In part 9 of this series on Math Academy I provided some feedback on those aspects of Math Academy unrelated to actually learning mathematics. In this post I express my opinions on the educational experience itself, as an active student in a Math Academy course. Again to keep things semi-organized, I’ll divide my comments up into multiple topics.
Inital course selection
As I’ve mentioned many times previously, my goal is to learn what an eigenvector is, and for that I’ll need to take the Linear Algebra course. However I was not so deluded as to think I could just jump into the course, since it had been a while since I’d seriously studied math.
I therefore elected to start with one of the Mathematical Foundations courses recommended for adult learners. Unfortunately my delusion, while not severe enough for me to start the Linear Algebra course right away, was severe enough to make me think I just needed to take the Mathematical Foundations III course, the immediate prerequisite for the Linear Algebra course.
That was a major mistake, as discussed in the next section. I did a cursory check of the “Overview,” “Outcomes,” and “Contents” sections in the Mathematical Foundations III course description. However, in retrospect it would have been nice to have some additional information that I could have used to make an initial assessment of course appropriateness even before taking the diagnostic exam.
For example, suppose that the Mathematical Foundations III course description had had a minimal discussion of prerequisites, like “Among other things, this course assumes that you know how to factor polynomials using synthetic division, how to use trigometric identities like the law of sines, and how to differentiate the product or quotient of two functions.”
Then I would have immediately known that the Mathematical Foundations III course was too advanced for me, and I should take the diagnostic exam for the Mathematical Foundations II course instead. (Yes, I could have also just looked at the content for Mathematical Foundations II to see what that course taught, but I was being lazy and didn’t think to do that.)
The diagnostic exam
Instead I started taking the diagnostic exam for the Mathematical Foundations III course. It was a brutal and dispiriting experience, as I found myself being able to answer only a small fraction of the questions correctly. For most questions I didn’t even know how to go about finding an answer.
At this point I was internally shouting, “Stop! Let me cancel this exam and go back to square one!” However, I didn’t see any obvious way to stop the exam and switch to the diagnostic assessment for Mathematical Foundations II. So I soldiered on, finished the exam, and then was thrown into the beginning of Mathematical Foundations III.
As with the diagnostic exam, I just wanted to stop, go back, and try the Mathematical Foundations II course instead. Unfortunately, I couldn’t figure out a way to do that from the main course screen. It wasn’t until I got an email announcing the results of the diagnostic exam that I was enlightened: you can change courses by clicking on the profile icon in the upper right, selecting “settings,” and then selecting “Course” from the “Settings” page.
I then took the diagnostic exam for Mathematical Foundations II, did much better, and was soon embarked on my math learning journey.
Lessons and questions
Once I started the Mathematical Foundations II course, I found I really liked the way Math Academy organizes lessons and the accompanying questions. The lessons are relatively short, so I can do them whenever I feel like; I’ve typically been doing a couple in the morning before breakfast, a couple during lunch, and then more in the evening after dinner.
I originally planned to accumulate 50 XP a day, seven days a week. (I thought it important to do some work every day to not get out of the habit.) As it happened, I did over 90 XP every day for the first few weeks before cutting back (as discussed below).
I did more than anticipated for two reasons. First, I got a little competitive when looking at the league standings; more on this below. I also found doing the lessons and exercises to be a fun alternative to scrolling through social media. You can get a good sense of my progress by looking at the activity log generated by the Math Academy system; it covers the time from when I started until just after I completed the Mathematical Foundations II course, almost a month in total. (However, as noted below, I came to rethink my practice of overdoing things like this.)
My working method when doing lessons has been to read through the first example in each lesson, and then to try to work through the second example myself before reading the explanation for it. I found this to be a useful low-stress way to prepare for the first question following the examples. The questions themselves are relatively straightforward to answer if I understand the examples. I always try to work through the question first before looking at the multiple choice answers.
This generally works well: Either I get the answer right or I screw up something, choose the wrong answer, and get corrected. However, there have been occasions when I got an initial answer, didn’t see it listed among the five choices, and went back to try to understand what I did wrong. Some may consider this a bit of a cheat, but it’s inherent to a system organized around multiple choice answers.
(The questions for which you need to type in something avoid this problem, but as noted in my previous post they have their own problems when trying to access the system on a tablet. Also, in cases where there are multiple values in the answer—e.g., you need to find two values a and b, instead of a single value a—the Math Academy system tries to mitigate the problem of students just looking at the multiple choices and then trying to verify each one. It does this by requiring the student to enter or choose a sum or product of the answer values—e.g., a + b—rather than presenting the values themselves in the multiple choices.)
The questions are generally variations on a theme—sometimes very minor variations. Given that, I can see why some people would hate hate hate this way of instruction. There seems to be a macho attitude on the part of some students that learning mathematics is not supposed to be easy, that one has to conquer the exercises in textbooks like “Baby Rudin” and “Papa Rudin” before one can consider oneself mathematically knowledgeable. It’s like a hazing ritual, in which each generation has to suffer in order for the prior generation to consider their own suffering justified and worthwhile. The people who created the Math Academy clearly do not share this attitude, and I for one am very glad of it.
When I answered a question incorrectly, most of the time it was because I messed up something in working the problem: I misread the question, made a sign error, or screwed up when copying a term from step to step. The diagnostic exam has a “I made a silly mistake” option that allows you to retry a question, but the regular course questions do not. I think this is the correct approach: if you make silly mistakes on a regular basis then your goal should be to train yourself to be more careful in working a problem, including checking the result.
I don’t guess at answers, but there have been times when I accidentally got the correct answer: I didn’t quite know how to work the problem, but I stumbled on the right answer anyway. There’s no option to tell the system “I got this right, but I really didn’t know how to do it.” I originally thought it might be a good option to add, if for no other reason than to get a needed extra review, but this happens so infrequently that I doubt it’s worth it.
Reviews
I’ve previously tried out spaced repetition systems, most notably Anki, for learning various topics. I soon found myself overwhelmed by the sheer number of items I needed to review each day, a number so large that I eventually gave up in despair—it was sheer tedium to work through them all, especially when first learning the information.
Fortunately, Math Academy does not have that problem. On average I had to do less than three reviews of topics each day; a few days there were no reviews at all, and on a couple days there were many as six to eight. Each review topic had just a few questions—plus I’ve noted that the review ends early if you answer the first three questions correctly.
I consider that amount of review quite reasonable. The Math Academy Way claims that the Math Academy system leverages the hierarchical nature of mathematics to reduce the amount of needed review. In my experience, that claim is justified.
However, there are cases where I might in fact like to have more reviews, namely when it comes to learning sets of related facts. A good example is the set of standard angles on the unit circle and the values of trigonometric functions for those angles: the sine, cosine and tangent of 30 degrees (π/6), the sine, cosine, and tangent of 45 degrees (π/4), and so on around the circle. I found myself unable to recall all of those values instantly, and had to resort to calculating them in some cases. (For example, the sine, cosine, and tangent of 150 degrees can be easily derived from the values for 30 degrees.)
I’d rather not slow myself down by having to derive some of these on the fly. I’d rather be able to recall them instantly, just as I can instantly recall that 4 * 5 = 20. I could certainly put those facts into Anki or a similar spaced repetition system, but it might be nice to have the Math Academy system allow for special reviews of facts like that. (It appears that Math Academy may be planning something along this line, based on tweets from some of its employees.)
Quizzes
I had to do less than one quiz a day, of which some were retakes due to my not doing so well the first time. As with reviews, I consider this an acceptable number.
I found the quizzes themselves to be a bit stressful due to the time limit. On several occasions I had to leave one or two questions unanswered. On other occasions I rushed through problems and didn’t check them properly. As as result I scored lower than I would have liked several times, with the system reacting by having me take the same quiz again. Of the 15 unique quizzes I was presented during the Mathematical Foundations II course, I had to retake five.
Despite raising my stress level a bit, I thought the quizzes were useful and reasonable. I think the key to doing well on them is to develop “automaticity” in the sense used in The Math Academy Way, so that you can solve the presented problems without having to think too much about exactly how they should be solved.
Leagues and leaderboards
The leagues and associated leaderboards are an optional feature of the Math Academy system. I left them turned on initially because I wanted to see how they worked and how I compared with others. I was able to quickly advance through the lower leagues, often being the high scorer within my group. When this series started I was in the Platinum League (the middle league in the list of league ranked by exclusivity) and was about to be promoted into the Sapphire League. I found this aspect of the system to be fun and motivating, at least for a time—often I would do a lesson or two more just to improve my standings within my league.
Speaking of motivation: There isn’t much reward within the system itself for ranking high within your league, or for being promoted from one league to another. The period of competition simply ends and then you’re in a new league. It might be nice to have even a simple “Congratulations on being promoted!” message, or a “Hooray! You ranked first in your group!” Maybe the Math Academy folks think such things are unneeded or even undesirable, or maybe they just haven’t gotten around to implementing frills like this.
The flip side of the league and leaderboard gamification is that it encourages people to pile up XP simply for the sake of advancing in the leagues. You see this phenomenon on X as well, with people posting their cumulative XP. I don’t really need to be chasing the 100 XP a day mark in order to meet my own learning goals: 50 XP a day would have me completing the Mathematical Foundations III course by July, and then I’d have the rest of the year to finish the Linear Algebra course.
So, after I got promoted into the Sapphire League, I turned off the league feature. I’ve settled into a routine I can maintain, no longer have the distraction of invidious comparison with others competing for promotion, and have recovered some of my spare time for other things I enjoy.
Measuring course progress
Now that I’ve moved on to the Mathematical Foundations III course, one source of frustration to me is figuring out how I’m progressing within the course relative to my goal date for finishing it. (Note: The numbers I quote in this section are from when I first wrote it, a few days before publishing this post. I’ve progressed further since then.)
In the “XP Goals” section of the “Settings” section of the website, I’ve told the system that my goal is to do at least 50 XP each and every day of the week. The system then tells me on that same pqge that “At a pace of 350 XP per week, it’s estimated that Mathematical Foundations III will be completed by late May.”
However, if I go to the main learning screen, where the lessons are, it tells me that I am 24% through Mathematical Foundations III and that “Estimated completion is mid-July.” If I hover the cursor over that statement, it expands into a subwindow claiming that at a (boldfaced) pace of 49 XP I will indeed finish in mid-July. And if I hover over the ”24%” in a circle (next to where the page displays the course title), that subwindow also claims my pace is 49 XP a day. It further claims that I started the course on February 6 (correct), have a goal of ending the course on June 30 (also correct), and have done 691 XP in the course thus far (also correct as far as I know, from looking at my activity log).
Then it goes on to tell me that my “expected” progress should be 1260 XP, and that I’m therefore 569 XP behind schedule. It also tells me that the “total days” is 26 (with a parenthetical comment, “non-holiday weekdays”.)
This is all very confusing. Where does this figure of 49 XP come from? If I look at the activity log for the 8 days that I’ve been working on Mathematical Foundations III, I’ve been averaging 67 XP a day. And where does the “26 days” come from? There are a lot more than 26 days between now and June 30, about 50 all told if my calculations are correct, not even counting weekends and holidays.
(As an aside, The Math Academy Way assumes a 5-day week in its examples of learning speed and related topics. I’m wondering if that’s carried over into the Math Academy system, even in the case where a student—namely me—has indicated their intention to study every day.)
I’m just asking for a reasonable estimate of when I’ll be able to complete Mathematical Foundations III, and what pace I’ll need to maintain in order to make my goal date. Right now I’m not getting that, and I have limited trust in the numbers that the Math Academy web interface is showing me.
Course completion and results
I finished the Mathematical Foundations II course as my first month with Math Academy was nearing its end. As with being promoted to a higher league, this was a non-event as far as the user interface was concerned: I simply looked at the page one day and noticed that it was now showing my being in the Mathematical Foundations III course. (This was presumably because I had enrolled in Mathematical Foundations III originally, and so the course was in effect waiting for me to finish Mathematical Foundations II.)
The transition between courses was about as exciting as watching your car’s odometer roll over at the next 10,000 or 100,000 mark—especially when you get distracted and miss the exact time when it rolls over. I put many hours into working through all the course lessons, and it would be nice if the system would take note of that.
However, I should note that I did get a transcript and a certificate of completion for the Mathematical Foundations II course, as well as documents providing a course overview, a detailed list of course contents, and the accreditation letter for Math Academy itself. (Note to Math Academy folks: the course overview document has a typo: “Upon completeing . . .”)
These aren’t much use to me in my current situation (except to show skeptics that I really did use Math Academy before reviewing it), but they’re nice to have. I guess someone could use the certificates and transcripts to show a potential employer the level of mathematics they have learned (assuming the job requires using math as a core component). But, if I were an employer, could I really trust such evidence? After all, it’s possible that the student just breezed through the courses using an LLM to come up with answers.
Speaking of LLMs, that’s one of the topics I’ll be discussing in part 11 of this series, in which I offer my final thoughts on Math Academy.