Math Ought to Make Sense: 3 Ways to Promote Meaningful Math Learning
I heard one of my favorite phrases the other day in class. Two young women raised their hands during a partner task and blurted in frustration (and in unison): "This answer just doesn't make any sense!"
I immediately stuck out my clenched hand for a congratulatory fist bump: "I'm sorry you are frustrated, but I love the thinking you are doing...I wonder if we can make it make sense?"
These young women were solving a quadratic equation to find the speed of a roller coaster. The mathematical process that they were chugging through is fairly complex and involves a few different equations. At several point is in the journey, they were prompted to show work and write about their thinking. Despite all this, these two budding mathematicians didn't just get an answer and move on. They looked at what they had on paper and asked that beautiful mathematical question: "Does this make sense?"
I think one of the great missed opportunities of the math curriculum that I experienced as a high school student was that it provided very little incentive for me to do this kind of thinking work. For me, solving a quadratic equation in Algebra 1 usually meant that I had found the elusive "x" and it was time to go on to the next problem. This value did not have meaning. There was usually no sense to make.
Making sense of an experience is a pretty important skill in life. Good scientists are critical of the data they get throughout an experiment: "Does this value make sense with what we already know about what we're studying?" My co-workers create systems for evaluating a plan as it is being implemented: "Does it make sense for us to keep doing this?" Successful friendships rely on this habit of mind: "Before we keep going, does this make sense to you?"
So how can math teachers promote this habit in our classes?
1. Students talk about math with one another. The partner work activity allowed the students in my class to voice questions and skepticism in a low-stakes way with feedback. The young women had been conversing with one another for a few minutes before they called me over. Neither was confident in her own understanding before this peer-to-peer conversation, but both developed confidence as they discussed.
2. Students work in an engaging context. The roller coaster problem helped create a "need to know" in my students. They like roller coasters and they have an intuition about them. The "non-sensical" answer was that the roller coast was traveling as fast at the end of the ride as it had been at the beginning. My students knew this could not be the case even though that's what popped out of their process. If we'd been talking about apples or "x's" they probably would not have thought — or cared — to question the number.
3. Students develop meaning-making habits. I obsess about things like labeling units and developing problems with real values (an obsession that has gotten much easier and much more likely to lead down the rabbit hole in the era of big data like StatCast). Students start to expect cars to travel around 30 miles per hour and dogs to weigh around 50 lbs. We hold one another accountable for using precise language: "meters per second" never "m slash s."
As a young teacher, I dreaded that moment in class when a student's frustration boiled over into a shout: "This doesn't make any sense!" At this point I've started to embrace those as moments where real thinking is going on and as opportunities for real learning.
I'm curious to hear how you think and talk about sense-making with your students.
Photo 1: A student engaged in making sense of mathematics by Mary Conroy Almada
Photo 2: Symbolic logic is beautiful and meaningful as long as students are empowered to decipher it by Roman Mager https://unsplash.com/