Communicating Mathematical Ideas

If someone were to ask me the which college class I have learned the most from, I would have to say my answer would be MTH 210 Communicating in Mathematics. The course has immensely helped me understand the importance of clarity and the difficulty that comes with trying to communicate your thought process and ideas. As a future teacher, I must recognize that no matter what career path my students choose to pursue, the ability to communicate their thought process and ideas will be a vital skill. Thus, if one of the purposes of education is to prepare students for their life after school, I feel as if it necessary to foster this skill in the classroom.

In my experiences as a Math Center Tutor at GVSU, I have found that the questions I most enjoy helping students with are what many professors are calling a journal entry. Essentially, the student is given a prompt where there is not enough provided information to compute a numerical value and they are asked to find an arbitrary solution and explain the how and the why of it. For example, a journal entry prompt I have seen is something to the effect of "If a > 0, for what values of b will
ax^2 + bx + c have at least one real solution?" What makes these questions great, in my opinion, is that it asks students to apply what they have learned in a conceptual sense and write about it.

Yes, math and writing can be integrated!

Often times, when I am helping a student with this type of problem in the tutoring center, what the student needs help with is not arriving at the solution, but rather communicating their computational work. My advice to them for how to proceed is to write down their thinking process in way that someone in their class who does not understand the concept quite yet could read it and comprehend it. Ron Swanson does a terrific job of depicting the facial expression I receive from students after saying this.

Ok, but...um...how do I do that exactly?

Sometimes the reaction stems from viewing the task as "stupid" or a lot of work, but I have a few students genuinely wonder exactly how they can put what in their mind on paper as they either believe the computational work constitutes a translation of the thinking process or they have difficulties with the translation from mind to paper. In response, I say that sentences in their entry can be formed by saying that "(insert known information) tells me that (insert conclusion)" or "since we want to (insert objective), we know that (insert way to show it)." For example, related to the sample prompt above, a sentence in the journal entry would be "Since we want the polynomial to have at least once real solution, we know that the determinant must be greater than or equal to zero; thus, b^2 - 4ac >= 0." This is definitely a challenge for some students, so in my future classroom, I plan to foster communication skills through a variation of the concept of journal entries: student blogging. For more on this idea, see this previous post of mine. Just like journal entries, student blogging has students explaining the how and the why, thus developing their written communication skills. Below is one of the examples in the linked post.

However, communication of thoughts and ideas is not only written, but also verbal. To help foster verbal communication, not only do I plan to infuse group work into my lesson plans, but I am considering to incorporate what the teacher I am currently observing calls "board problems." What she does is require that once a marking period (quarters at this school), each student goes up to the whiteboard and explains the how and the why of a "challenge" problem in the homework assignment.

A written computational solution is written on the board,
but a verbal explanation is required as well.

Students have the choice of deciding when in the marking period they can present their board problem. The benefit in doing it this way is that it does not harm the student if the board problem was to be preassigned and the concept did not go over well. I realize that giving student's choice is important, however, I am not quite sure that if I use this in my classroom I will let the student decide when they present. I want students to realize they can, in fact, learn from their mistakes, learn more from wrong answers than right answers, and become unafraid to share their work and thought process. Any suggestions on which way to go are definitely appreciated!

The benefits of incorporating journal entries/student blogging and board problems in the classroom goes beyond the development of communication skills. In using these, the teacher has multiple types of formative assessments that allows he/she to get a deeper and more accurate look into the student's understanding of the content, both computationally and, more importantly, conceptually. So with this win-win situation, other than the fact that it might be more work on the teacher's end, I cannot find a reason to not include these in my future classroom.

For more ways to incorporate discourse and writing in a math classroom, take a look at this presentation on the topic from a couple Michigan math teachers.