# MATHEMATICS – Construct Viable Arguments & Critique Reasoning of Others

by Mrs. Jessican Mangini

This month, I will continue discussing the Mathematical Practice Standards which are present in every grade and work in conjunction with the Connecticut Core Content Standards. These standards determine how students will demonstrate their mathematical knowledge.

To refresh, here are the 8 Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

This month I will highlight Standard 3- Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They justify their conclusions, communicate them to others, and respond to the arguments of others. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Source: Inside Mathematics

In my opinion, this practice standard embodies one of the most important things in the entire Common Core.

First, the practices of constructing arguments and critiquing reasoning represent a cross-disciplinary approach and are as relevant in mathematics as they are in English language arts, science, and social studies. This standard helps break down and support the understanding that communication, justification, and critique are essential components of learning.

This practice is easy to implement and embrace because it represents a shift away from the multiple-choice, skill-based, rule-based focus of years past. Test-prep memorization and regurgitation have little to do with thinking and reasoning and cheat students out of the opportunities consistently found in highly effective schools.

But the primary reason this practice is so important is its implications for teaching. In order for students to construct viable arguments, they must be asked why or be expected to explain their thinking or be directed to convince the class. In a classroom where the shared expectation is that all answers need to be justified, students consistently construct arguments. They communicate their understanding and demonstrate their thinking process. No longer is it acceptable to stop at “83” or “9 square centimeters.” Instead, they individually and publicly justify why the answer is 83 or 9, thereby giving the teacher and the entire class an opportunity to discuss alternative explanations. (source: Inside Mathematics)

But the primary reason this practice is so important is its implications for teaching. In order for students to construct viable arguments, they must be asked why or be expected to explain their thinking or be directed to convince the class. In a classroom where the shared expectation is that all answers need to be justified, students consistently construct arguments. They communicate their understanding and demonstrate their thinking process. No longer is it acceptable to stop at “83” or “9 square centimeters.” Instead, they individually and publicly justify why the answer is 83 or 9, thereby giving the teacher and the entire class an opportunity to discuss alternative explanations.

Similarly, in order for students to critique the reasoning of others, they must be in a classroom where reasoning is made public and open to review and comment. A classroom culture that values critique rather than the one right way to get the one right answer is a culture in which students are far more actively engaged in their learning. When you walk into a math classroom at Tashua, you will clearly see this approach used!!

Again, in place of a single approach or justification, many approaches and justifications surface, thereby strengthening everyone’s learning. In addition, given the learning opportunities of mistakes and misconceptions; critiquing others’ reasoning helps address common mistakes and misconceptions, also to everyone’s benefit. It is a joy to observe a class focus on the mathematical reasoning of three different groups of students and then discuss approaches: what was the same, what was different, what was correct, what was flawed. What better preparation for life and citizenship could we hope for?

Source: Steve Leinwand (@steve_leinwand),

American Institutes for Research, author of Accessible Mathematics.

In this standard, students think: “I can explain my thinking and consider the respond to the mathematical thinking of others.”

When helping your child complete homework here are some great questions you ask them to tap into this mathematical practice:

1) Why did you use (a graph) to solve it?

2) How did you get (that equation)?

3) How can you prove that your answer is correct?

4) What math language will help you prove your answer?

5) What example can prove or disprove your argument?

*December 2015 Tiger Times*