We’ve spent a lot of time at Mentorhood Math creating a curriculum that teaches math through concrete, relatable methods, because we believe that’s the best strategy to building mathematical intuition. When it comes to building a solid foundation of understanding that gives our learners the ability and the confidence to advance to higher topics, we believe that teaching slowly, systematically, and intuitively is far superior than teaching through repetition, rote training, or spoon feeding.
Spoon-feeding and rote training give students mechanical answers to math problems, and repetition reinforces its memorization. Students have the option to copy and repeat the solutions without really understanding what they are doing, or without being able to extrapolate the principles to slightly different situations.
At Mentorhood, we want students to understand the problem, know how to arrive at the solution, and proceed to calculate it.
If repetition can take the thinking out of the equation, should it be done away with all together? Is it destroying our children’s ability to think through math for themselves?
Building Mechanical Understanding
As stated above, repetition reinforces the ability to repeat mechanical answers without spending too much time thinking about it. Repetition ingrains mathematical processes into a student’s mind and body so that they can repeat the mechanics with almost no hesitation.
This can be highly advantageous―so long as you don’t skip the thinking from the start.
The Path to Mastery
Mentorhood believes that understanding is the first step on the path to mastery, but that memorization falls on the path too. We believe that repetition should be something students work towards, not work from. We don’t train through memorization; we reinforce through it.
Our three-step approach to teaching mastery in a topic begins understanding and builds up in skillfulness:
Teach the students to know how to solve a question, using our CPA approach, working from concrete representations all the way to abstract numbers. This will build the students’ understanding of the problem and the solution.
Help the students practice remembering how to solve a question, especially with slightly modified scenarios, to strengthen the accuracy of their approach to the solution.
Train the students to memorize the solution to that type of problem, or to a handful of common problems, improving their speed.
Building their skillfulness from understanding to memorization not only makes them more efficient calculators, but gives them confidence that they have mastered the topic. They understand what they are doing so well, and how to apply it to different situations, that they don’t need to think about it anymore, and can focus on efficiency instead. They know that if required, they could pull out the knowledge underneath their memorized solution at the drop of a hat.
It’s All About the Numbers
Math really is a numbers game. It’s about speaking numbers, breathing numbers, and drawing inferences from how numbers operate and interact with each other in the real world. Math is the language of numbers, and like learning any new language, sometimes you have to memorize a few new words.
We believe in first helping students understand that mathematical questions come from tangible problems that they can relate to in their lives. We seek to help learners develop intuitive visualizations so that they can see how numbers relate to one another. And we believe that once understanding is achieved, the next step on the path to true mastery is expediency.
The more expedient students become at arriving at solutions that they have calculated over and over (rather than answers that have been fed to them), the more confident and quick they will be in integrating those calculations into more complex topics. They will be quicker to recognize patterns and relationships because their memory banks will be full of similar problem and solution sets that they have seen before. This can make them even more capable of drawing inferences and conclusions based on the numbers they see.
It Starts With Understanding
That’s the difference between our approach and the traditional methods of rote training. Memorizing through rote training fails to link these numbers into relationship with one another because questions-and-answers are memorized individually, and are stored in little memory banks as isolated instances.
Memorizing based on understanding allows students to draw on a well of related information that bubbles up into memorized answers. It allows students to tap into an interconnected network of mathematical relationships underneath the surface of what they can repeat without hesitation.
So, is there room for memorization at an institution that stresses learning through understanding and building intuition? Absolutely. Memorization is just one more step towards building a more thorough and robust mastery of mathematics. And with enough time and the right understanding-based approach, memorized math can feel just as reliable and comfortable as intuition, too.