I tutor math. It’s one of my favorite things to spend time doing. I’ve tutored people at various stages of their development, but there’s one foundational thing that is very difficult to teach people unless you have the luxury of spending at least a year with them.
Learning math is not about memorizing processes for solving problems. It’s not about finding x. It’s not even about being able to figure out how old Sally is if she’s twice as old as her brother Bill who is 6 years younger than 13 year old Nitish.
Learning math is about learning how to think flexibly and creatively about problems you face in the world. It is about exercising the part of your brain that can turn observation into insight. It is about building a toolkit that will serve you well in any leadership role, which is especially true if you pursue business, engineering, or science.
Unfortunately, so many of the students that I tutor suffer from the same problem. They have all learned that the way to learn math is to memorize it. This works when you are memorizing that 7 x 6 = 42. This works when you memorize that the circumference of a circle is the product of pi and its diameter. For most people, this stops working in either high school or college.
When this stops working, the A+ students diverge. Some of them have been relying on memorizing everything, and some of them have been struggling with the concepts and trying to learn the underlying foundations. The former group get really frustrated at this stage of their careers, some of them having chosen to pursue medicine or engineering. They really, really struggle. Unfortunately, they’ve already undertaken such a heavy courseload that depends on a core mathematical skill that it’s very difficult to re-teach oneself. Often, these students aren’t aware that this shift has occurred.
A friend of mine once taught a Linear Algebra course. As part of his tests, he would ask his students to explain what a basis of a vector space is. Many students would protest that math is not about memorizing vocabulary, but is rather about being able to do the useful mathematics. This is an important case study, because these students clearly believe they are learning math the correct way. However, I would argue that they have not actually learned the math, they have simply memorized the process by which one can compute the basis of a vector space. As my friend so aptly put it: “If you don’t know what a basis of a vector space is, I see no reason to ask you to compute one.” That is, at its core, the problem of many who are taught mathematics today. We are told to memorize process, and never to learn what we’re actually doing.
Perhaps a more accessible example is a student whose mother frantically contacted me one summer, saying her son was going into 8th grade and was woefully unprepared to begin thinking about high school math. She wanted me to help, and I was excited to lend a hand. It became clear though, that this student had been put through years of math that he had not understood. When we worked on word problems, it was clear that he had no fundamental understanding of what mathematical operators like adding and multiplying meant. For example, this student was stumped by the following word problem: “Joel will be paid $5.00 every week for 10 weeks. After 8 weeks, how much money will Joel have?” He knew that he probably had to do something with the numbers that were given to him, so he tried adding 10 and 5 to get 15, and asked if that was the correct answer. So I spent the rest of the lesson visually explaining to him how multiplication is really like adding a number to itself a certain number of times. But because of that, we made zero progress on his homework sheets that day. So what should we do? Should we make sure his homework got done, or should we spend time teaching him things he should have learned years prior? See the thing is, he could easily tell me the answer to 17 x 35, he had long ago memorized the process for multiplication. But he fundamentally had no idea what multiplying two numbers together meant.
The solution to this problem is to weight the understanding earlier in childhood education. We become too enamored with “child genius” when a 3.5 year old already knows their multiplication tables. Our teachers are under such pressure to ensure that every child graduating second grade has memorized their multiplication tables (my sister had to be able to multiply 60 pairs of numbers in 60 seconds) that there is no time allowed for ensuring proper understanding. This propagates for the rest of these students’ careers.
Common Core, beloved as it is, is sometimes approaching this type of model. This is why many who memorized their way through elementary math are so opposed to it. It seems like children are being taught an unnecessarily difficult process for what they feel should be a very simple thing. However, I think many implementations of common core math are actually overall going to be beneficial to students’ learning.
So if you’re a parent helping a child with their homework, don’t urge them to memorize the process for figuring out how long it will take train A to meet train B in city X. Help them develop mental models for how the world works, and then teach them how math is a powerful tool that can help them describe that model of the world.
If you’re the student yourself, I would urge you to hold yourself accountable for the same.
Oh, and be nice to your teachers. They’re overworked, underpaid, underappreciated, and have inherited the terrible system described above.