Struggle Does Not Imply Inability

If you do poorly in a math class, it doesn’t necessarily mean that you are incapable of learning that level of math. There are a number of reasons that could be the root cause of your struggle.

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Struggle does not imply inability. If you do poorly in a math class, it doesn’t necessarily mean that you are incapable of learning that level of math. There are a number of reasons that could be the root cause of your struggle.

While it’s true that everyone’s mathematical potential has a limit, in practice the ceilings we hit rarely represent our true abstraction ceilings. All sorts of factors can artificially lower our ceilings, such as missing foundations, ineffective practice habits, inability or unwillingness to engage in additional practice, or lack of motivation.

1) Struggle can be caused by MISSING FOUNDATIONS.

When people age, they accumulate biological damage that eventually reaches a tipping point and leads to a cascade of catastrophic health issues. The same thing happens to students learning mathematics.

Students accumulate weaknesses and knowledge gaps as they progress through math – even a grade of B+ or A- means that there are things in the course that the student never completely grasped, much less mastered. Additionally, gaps can be created if a student takes a course that is not comprehensive and does not cover some topics that are assumed to be prior knowledge in higher-level courses. Once a student has accumulated a critical number of gaps (and by the way, a gap begets more gaps), then the student is doomed to struggle unless proper remediation is enacted to fill in those gaps.

Remediation is extremely difficult to accomplish outside the context of an adaptive, automated learning system. It rarely happens in the classroom – teachers just don’t have the bandwidth to spend enough time with each student to figure out exactly which pieces of foundational knowledge are missing. And while remediation can often be performed by a skilled tutor, it generally requires many tutoring sessions over a long period of time, continuing indefinitely into the future to prevent new gaps from forming, which makes it prohibitively expensive for most families.

Students usually stop taking math classes once they amass a critical number of knowledge gaps. The usual sequence of events starts with students trying to imitate procedures cookbook-style, without really understanding what’s going on, because they can’t intuitively grasp any of the new material that they’re being taught. Soon after that, they find themselves unable to solve any problems that involve critical thinking or many steps.

It’s similar to how professional athletes usually retire not because they’re too old, but because they’ve accumulated too many injuries. As Indiana Jones once put it: “it’s not the years, it’s the mileage.” Or as math writer/cartoonist Ben Orlin humorously described, it’s the “law of the broken futon”: a single missing part can, over time, warp an entire futon and render it unusable.

Students will almost assuredly accumulate these deficits in traditional classrooms. It’s only the most gifted and motivated students who are able and willing to identify and “self-repair” their gaps on their own.

• In traditional classrooms, students often get stuck on foundational topics but are required to complete homework on more advanced topics, leading them to "scrape by" without really understanding the subject matter.
• Students also do not review material learned in previous years, and often do not even review material from the course that they're in unless they are preparing for a test. This leads them to quickly forget what they've learned, requiring re-learning scratch if and when those topics show up again in the future.
• Often, traditional courses are not even comprehensive! It's not uncommon for instructors to run out of time before the end of the year and skip sections of the textbook.

2) Struggle can be caused by INEFFECTIVE PRACTICE.

Effective learning feels like a workout with a personal trainer. It should center around deliberate practice, a type of active learning in which individualized training activities are specially chosen to improve specific aspects of performance through repetition and successive refinement. Below are some key points:

• Effective learning is active, not passive. It is not effective to attempt to learn by passively watching videos, attending lectures, reading books, or re-reading notes.
• Deliberate practice requires repeatedly practicing skills that are beyond one's repertoire. However, this tends to be more effortful and less enjoyable, which can mislead non-experts to practice within their level of comfort.
• Classroom activities that are enjoyable, collaborative, and non-repetitive (such as group discussions and freeform/unstructured project-based or discovery learning) can sometimes be useful for increasing student motivation and softening the discomfort associated with deliberate practice -- but they are only supplements, not substitutes, for deliberate practice.
• Deliberate practice must be a part of a consistent routine. The power of deliberate practice comes from compounding of incremental improvements over a longer period of time. It is not a "quick fix" like cramming before an exam.

3) Struggle can be caused by INSUFFICIENT PRACTICE.

Struggle can be caused by needing more practice than other students (or, equivalently, the pace of the class might be too fast). This is not necessarily a catastrophic issue in itself because it can usually be remedied by engaging in further practice. However, it can cause problems if coupled with other factors such as the following:

• The instructional material is not highly scaffolded.
• Few practice problems are available.
• Exam problems are substantially different from homework problems.
• The additional practice required exceeds the amount of effort that you are willing to put forth to learn the material.

4) Struggle can be caused by LACK OF MOTIVATION.

Properly motivated students are usually driven by one or more of the following factors:

• They are intrinsically interested in the material. Some students truly love math and see beauty in the way various mathematical ideas fit together and give rise to new perspectives.
• The material is highly relevant to their future goals. For instance, an aspiring rocket scientist might not love math but might be motivated to learn it because of how useful it is for getting rockets into space. Likewise, an aspiring doctor might not love math but might be required to evidence a baseline level of mathematical knowledge when applying to medical school. Even students who do not have specific future goals might feel strongly about keeping potential career doors open which would otherwise be shut by not learning enough math.
• They enjoy competing in mathematical exams and science fairs. Some students have neutral feelings about math, but find that they are good at it, and that they enjoy learning more advanced mathematics to provide a competitive edge in exams and science fairs.
• Their parents have motivated them with a meaningful extrinsic reward. Sometimes, a student may not fall into any categories above, but their parents (often rightfully so) want them to fully take advantage of any opportunities to learn math while they are still in school. For some students, this may mean learning the basic math they need to get by in life after school; for other students, this may mean learning more advanced math to open a wide variety of career doors. If a student is highly interested in other activities like reading novels, playing video games, or even something as simple as going out for dessert, offering them extrinsic rewards in return for meeting checkpoints in their math learning can often provide sufficient motivation to keep them from "checking out" during learning.

If a student is not driven by any of the motivational factors above, they may “check out” or otherwise struggle due to a lack of interest in learning the material.

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