Do you learn by seeing, or by hearing? By reading, or by doing, perhaps? The leading theories today support that everyone learns across all the classic learning styles; the best learning takes place when multiple senses and actions are engaged.
But what about remembering? How do we really make something “stick”?
As we’ve been exploring, the lessons that tend to lodge themselves most effectively in our brains are the ones we’ve internalized. This might mean that we care about them in some capacity, that we’ve attached meaning to them, or that we’ve found them useful or surprising. We also discovered that it’s easier for novel information to take root when we connect it to old ideas we’re really familiar with.
This, at the very least, helps us to grasp a simple concept or take a first pass at a more complex one. But what about when analogies fail and we’re really venturing into uncharted territory?
With some tools, the best way to learn how to use it is to, well, use it.
Application Generation
There is something to be said for rote practice, to a degree. How is a student supposed to learn what order to do BEDMAS/PEMDAS questions without doing several example questions that get them in the habit of doing multiplication before subtraction? Arguably, a student should first understand the intuition behind brackets being a notational priority. Beyond that, the student should be exposed to several permutations of questions that pose different challenges to properly employing order of operations. The student must use the BEDMAS “muscle”, building those neuropathways similar to muscle memory, and they must also use it from multiple angles, allowing them to understand how the tool applies to all kinds of situations.
An analogous case can be made for internalizing topics that involve more than straight memorization. No educator will argue that an audible lecture, or even robust visual explanation, on a brand new, complicated topic will be enough for mastery. To store that mathematical tool into their mind’s toolbox, a student must practice. This helps, at the very least, to drill in surface-level memorization. For next level internalization, a student should use that tool in a set of applications as varied as possible.
What exactly do we mean by application? If you’re looking for a two-word answer, then “word problems” will be about eighty percent sufficient. Word problems are fantastic frameworks that force students to identify key information, holes they need to solve for, and possible plans of attack. Word problems are also brilliant tools for tackling multiple angles of a problem. What was given in one question can be left for solving in another, or buried beneath a second calculation step.
This kind of application approach isn’t limited to word problems, however, which is where the other soft twenty percent comes in. Process-of-elimination questions for calculating angles and side lengths, for instance, are often very effectively visually and arithmetically presented.
The Power of Varied Application
The point is, the more diverse the kinds of questions students learn to solve off of a given concept, the more thoroughly they understand that concept and its power. A really neat side benefit is that looking at an idea more three-dimensionally can be powerfully fertile ground for epiphanies and “aha” moments.
Let’s take a simple example and see how guidance through the right kinds of application challenges can develop foster a robust understanding.
A student is working on division skills, and has a pretty good grasp on the arithmetic. Take a large number, use a long division bracket and your times tables, and break that number down into a “number of items per group” answer. Watch how we can develop a more sophisticated understanding of the power of division without introducing any other arithmetic tools other than multiplication and division.
Level I: Simple division.
Ling has 42 cinnamon hearts she wants to give away as valentines. She has 6 friends. How many cinnamon hearts will each friend get?
Level II: Reverse division (multiplication) and regrouping.
Ling is planning to give all of her cinnamon hearts to her 6 friends as valentines. Each friend will get 7 hearts. She would like to keep some for herself. How many cinnamon hearts will each person get?
Level III: Multiplication, regrouping, and an intuitive understanding for the remainder.
Ling wants to give 7 cinnamon hearts to each of her 6 friends. The candy store sells them in packs of 10. How many packs will she need to buy? Will she have any left over?
Level IV: Multiple division combinations (factoring).
Ling buys 6 packs of 10 cinnamon hearts. She’s not sure she wants to give them to all of her friends, but when she does decide how many people to give them to, she wants to give them each the same amount. What are all the numbers of friends she could give her hearts to so that each friend gets the same amount with no leftovers?
Level V: Multiplication combinations (LCM).
Ling has 8 friends. Cinnamon hearts are sold in packs of 6. What’s the minimum number of packs she needs to buy in order to give all 8 friends an equal number of cinnamon hearts with no leftovers?
…and the list could go on.
Varied application allows students to view their newly minted tools in different light, apply them to different circumstances, and see how they respond to different demands. This helps solidify the tool’s usefulness and begins to develop the intuition that will trigger when that tool needs to be used again. This helps students identify immediately, when they see a new question, what kinds of operations they can use to tackle it. “Oh, this question has a division component.” “Oh! This question is really about fractions.” “This boils down to a simple subtraction problem.”
Back to Familiar Connections
This intuition allows students to do exactly what we talked about last time: wade through complex concepts to pick out aspects that connect to what they are very familiar with. When these tools get added to the toolbox and students have a robust understanding of how and when to use them, that’s when you know they’ve really internalized the learning.
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