To improve in Math, why precisely must most students slog at exercises, unlike driving that AI still can't automate?

Question

Too many parents ask why students need to buckle to problems to succeed at math — like end of chapter questions in textbooks. Bewildered, they analogize to lasting skills that — after acquisition — they can accomplish and remember for life, without slogging or practice.

What distinguishes math from these lifelong competencies? Why can you lose math, if you don't use it — but not other lifetime competences?

Most parents analogize to driving, which AI still fails to automate. But most people don't lose driving skills.

Machine-learning systems can be duped or confounded by situations they haven’t seen before. A self-driving car gets flummoxed by a scenario that a human driver could handle easily. An AI system laboriously trained to carry out one task (identifying cats, say) has to be taught all over again to do something else (identifying dogs). In the process, it’s liable to lose some of the expertise it had in the original task. Computer scientists call this problem “catastrophic forgetting.

After issuing licenses, Canada and USA aren't required to test drivers again until elderliness!

RoadSafetyBC, the B.C. government’s agency responsible for road safety, mandates persons 80 and above to renew their driver’s licence every two years. They are required to get a medical examination report to be completed by their physicians. [...] Those that fail the medical exams, however, may end up having to take the Enhanced Road Assessment to determine whether they are fit to drive. It is free of charge but it takes a 90 minutes to complete and includes a pre-trip vehicle orientation, a 45-minute on-road drive and a post-trip review.

DMV does not take away your driver’s license when you reach a certain age. Your mental and/or physical condition or your inability to follow traffic laws and rules regardless of age determines whether your license is renewed, restricted, suspended, or revoked.

After acquiring their licence, most adult drivers don't need to practice or train their driving deliberately with an instructor! Most still drive competently without incident!

Some parents analogize that most humans never forget competencies of muscle memory like swimming, or implicit memory like how to ride a bike — even if they haven't performed it in years!

Some parents analogize to cooking as most people haven't attended culinary school, but most can still cook well enough.

Most people haven't studied cooking professionally, and can't cook like Michelin Star chefs. Most people don't seek to advance or hone their cooking, but most people can still cook to survive!

Answer 1

The analogy to driving a car or riding a bike is a bad one. I agree that once you learn those skills, is not unreasonable to spend some years without doing those and still have the skill learned fairly well. I was perfectly able to ride a bike after many years of not riding one. I got my driver’s license at $17$, did not own a car until $9$ years later, and very rarely drove during those $9$ years, but I had no real problem driving after I owned my first car.

I suggest to the OP that you respond to anyone who makes an analogy to riding a bike or driving a car with other skills: playing a musical instrument and learning a foreign language. Anyone who learned to play a violin or speak a foreign language recognizes that after years of not practicing these, your abilities will be pretty rusty. I am reminded about this music stack exchange question on the topic of practicing scales. And it is a common phenomenon for Chinese and Japanese speakers to have their ability to write characters get degraded (character amnesia) now that “writing” is done so often on screens with autocomplete available. See here for Japanese and here for Chinese.

If it makes parents of math students feel better, you can tell them that even mathematicians may forget some areas of math that they learned as students but never use in their own work. For instance, those who work in analysis may forget things they once knew about finite groups (conjugacy classes for dihedral groups or the Sylow theorems, say) and those who work in algebra may forget basic theorems from measure theory. High school math teachers will forget the calculus they once knew if they only teach other topics and stop engaging with calculus.

A main reason students need to practice the math they learn in each new math class is that it contains new mathematical topics, so they can’t rely on only knowing math from earlier classes. Going back to driving cars, a better analogy there would be driving different types of vehicles: an automatic car, a standard (stick shift) car, a limousine, a motorcycle, and a truck. Do the parents think that being able to drive an automatic car makes you qualified to start driving a stick shift car right away, or that being able to drive a stick shift car makes you qualified to drive a truck without training? There is a good reason that a driver’s license is limited to a certain class of vehicles! And do they think a pilot licensed to fly one type of plane should be able to get into any other type of plane and fly it with no practice? Surely not. In fact, even flying the same type of plane requires regular practice: pilots are not allowed to fly a commercial jet without going back to something like a flight simulator for practice if they have not logged enough monthly flight hours. This became especially notable when covid shut down so much airline activity.See here.

Answer 2

Generally, when we talk about improving a skill, we tend to refer to skills that are physical-based. For instance, a professional basketball player may be a strong three-point shooter because they go to the gym, shoot a lot of three-pointers, and make adjustments to improve their results. Also, marching bands learn halftime shows and each member has to memorize the music and where they are supposed to be on the field at any given time, which they improve at through repetition during rehearsal.

Paul Halmos famously said "The only way to learn mathematics is to do mathematics.", which I interpret as although mathematics is inherently an abstract discipline, there is still some physical component to it. Practice is a significant part of improving at mathematics, because solely watching someone else do math problems such as the teacher or some stranger in a YouTube video is only sustainable for short-term and not long-term internalization. At some point, the student needs to roll up their sleeves and work on solving math problems. Hence, if you look at how to improve a math skill through a lens similar to the examples mentioned above, learning math is not so different from learning other skills as one might think.

The challenges with helping students improve at mathematics are:

1. as mentioned by Guest, the human brain needs to work harder to understand mathematics since it is an abstract discipline.
2. having a system where the students not only have the opportunity to practice mathematics, but practice efficiently.

To facilitate overcoming these obstacles, I minimize lecturing and have my students work on carefully-chosen activities that allow them to see how the concepts unfold and build off of each other, even if I have to break down a more complicated problem into multiple parts so they can pull through. Then we discuss the main idea of each activity where I try to get several students to state the main idea. They need to "own" the lesson and be able to explain the concept(s) in their own words because the person who does the talking does the learning.

This sets them up to complete the regularly assigned homework. I justify to my students that they need to practice as soon as possible because the main ideas of the lesson are reinforced most effectively when the students work on the practice problems the same day, as opposed to several days later when they may have forgotten everything. This also allows them to discover sooner which concept(s) they need to spend more time practicing (homework is assigned online so instant feedback is given), as opposed to discovering this the day before the assessment. Instant feedback is critical when working on practice problems. For instance, if a student were to solve 10 quadratic equations and unknowingly solve all 10 of them incorrectly due to the lack of instant feedback, then recovering from that involves reteaching the skill, which puts the student in a poor position since they have to undo the muscle memory from the 10 incorrect attempts.

Eventually, the summative assessment takes place and then we move on to the next unit. However, the concepts need to be revisited at some point in the future so that the students do not forget them. One way I achieve this is by allowing my students to retake assessments if they are not satisfied which their performance because this is a form of "doing math"; these students are taking it upon themselves to learn from their mistakes, thus showing that they understand how learning mathematics is a continuous process that does not end with taking a test.

Despite my carefully organized system, it still may not be enough because the students are ultimately responsible for their learning. Their views towards mathematics, their motivation to succeed, and the busy nature of their lives all affect their willingness or how much time they have to practice efficiently. Nevertheless, doing exercises and problems is fundamentally what allows students to "do math", and engaging in these activities efficiently allows students to learn math.

Answer 3

Our minds are not well adapted to math. It is an abstract activity. And requires significant practice to sink in. We are not rule based systems. Not flawless computers. Our brains are optimized for social interactions and physical control of our bodies. Not for math, although they are plastic enough to have some ability in math, it is not their strong suit.

All that said, your question seems a bit muddled in asking why need to do end of chapter exercises which are clearly part of the initial learning process, and then why you don't retain some unpracticed math techniques many years later, like bike riding.

Moreover, it is true you may not recall some math technique years later, if unpracticed, you can probably relearn it much easier if needed than someone learning it the first time.

Finally, it is not just math per se, but other calculational topics like chemistry and physics that benefit from exercises.