It’s a hard truth that some people have more advantageous cognitive differences than others — e.g., higher working memory capacity, higher generalization ability, slower forgetting rate. However, there are two sources of hope: 1) automaticity can effectively turn your long-term memory into an extension of your working memory, and 2) many sources of friction in the learning process can be not only remedied but also exploited to increase learning speed beyond the status quo.
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The purpose of this article is to lift up the spirits of a reader who, after reading Chapter 7. Myths & Realities about Individual Differences in The Math Academy Way (these parts in particular), is struggling to cope with the depressing reality that some people have more advantageous cognitive differences than others (e.g., higher working memory capacity, higher generalization ability, slower forgetting rate).
I’ll present two sources of hope.
Hope #1: Automaticity
Research indicates that the best way to improve your problem-solving ability in any domain is simply by acquiring more foundational skills in that domain.
The way you increase your ability to make mental leaps is not actually by jumping farther, but rather, by building bridges that reduce the distance you need to jump.
What makes this so hopeful? Well, it means that even if someone can’t mentally jump as far as another person, they can still go further, and solve more advanced problems, solely on the basis of bridge-building.
Scientifically, what this amounts to is: by developing automaticity on your lower-level skills, you can effectively turn your long-term memory into an extension of your working memory.
It’s kind of like how in software, you can make a little processing power go a long way if you get the caching right.
As summarized by Anderson (1987):
- “Chase and Ericsson (1982) showed that experience in a domain can increase capacity for that domain. Their analysis implied that what was happening is that storage of new information in long-term memory, became so reliable that long-term became an effective extension of short-term memory.”
And here’s a direct quote from Chase & Ericsson (1982):
- “The major theoretical point we wanted to make here is that one important component of skilled performance is the rapid access to a sizable set of knowledge structures that have been stored in directly retrievable locations in long-term memory. We have argued that these ingredients produce an effective increase in the working memory capacity for that knowledge base.”
Reber & Kotovsky (1997) actually did some experiments on this and found that indeed, the impact of working memory capacity on task performance was diminished after the task was learned to a sufficient level of performance:
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*“Participants solving the Balls and Boxes puzzle for the first time were slowed in proportion to the level of working memory (WM) reduction resulting from a concurrent secondary task.
On a second and still challenging solution of the same puzzle, performance was greatly improved, and the same WM load did not impair problem-solving efficiency.
Thus, the effect of WM capacity reduction was selective for the first solution of the puzzle, indicating that learning to solve the puzzle, a vital part of the first solution, is slowed by the secondary WM-loading task.”*
More generally, as Unsworth & Engle (2005) have explained:
- ”..[I]ndividual differences in WM capacity occur in tasks requiring some form of control, with little difference appearing on tasks that required relatively automatic processing.”
In addition to behavioral studies, this phenomenon can be physically observed in neuroimaging. Developing automaticity on skills empowers you to perform them without disrupting background thought processes (so you can keep the “big picture” in mind as you carry out lower-level technical details).
At a physical level in the brain, automaticity involves developing strategic neural connections that reduce the amount of effort that the brain has to expend to activate patterns of neurons.
Researchers have observed this in functional magnetic resonance imaging (fMRI) brain scans of participants performing tasks with and without automaticity (Shamloo & Helie, 2016). When a participant is at wakeful rest, not focusing on a task that demands their attention, there is a baseline level of activity in a network of connected regions known as the default mode network (DMN). The DMN represents background thinking processes, and people who have developed automaticity can perform tasks without disrupting those processes:
- “The DMN is a network of connected regions that is active when participants are not engaged in an external task and inhibited when focusing on an attentionally demanding task… at the automatic stage (unlike early stages of categorization), participants do not need to disrupt their background thinking process after stimulus presentation: Participants can continue day dreaming, and nonetheless perform the task well.”
When an external task requires lots of focus, it inhibits the DMN: brain activity in the DMN is reduced because the brain has to redirect lots of effort towards supporting activity in task-specific regions. But when the brain develops automaticity on the task, it increases connectivity between the DMN and task-specific regions, and performing the task does not inhibit the DMN as much:
- ”…[S]ome DMN regions are deactivated in initial training but not after automaticity has developed. There is also a significant decrease in DMN deactivation after extensive practice.
…
The results show increased functional connectivity with both DMN and non-DMN regions after the development of automaticity, and a decrease in functional connectivity between the medial prefrontal cortex and ventromedial orbitofrontal cortex. Together, these results further support the hypothesis of a strategy shift in automatic categorization and bridge the cognitive and neuroscientific conceptions of automaticity in showing that the reduced need for cognitive resources in automatic processing is accompanied by a disinhibition of the DMN and stronger functional connectivity between DMN and task-related brain regions.”
In other words, automaticity is achieved by the formation of neural connections that promote more efficient neural processing, and the end result is that those connections reduce the amount of effort that the brain has to expend to do the task, thereby freeing up the brain to simultaneously allocate more effort to background thinking processes.
Hope #2: Learning Efficiency Speedups
Most people practice ineffectively and consequently do not reach anywhere close to their personal maximum learning rate.
I covered this in detail in the post Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think.
The idea is that individual cognitive differences put a soft limit on how much math someone will be able to learn –
BUT, in practice, few people actually reach this limit because they get knocked off course early on by factors such as
- missing foundations,
- ineffective practice habits,
- inability or unwillingness to engage in additional practice, or
- lack of motivation.
As I described in detail in that post, many of these things CAN be remedied.
And not only remedied, but also exploited to increase learning speed beyond the status quo.
For instance, it’s a problem when classes don’t review previously-learned material. Students constantly forget things to the point of continually having to re-learn them almost from scratch, which introduces lots of friction into the learning process.
You can reduce that friction by, well, reviewing previously-learned material. Any teacher worth their salt knows that.
BUT there is still plenty more room to improve!
Review is better than no review… but what’s BEST is to optimize the review process so that
- you are reviewing only what you absolutely need to, and
- you are selecting learning tasks that minimize the amount of time you have to spend reviewing, to knock out all the review you need to do.
At Math Academy, I built an automated task selection system that leverages the power of “encompassings” to ensure that students spend most of their time learning new material while simultaneously making sure they keep getting practice on things they’ve previously learned.
The idea is that we are often able to have students knock out review by learning something new instead.
For instance, if a student learned how to solve ax=b equations yesterday, and they’re due for a review today… let’s just learn the new topic $ax+b=c$ equations instead!
Solving ax+b=c “encompasses” solving $ax=b$ as a component skill, so it provides the review that’s needed – all while the student is learning something new.
(And whenever we can’t “knock out” all a student’s due reviews with new material, we can still compress them into a much smaller set of review tasks. Instead of having to review 10 topics, you might just have to review 2 topics that collectively encompass all those 10.)
There are a number of other instances where you can take a remedy, lean into it further, and turn it into an exploit.
For instance, instructional scaffolding: some is better than none, but more is better!
All the sources of friction I wrote about here can be seriously exploited, turning them into massive speedups.
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