In our last two articles, we’ve been exploring the idea that a Bottom-Up Approach to teaching math can do more for lasting skill mastery than the more traditional Top-Down one. The system that will take the best advantage of this approach will have strategies related to What, How and When to Practice. Our last article explored the When piece of this. Today, we will dive into the What piece.
What do I practice? Have you heard the quote: “I didn’t have time to write a short letter, so I wrote a long one instead”? It takes time and attention to pare something down to its essentials, yet doing so is vitally worth it. In math especially, it is so much better to understand a few concepts deeply than a whole slew of things not so well. Knowing these most important skills totally solidly will help you make sense of any other skills you happen to come across, and they will really hold you back if you try to move forward without them.
The lesson here is to limit your “work in progress.” Just as there were vast amounts of research supporting the phenomenon of “spaced repetition” from our last article, there is extensive research showing that limiting the things you’re trying to accomplish—be they tasks around the office or math skills on their way to mastery—has an important impact on the results you’re able to achieve. See our footnotes section below for some examples.
Even without reading the science to back it up, you KNOW this. There is a limit to how many pieces of information your brain can hold at a time. As skills are transitioning from short-term memory to long-term knowledge, it is counter-productive to give the brain too much information to hold onto. And in a sequential subject like math, adding on new skills before former ones take hold just sets a student up for failure.
Building a house on wet sand is catastrophic and so is building complicated concepts on shaky understanding. Choosing a finite amount of skills to work on to mastery is one of the most important things you can do to work smart not hard, otherwise you’ll be building a very large house on very wet sand. Here at Math For Keeps, we see this all the time, as even students who have gotten as far as Calculus struggle because of a weak foundation in the Basics and in Algebra 1.
So what does it look like to put the What piece into practice? There are two sub-questions to the What Question: HOW MANY? and WHICH? How many skills to choose and which ones depends on many factors but the general rule of thumb for HOW MANY is to think in terms of micro-skills as opposed to topics or macro-skills, and to have a window of time within which your task is to master a finite number of them.
An example of a macro-skill is “solving equations”. An example of a micro-skill is “solving equations in a single variable with a fractional coefficient and no negatives”.
The general rule of thumb for WHICH is to think in terms of which skills show up over and over again? Which ones will render you stuck and incapable of learning the next level of skills if you don’t master them? An example of a foundational skill is multiplying fractions. There are countless subsequent skills that rely on this skill.
In summary, choose a finite amount of skills to work on in a finite amount of time, and practice those skills all the way to mastery. Rinse and repeat.
Stay tuned for our next article where we’ll talk more about the How piece of a good practice-to-mastery, Bottom-Up approach to learning math.
- Math teaching in US: A mile wide and an inch deep.
- This is a great post in a series of great posts about limiting your work in progress:
- Read how multitasking is counterproductive.