By Clayton Edwards, Posted November 10, 2014 – 

Mathematical understanding is paramount in ensuring that students actually achieve the standards being presented. Any student can nod his or her head, give a thumbs up, and/or write down a few correct numerical answers to indicate understanding. Chances are, within all three scenarios, a majority of the class is not understanding the content.

I have had the best math students tell me that they understand something, and then when pushed, I find that they, in fact, do not understand. Students may convey that they understand even when they do not because they are embarrassed to admit their lack of knowledge, or they truly do think they understand. If students can present a clear and concise argument for their answers to peers or teachers, odds are they understand. The process of developing mathematical understanding isn’t easy to implement in the classroom, but the results far outweigh the struggle. Here are 3 ways to get started.

1. Model Understanding

When I first introduce the idea of mathematical understanding to my sixth graders, they immediately think of showing their work, which is not exactly what I mean. When I show them an example of a task or assignment, jaws drop around the room. What they are about to encounter has never entered their realm of possibility. 

One way we practice writing in mathematics class, for example, is through a whole-class-assigned question. Two times a week, we work on a question in which students are expected to show their understanding of the solution in writing and then explain it to three people. I give them options of how to show their understanding. These include, but are not limited to, a paragraph, labeled diagrams, and explaining the significance of numerical answers with labels. The progression usually goes from writing down numerical answers and slowly moves to the final product after weeks of work. Students then begin to understand how much I am expecting them to explain. Asking, “Why did you use a division problem?” is something most students have never thought about. They just “know” to use a division problem, but I ask that they be able to state why. After many examples and nonexamples, this idea is accomplished. 

We use this same process to model mathematical conversations. During the whole-class questions, students are expected to discuss explanations and findings with three other students. As these conversations occur, I stop the entire class, talk about positives and negatives that I see, and ask the class what could happen differently. One common problem with these conversations is that the students on the receiving end will nod their heads as if they understand when they do not. We discuss how these two-way conversations should involve everyone. If something is not understood, clarifying questions must be asked. The person who is listening to an explanation needs to leave the conversation understanding the presenter’s point of view, and the presenter needs to leave the conversation knowing if what he or she said makes sense. 

2. Give Students Opportunities to Write and Talk

Students will not instinctively start writing and talking about math if the opportunity is not given. The more opportunities presented, the more this practice will become the norm. The examples below can be implemented in your classroom to start the process.

A. Assign Fewer Problems

Students can turn in correct answers to a page of 100 problems, and the teacher may still have no idea if they really know what they are doing. Instead of 100, choose two problems and ask students to show their understanding in multiple ways. After the written part is finished, each student must find two other students to discuss their findings. At the end of the class period, the teacher picks a random student to explain one method of understanding. The simplest problem can be turned into a good discussion if that is consistently expected.

Here is a simple proportion problem:

Six students have 18 pencils. If the proportionality holds true, how many pencils would 14 students have?

Although it is not a very enticing question, you can still drive home the mathematical understanding.

B. Provide Problem-Solving Tasks

This is an idea I have used since discovering Dan Meyer’s blog. I have taught every one of his tasks in my classroom. My students discuss problems with one another in groups, and they are each expected to write a detailed synopsis of the solution, how they know they are correct, and everything it took to get there. I would start with his resource first. You can always make changes or add/subtract questions and information to your liking. Here is an example of one of the tasks and what is expected.

http://threeacts.mrmeyer.com/leakyfaucet/

C. Use Whole-Class Questions

I already described the process for using a whole-class question above, but here is a video example to depict more of the writing and the discussing.

D. Implement Estimation 180 Questions

This website created by Andrew Stadel shows a picture in which students have to estimate something about an everyday object or person. We often use one of these images at the conclusion of class with the same goal: Show your understanding (written) and explain your thinking (verbal). Not everyone is always correct, but that is part of the process. If you can explain your thinking (even if you are wrong), fixing the misconception is a lot easier. Here is an example with student work

http://www.estimation180.com/day-7.html

These are only a few examples of how to get students writing and talking. It is hoped the message is clear that this should happen with every activity in your class. Mathematical understanding must become commonplace and not just for special occasions. 

Come back to Blogarithm in two weeks for the finale of this post. Additional topics will include the amount of time necessary for your students to have success thinking, writing, and speaking mathematically; the connection these justification skills have with the CommonCore Standards for Mathematical Practice; and some closing thoughts drawn from my own experiences in the classroom.

How to Get Started (Written and Verbal Mathematical Understanding: Part 1)

Comment or question? Join the discussion by responding below.

  Clayton Edwards, @doctor_math and cedwards@spartanpride.net, is a middle level mathematics instructor at Grundy Center Middle School in Grundy Center, Iowa. He is interested in the mathematical learning of all students of varying ability levels through self-pacing, task-based instruction, and other methods.