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.

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.