Mathematical Thinking vs. the Right Answer

One of the things that I try to impress upon my students is that being “good” at math is really about the thinking skills you use to solve problems, versus solving problems just to get a right answer.  Of course, getting “correct” answers is a nice side effect of good thinking skills, but I believe that coming up with solutions (right or “wrong”) that are based on good thinking skills is the best side effect.

October was an exciting and interesting month because of the type of work that my seniors were beginning to do in Pre-Calculus. Chapter 1 covers a lot of different material and the knowledge differential of my students is ALL OVER THE PLACE (which is normal).  But the cool thing was that my students were getting used to trying to solve problems for the sake of knowledge, and not for the correct answer.  I noticed that there was an increase in the type of clarifying questions my students would ask.  They started to say, “Are we just using thinking skills for this activity?” at the beginning of the problem and I LOVED that.  Also, my students were becoming more comfortable getting up in front of each other to give their opinion of WHY their answers made sense.  Because they are my high school students, they still tended to “joke around” a lot, but they did NOT ridicule each other for having different answers. Instead, they tried to win their classmates over to their view of the solution, even when they had the “wrong” answer.  

When I compare the type of learning going on in my Pre-Calculus class to the type of learning in my Algebra 2 class, the difference is HUGE.  My Algebra 2 students are still looking for the right answer, and are not used to working at all; such behavior may be pandemic related, it may just be a “class personality” issue, or it may be because I had the Pre-Calculus students last year for Algebra (this is the first year that I have my Algebra 2 students).  As a matter of fact, because all Detroit Public Schools Community District students were virtual in 2020-21, this group of students did not get to know me in person. Our school is so small (under 250 total students) that the teachers and students all get to know each other when the students are in the 9th grade, even if they are not in our classes yet. This was very disconcerting for me, because I am “that teacher” – the one that the students are excited to get, even though I am “hard” (at least that is what the kids tell me). Therefore, my reputation precedes me, and the students in my school know that I care, even though I will work them hard. Anyway… I digress.

Fast forward to November and December and my Pre-Calculus students have truly begun to embrace the thinking process!  The majority of them are starting to work for the sake of knowledge, and I know this because when we finish activities and share out, they now say something to the effect of, “What I saw was this,” or “This is the answer that I worked out because of…,” or even statements like, “Ms. Scott, listen to my explanation, then listen to Josh’s explanation. Which one makes more sense to you?” They even realize that I am not the absolute authority in math now. Before they would have said, “Who has the right answer, me or Josh?”, or “Just tell us the right answer.”

Now it is January, and we have had 3 weeks of virtual learning.  What I have noticed is that my Pre-Calculus class has great attendance – 85-95%, while my Algebra 2 class has about 50% attendance.  Sometimes the Pre-Calculus students show up late, but they show up AND they work.  They still go at each other for their differences in thinking.

I have one student who has really made some changes. He frequently comes to school tired because he started working at McDonald’s in November, and if he’s not making hamburgers, he’s at the basketball court. But he has showed up to class every day ready to participate. He speaks his mind and gets to work!!!! When he speaks up and uses his thinking skills, the class listens, and I know that makes him feel as though he adds value to our discussion. I attribute these changes to a class focus on mathematical thinking skills. He is exemplary, but all of my students are starting to think mathematically on a regular basis.

During this virtual period, my students have worked with me to have a class where learning takes place and where respect for the thinking of all students is the rule. Even as we review for their chapter test (I’m not giving a final exam!), they are paying attention during group Desmos time and working with each other to gain a deeper understanding of the math.  I am excited for what next semester will bring (at least until senior-itis kicks in…).

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