Throughout this series, we’ve been exploring how the teaching techniques we use at Mentorhood Math, inspired by the highly successful Singapore Math method, focus on learning through intuition.
We refer to our students as “learners,” subtly encouraging them to see themselves as in charge of their own learning, rather than subordinate to instruction. We take our time, solidifying basic concepts before moving on to more advanced ones. We utilize visual tools, concrete objects, and fun to make learning more accessible and relatable to young minds. All of this helps students internalize what they are learning.
But what really is the difference between CPA learning (working with Concrete, Pictorial, and eventually Abstract representations of math) and rote training, also known as learning through repetition or spoon feeding? Are the differences that major? Are the advantages truly significant?
Leaving More than Potential
In much of the world, math is told rather than taught. Instructors state a new type of problem that the class will be tackling, and then go on to explain a series of mechanical steps to solve the problem.
Often, the example problems are stated in abstract, mathematical terms and are hardly connected to the real world at all. Ever heard a child say, “But when am I going to use this?” Perhaps you’ve said it yourself.
Once the mechanical steps have been sufficiently memorized by a sufficient percentage of the class, the real keeners are given the opportunity to apply the mechanics to word problems. Students have the added challenge of picking out relevant details, determining which operation they should use, and proceeding with the calculations.
The rest of the class is left stumbling, puzzling over which number goes where in the formula and trying not to invert steps while asking themselves in frustration, “Why on earth is Joan wasting time calculating where her baseball will land at all? Just hit it!”
Rote learning leaves more than potential for understanding, connection to the real world, and appreciation for math in daily life on the plate. Rote learning often leaves a lot of its learners behind.
Lap It Up or Be Lapped
In much of the world, courses race through math concepts at top speed. This is understandable: regional, national, and global mathematical standards have ever-growing standardized lists of concepts that should be mastered in order to compare the performance of one particular group of students to another.
This causes an interesting dynamic of quantity over quality in the classroom: teachers must emphasize getting through a certain number of topics and hope the students keep up.
As a result, math is often taught abstractly, with mechanics and symbols and numbers and letters being the only experience with math many students have. Teachers are tasked with connecting math to the real world, but this is seen almost as superfluous to knowing the mechanics. Students learn for the sake of learning—or, more accurately, memorize for the sake of learning, without understanding why. Those that do not have a naturally abstract mind will have trouble following and executing, while those that can pick up mechanics are at an advantage. This is why we see students label themselves into two groups: “good at math” and “bad at math.”
The problem with rote or mechanical training is that it takes independent decision making out of mathematical discovery. Students are given a problem and a set of steps to solve it and taught to look for identical circumstances in which to put their memorized steps into practice.
This is different from the way that math was first developed. People started with real world problems that they needed solutions for. They discovered universal mathematical truths as they came up against dilemmas, and used abstract language—numbers and symbols—as a way to efficiently represent and extrapolate these truths.
Abstract math language evolved as an efficient way to represent real-world needs for math. In short, it was an eventual destination on the journey of mathematical inquiry. With rote learning, this “destination” is used as a starting point. Students rehearse answers without any real investment in the problem: they are spoon fed, when they don’t even know they are hungry.
The Link Between Independence and Discovery Math is about discovery. It’s about representing the world through awe-inspiring universal truths and finding information that we need to make decisions. It’s about solving problems one dilemma at a time, using arithmetic tools at our disposal to learn and create. Taking an intuitive approach to math enables us to make decisions along the way that we think will drive us closer to the answer we’re looking for.
At Mentorhood Math, time is on our side. We with little mathematicians to understand the simple problems they face in class. We make sure that they are involved every step of the way in determining how a problem should be solved, and we don’t leave students behind.
Students who have studied at Mentorhood Math are gifted with understanding on a very foundational level, so that they are not only set up to pick up more advanced concepts, but they also understand the why they are doing what they are doing. We are equipping them with tools and growing their confidence that they can apply the right tool for the job.