The Greatest Educational Life Hack: Learning Math Ahead of Time

by Justin Skycak (@justinskycak) on January 13, 2024

Learning math early guards you against numerous academic risks and opens all kinds of doors to career opportunities.

This post is part of the book The Math Academy Way (Working Draft, Jan 2024). Suggested citation: Skycak, J., advised by Roberts, J. (2024). The Greatest Educational Life Hack: Learning Math Ahead of Time. In The Math Academy Way (Working Draft, Jan 2024). https://justinmath.com/the-greatest-educational-life-hack-learning-math-ahead-of-time/

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Why learn math ahead of time?

Because it guards you against numerous academic risks and opens all kinds of doors to career opportunities.

Minimizing Risk

You know how, when you take a language class, there’s often a couple kids who speak the language at home and think the class is super easy?

You can do that with math.

When you pre-learn the material in a math course before taking it at school or college, you’re basically guaranteed an A in the class.

You guard yourself against all sorts of risks such as the course

moving too quickly,
brushing over concepts,
explaining things poorly,
assuming knowledge of important but frequently missing prerequisite material,
not offering enough practice opportunities,
...

There are a hundred different ways to teach a class poorly, and most classes suffer in at least a handful of those aspects.

This is especially helpful at university, when lectures are often unsuitable for a first introduction to a topic.

But if you pre-learn the material, you’re not depending on the teacher to teach it to you, which means you’re immune to even the worst teaching.

Opening Doors

Of course, the natural objection is “won’t you be bored in class?” – but if you do super well in advanced classes, especially at university, then that opens all kinds of doors to recommendations for internships, research projects with professors, etc.

It doesn’t matter whether you’re doing super well because you’re learning in real-time or because you pre-learned the material.

When you blow a course out of the water while also interacting with the professor, that sets you up for a great recommendation letter – which is vital not just for high schoolers applying to college, but also for college students applying to summer research programs and graduate schools.

Plus, it can open the door to working on a research project with the professor, or having them connect you to jobs, internships, and other opportunities with people in their network.

Basically, you can use pre-learning to kick off a virtuous cycle.

Even if you aren’t a genius, you appear to be one in everyone else’s eyes, and consequently you get a ticket to those opportunities reserved for top students.

Students who receive and capitalize on these opportunities can launch themselves into some of the most interesting, meaningful, and lucrative careers that are notoriously difficult to break into.

Why Stop at Pre-Learning One Year Ahead?

Many people think calculus is the “end of the road” for math, and that it doesn’t matter if you get there many years ahead of schedule.

But that’s far from the truth!

There are even more university-level math courses above calculus than there are high school courses below calculus.

After a single-variable calculus course (like AP Calculus BC), most serious students who study quantitative majors like math, physics, engineering, and economics have to take core “engineering math” courses including Linear Algebra, Multivariable Calculus, Differential Equations, and Probability & Statistics (the advanced calculus-based version, not the simpler algebra-based version like AP Statistics). Beyond those core “engineering math” courses, different majors include plenty of specialized courses that branch off in various ways.

There are so many courses that a student could not fit them all into a standard 4-year undergraduate course load even if they overloaded their schedule every year – however, the more of these courses a student is able to take, the more academic opportunities and career doors are open to them in the future.

(While it’s true that students don’t need to know much beyond algebra to get a job in fields like computer science, medicine, etc. – the people in such fields who do also know advanced math are extra valuable and in demand because they can work on projects that combine domain expertise and math.)

Maximizing Reward: Learning A Lot of Advanced Math Ahead of Time

When a student learns a lot of advanced math ahead of time, they unlock the opportunity to delve into a wide variety of specialized fields that are usually reserved for graduates with strong mathematical foundations.

This fast-tracks them towards discovering their passions, developing valuable skills in those domains, and making professional contributions early in their career, which ultimately leads to higher levels of career accomplishment.

I’m not exaggerating here – this is actually backed up by research. On average, the faster you accelerate your learning, the sooner you get your career started, and the more you accomplish over the course of your career.

For instance, in a 40-year longitudinal study of thousands of mathematically precocious students, researchers Park, Lubinski, & Benbow (2013) concluded the following:

"The relationship between age at career onset and adult productivity, particularly in science, technology, engineering, and mathematics (STEM) fields, has been the focus of several researchers throughout the last century (Dennis, 1956; Lehman, 1946, 1953; Simonton, 1988, 1997; Zuckerman, 1977), and a consistent finding is that earlier career onset is related to greater productivity and accomplishments over the course of a career. All other things being equal, an earlier career start from [academic] acceleration will allow an individual to devote more time in early adulthood to creative production, and this will result in an increased level of accomplishment over the course of one's career.
...
[In this study] Mathematically precocious students who grade skipped were more likely to pursue advanced degrees and secure STEM accomplishments, reached these outcomes earlier, and accrued more citations and highly cited publications in STEM fields than their matched and retained intellectual peers."

Higher Math, Not Competition Math

To be clear: in all this discussion about learning advanced math, I’m talking about higher-grade math, not grade-level competition problems.

When a middle or high school teacher has a bright math student, and the teacher directs them towards competition math, it’s usually not because that’s the best option for the student. Rather, it’s the best option for the teacher. It gives the student something to do while creating minimal additional work for the teacher.

Competition math problems generally don’t require students to learn new fields of math. Rather, the difficulty comes from students needing to find clever tricks and insights to arrive at solutions using the mathematical tools that they have already learned. A student can wrestle with a competition problem for long periods of time, and all the teacher needs to do is give a hint once in a while and check the student’s work once they claim to have solved the problem.

But if you look at the kinds of math that most quantitative professionals (like rocket scientists and AI developers) use on a daily basis, those competition math tricks show up rarely, if ever. What does show up everywhere is university-level math subjects like linear algebra, multivariable calculus, differential equations, and (calculus-based) probability and statistics.

Given that most students who enjoy math end up applying math in some other field (as opposed to becoming pure mathematicians), it would be more productive for them to get a broad view of math as early as possible so that they can sooner apply it to projects in their field(s) of interest.

Of course, the countering view is that “students should go ‘deep’ with the math that they’ve already learned – they’ll learn the other math subjects when they’re ready.”

But, in practice, the second part of that claim is not true. There are so many other math subjects that even most math majors only learn a tiny slice of all the math that’s out there.

Students generally can’t learn other math subjects “on the job” after graduation, either – if you’re trying to solve cutting-edge problems that nobody has solved before, then there is no “known path” that can tell you what additional math you need. And to even realize that a field of math can help you solve your problem, you generally need to have learned a substantial amount of that field in the first place.

In practice, the only way for students to “learn the other math subjects when they’re ready” is to learn as much math as possible during school.

Developmental Appropriateness

Many people think that learning math early is not appropriate for students’ social/emotional and cognitive/academic development.

But the reality is that educational acceleration does not lead to adverse psychological consequences in capable students.

I realize that if you’re skeptical about this, you’re probably looking for some empirical evidence to the contrary, not just a logic/reasoning-based argument. So let’s dive into the research.

According to a study titled Academic Acceleration in Gifted Youth and Fruitless Concerns Regarding Psychological Well-Being: A 35-Year Longitudinal Study that followed thousands of accelerated students throughout their lives over the course of 35 years (Bernstein, Lubinski, & Benbow, 2021):

"The amount of educational acceleration did not covary with psychological well-being. Further, the psychological well-being of participants in both studies was above the average of national probability samples. Concerns about long-term social/emotional effects of acceleration for high-potential students appear to be unwarranted, as has been demonstrated for short-term effects.
...
These findings are consistent with research on the effects of academic acceleration on psychological well-being. That is, there is little evidence that academic acceleration has negative consequences on the psychological well-being of intellectually talented youth (Assouline et al., 2015; Benbow & Stanley, 1996; Colangelo et al., 2004; Gross, 2006; Robinson, 2004).
...
These findings do not support the frequently expressed concerns about the possible long-term social and emotional costs of acceleration by counselors, parents, and administrators. ... Those who were accelerated had few regrets for doing so. Indeed, if anything, they tended to wish that they had accelerated more."

Whether a student is ready for advanced mathematics depends solely on whether they have mastered the prerequisites. If a student has mastered prerequisites, then it is appropriate for them to continue learning advanced math early, and not appropriate to stunt their development by holding them back. As the study authors note:

"Many fear negative possibilities of moving a gifted child to a more advanced class. Yet it also is important to consider the negative possibilities of holding children back in classes aiming to teach subject matter that they have already mastered (Benbow & Stanley, 1996; Gross, 2006; Stanley, 2000). Choosing not to accelerate is as much of a decision as choosing to do so ...

This is particularly important given the extensive empirical literature showing positive effects of acceleration on academic achievement (Kulik & Kulik, 1984, 1992; Lubinski, 2016; Rogers, 2004; Steenbergen-Hu et al., 2016) and creativity (Park et al., 2013; Wai et al., 2010). ... Presenting students with an educational curriculum at the depth and pace with which they assimilate new knowledge is beneficial. Other studies have shown that academic acceleration tends to enhance professional and creative achievements before age 50 (Park et al., 2013; Wai et al., 2010)."

Numerous other studies on the long-term effects of educational acceleration have drawn similar conclusions. As Wai (2015) summarizes:

"...[F]or many decades there has been a large body of empirical work supporting educational acceleration for talented youths (Colangelo & Davis, 2003; Lubinski & Benbow, 2000; VanTassel-Baska, 1998). Although neglecting this evidence seems increasingly harder to do (Ceci, 2000; Stanley, 2000), putting research into practice has been challenging due to social and political forces surrounding educational policy and implementation (Benbow & Stanley, 1996; Gallagher, 2004; Stanley, 2000).
...
The educational implications of these studies are quite clear. They collectively show that the various forms of educational acceleration have a positive impact. The key is appropriate developmental placement (Lubinski & Benbow, 2000) both academically and socially. ... Educational acceleration is essentially appropriate pacing and placement that ensures advanced students are engaged in learning for life. Every student deserves to learn something new each day (Stanley, 2000). The evidence clearly supports allowing students who desire to be accelerated to do so, and does not support holding them back.
...
[T]he long-term studies reviewed here show that adults who had been accelerated in school achieved greater educational and occupational success and were satisfied with their choices and the impact of those choices in other areas of their lives."

As researcher James Borland (1989, pp.185) sums it up:

"Acceleration is one of the most curious phenomena in the field of education. I can think of no other issue in which there is such a gulf between what research has revealed and what most practitioners believe. The research on acceleration is so uniformly positive, the benefits of appropriate acceleration so unequivocal, that it is difficult to see how an educator could oppose it."

But Why Does the Myth of Developmental Inappropriateness Persist?

At this point, you might wonder why so many people think that academic acceleration is developmentally inappropriate given all the research to the contrary. What’s going on? Do they disagree with the research? Are they simply unaware?

It becomes pretty clear if you look at the incentives.

Acceleration requires extra work, but people typically don’t like to do extra work, so they will gladly rationalize that the extra work wouldn’t have really helped them anyway, even if their rationale is incorrect.

Now, you might ask, what about schools? Isn’t it in their job description to get their students to work? If accelerating their capable students will lead to beneficial outcomes, wouldn’t they be pushing it, or at least, not discouraging it?

The problem is that acceleration is also very inconvenient to schools.

In schools, each grade typically progresses through the math curriculum in lockstep, which means that accelerated students would need to be placed in above-grade courses. This can lead to major logistical challenges.

For instance, if above-grade course is not offered by the school (which would certainly be the case for accelerated 5th graders in elementary schools, 8th graders in middle schools, and 12th graders in high schools), then either

the students would need to take the class at another school (which introduces transportation, scheduling, and administrative issues) or
the school would need to hire a teacher who is capable of teaching the higher-grade material (and it's hard enough for schools to hire teachers who are capable of teaching grade-level mathematics).

And even if the above-grade course is offered by the school, there may be schedule conflicts with grade-level courses that mathematically accelerated students still need to take. (Course schedules are typically optimized to minimize conflicts within grade levels, but not across grade levels.)

Besides logistical issues, there are other factors that can disincentivize acceleration and lead the myth to be perpetuated out of convenience. As Steenbergen-Hu, Makel, & Olszewski-Kubilius (2016) describe:

"[E]ducation administrators may have perverse incentives to avoid acceleration. For example, although acceleration can often actually save schools money because students spend fewer years in school, it can also 'cost' schools money. Because school funding is often allocated based on headcounts and accelerated students spend fewer years in school, schools receive fewer dollars overall, or in the case of dual enrollment, may have to spend some of those dollars outside the district.

Similarly, in states that offer open enrollment, students could leave a district for one where their needs are better met. Moreover, in the age of accountability via test score performance, keeping students who could be accelerated with their same-age peers can boost average test scores, regardless of whether the students are learning."

Even in schools that do offer acceleration, typically only a small portion of students per grade are accelerated. Given how many logistical challenges and other disincentivizing factors there are, how few students are typically accelerated, and how easy it is to imagine a young student struggling socially when they are placed in a class with older students away from age-level friends, it is not surprising that the myth persists.

Some Clarifications in Response to Follow-Up Questions

I received some great follow-up questions about this post. Below are some points I’d like to clarify. Feel free to contact me if you have any additional questions that aren’t addressed here.

Won’t students lose their minds from being so far ahead that they have nothing to do in class? Sounds like you have to go to a specialized school for this to work.

Personally, learning advanced math ahead of time transformed my entire life for the better, and I didn’t go to a specialized school. I went to normal elementary, middle, and high schools and self-studied on my own. I often self-studied during normal classtime, but on the sly, hiding the fact that I was reading and working out problems from university math/physics books of paying attention during class.

When I got to college I placed out of a ton of math courses – more than you could place out of via a placement exam. I did this by taking initiative to reach out to the head of the math department. They worked with me as a “special case” student. My first year, courses I took included metric spaces / real analysis, abstract linear algebra, and topology. I took some grad courses my second year.

My second year was also when I fully lost interest in academia and started working as a data scientist, which I continued full-time throughout the rest of college while taking a minimum courseload. But if I had wanted to pursue math further, then there was still plenty of road ahead – I could have kept on taking grad courses and done some math research, and I could have graduated a year or two early.

References

Bernstein, B. O., Lubinski, D., & Benbow, C. P. (2021). Academic acceleration in gifted youth and fruitless concerns regarding psychological well-being: A 35-year longitudinal study. Journal of Educational Psychology, 113(4), 830.

Borland, J. H. (1989). Planning and Implementing Programs for the Gifted. New York: Teachers College Press.

Park, G., Lubinski, D., & Benbow, C. P. (2013). When less is more: Effects of grade skipping on adult STEM productivity among mathematically precocious adolescents. Journal of Educational Psychology, 105(1), 176.

Steenbergen-Hu, S., Makel, M. C., & Olszewski-Kubilius, P. (2016). What one hundred years of research says about the effects of ability grouping and acceleration on K–12 students’ academic achievement: Findings of two second-order meta-analyses. Review of Educational Research, 86(4), 849-899.

Wai, J. (2015). Long-term effects of educational acceleration. A nation empowered: Evidence trumps the excuses holding back America’s brightest students, 2, 73-83.

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