empirical evidence alone.” This example is used to emphasize the importance of teachers using “logical necessity” when teaching mathematics to students. Logical necessity is defined as “the condition for which conclusions follow necessarily from premises. The concept is central to deductive reasoning because it allows us to say that one mathematical statement or property implies another.”
Example 2: 4th Grade Sum of Doubled Numbers and Product of Squared Numbers – With this example, the teacher asked the class to take the number 4 and add it to itself and then take the same number and multiply it by itself. Then she asks if there is a relationship between these two operations. As students begin to brainstorm on this relationship they suggest that other numbers be examined also. The teacher further asks students what have they done in the past as they have investigated numbers. A student answers that they have used models to help them understand. Then, the teacher recommends that students use graph paper to make a model and “show” their results. The teacher models how she wants students to begin their investigation and instructs them to have a recorder write down anything that they can “find out about this number.” Students found that as numbers were doubled the sum was always an even number and that as numbers were multiplied by themselves (squared) the product followed a pattern of even, odd answers. Further they identified that as numbers were multiplied by themselves, the graph was a square that increased one row and one column larger as the multiple increased. The teacher noted how important it is to emphasize communication. Further, she noted the importance for the students to communicate with each other because their discoveries are so much more meaningful.
Reasoning and Proof in the Common Core Standards
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
This is a general standard established for all grade levels. It requires that a “mathematically proficient” students should be able to make conjectures and explore the truth of those conjectures. They should be able to analyze situations and use counterexamples. Students should be able to construct arguments that make sense, and listen to arguments of other students and determine whether those arguments make sense using questions to clarify and improve.
Within the Common Core Standards of every elementary grade level overview, the following general statement is made …. Mathematical Practice (3) Construct viable arguments and critique the reasoning of others. Otherwise Reasoning and Proof is not at the forefront of math teaching requirements.
Reasoning and proof does not seem to be at the forefront of all math teaching standards as suggested in the literature. Following are the few specifics inwhich the standards direct the use of “explain” or “reasoning”:
Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.