What is “mastery” in a mathematical topic?

This question was prompted by looking at Khan Academy's website to see how a comprehensive lecture series could be done and often I see the word, "mastery". To me, I'd think mastery is something like "being an expert in", and that in turn means "being extremely capable and knowledgeable"

However, when I look at what "mastery" means to Khan Academy, it seems to be completion of various topics and knowing computation. One of Khan's quotes in a Ted talk is, "Due to the constraints of time and curriculum a student may get only 60% right, or 80% right or even 95%... so it creates a Swiss cheese gap in the foundation of your understanding of that subject. " Also, he often gives the analogy of a house being built and how if, for the second story not to fall, it needs a solid first story, and that needs a solid foundation.

I'm probably being too nit-picky with word choice, but the more I look into the website, it seems that this is more "Completion Learning" rather than "Mastery Learning" because it not based on mathematical proof and rigor. Or, if it is mastery learning, it's mastery in the sense of computation and calculation of the basic ideas, but not of where those ideas come from. (And, btw, I'm not against computation and calculation but if that is what was meant then I'd rather have more accurate wording)

If I look at one of his proof videos (which would be more of what I'd think "mastery" is), like his video of "formal definition of limits - Part 4", from what I can tell, the algebraic manipulation seems to go the "wrong way" in terms of finding the $$\epsilon-\delta$$ relationship, and the final $$\left|x-a\right|<\delta$$ without the $$0<$$ on the left side. My rhetorical questions are "does that video show mastery from the teacher?" and "would a student master that topic based on that video?"

--

If I teach a topic, then how do I know if I'm teaching "mastery" of that topic? (or am I more honest in saying "computation" and "calculation" of a topic) Or if I see a student's work, how would I know if they have mastered the topic's concepts and logic, or are just repeating "plug'n'chug" math with a sophisticated kind of rote memorization?

• See my answer here. – Joel Reyes Noche Jul 9 at 14:06
• You might also want to look at the Wikipedia entry for mastery learning. – Joel Reyes Noche Jul 9 at 14:27
• "If I teach a topic, then how do I know if I'm teaching "mastery" of that topic?" — I'd say mastery is what one can acquire, not what can be taught. – Rusty Core Jul 9 at 19:00
• I doubt such a concept has a precise, useful meaning. It cannot mean something like "perfection" or "can perform operation X correctly under any circumstances", for under such a definition, even the best mathematicians haven't mastered basic addition. – YiFan Jul 10 at 8:59
• It means that they want to sell you a course – David Jul 13 at 11:54

2 Answers

1. I don't think fighting for the definition of a word makes most sense. After all, we could consider Robbie mastery versus Kahn Academy mastery (as different attributes).

2. For what it's worth, the Kahn Academy sense of mastery is a rather normal one. The idea comes from the theory of building block automaticity. That you have skills down to the sense that you can use them without struggle. Our minds are not silicon, they have very limited RAM. So if you struggle on basic reading, it becomes a chore to do the reading and then do higher processing. Similarly, if you struggle on basic manipulations, than new concepts in physics or math are hard to work on. Consider the issues for a kid working in calculus who is weak in trig or doesn't have an iconic recall of (-b +/-sqrt(bsq-4ac)/2a). [Own goal on me, if I flubbed that formula!]

Here is an example of Richard Feynman discussing mastery in the sense of manipulation skill:

So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review. I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”

Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.

What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.

Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors. Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!

1. I would also suggest that instead of debating the usage of the term mastery, you could also differentiate the subject. In other words, if I master Thomas Finney calculus. (I literally and not how millennials use that word did every homework problem in that text and scored perfect tests throughout high school and as far as I could tell on the AP exam.) Now I would say that was automaticity and ease of use and Kahn-Feymnanian mastery OF the 1980s AP calculus class. Of course if you want to say there should be some OTHER calculus course (with a bunch of real analysis thrown into the frosh calc course instead of waiting until junior year), then fine. But you're really debating the subject, not the concept of mastery. After all mastering high school geometry certainly involved lots of proofs since that course was mostly proving geometric propositions (was actually weak on mensuration practice, I thought.)

2. Also, I think there are some reasonable differences in the depth of mastery expected of a student and of a teacher. Both should have acquired automaticity at doing the homework drill problems. But the teacher actually benefits from knowing things not really even PART of the course for the student. Advanced courses that use the skills from the course in discussion. The history of the development of the subject. And applications. But it's not reasonable to expect the same thing of the student. He just needs to master the subject itself. Not a bunch of related matters outside the scope.

3. I don't think nitpicking a mistake in one of his videos is relevant to the concept of student mastery and/or the preferred definition/usage of the term. They are really not connected.

• btw, the numbers organize my replies; they're not referring to yours (1) This is a great answer even if I disagree with it. At the heart of the disagreement is that defining words is important since it leads to what is the expected outcome of the work of teacher and student, If I say you're going to learn "algebra" and expectation is learning how graphing linear equations then you'll be confused when you're introduced to groups, rings, and fields. "Algebra" has different definitions and you'd only know which one unless it's defined or if you can see examples that determine which one it is. – Robbie_P Jul 12 at 2:41
• (2) I have learned, from this question, that "mastery" (in terms of mastery learning) seems to be more about practice and knowledge, rather than having very deep understanding of the nuances, caveats, and pitfalls of the theory. To me, the latter is "mastery", and if it's not then what would be the word to describe that? I don't have anything against practice and knowledge, but I simply call that what it is: "practice and knowledge"; I wouldn't call it "mastery". But it seems that might just be personal taste in word choice. – Robbie_P Jul 12 at 2:44
• (3) I do take a major issue with your number 5. I don't see how a student's mastery isn't fundamentally linked to a teacher's mistakes. If "mastery learning" is about practice and knowledge, when what happens when it's taught incorrectly? If a teacher makes a mistake, then it seems that the student would just repeat those same mistakes in their practice. As for knowledge, wouldn't it show that what they're learning is incorrect? – Robbie_P Jul 12 at 2:51
• (4) Consider that university profs usually have a PhD is because they need to know their subjects quite extensively. Although they may or may not be the best teachers, I think it's safe to say they are experts/masters in their subject. Of course, students would need to practice on their own and learn knowledge from their profs, but it seems that the quality of that learning is partially based on how much their educator knows, especially when it comes to fundamental concepts. If fundamental concepts are taught incorrectly, what kind of standards are we setting for educators and students? – Robbie_P Jul 12 at 2:59

The Kumon Math Program defines mastery as

The ability to apply a skill or concept with the accuracy, speed and confidence to demonstrate a total command of the material. Mastery means that a skill or concept has become second nature and will be permanently retained. Mastery is the prerequisite for advancement and the goal of Kumon study. [source]

Also,

Speed + Accuracy = Mastery
Material should be completed with a perfect score within a prescribed period of time. [source]

This "prescribed period of time" is called the Standard Completion Time, which is

A prescribed number of minutes to complete a particular worksheet with a perfect (or near perfect) score. Kumon determines-and continually reevaluates-SCT based on the actual performance of tens of thousands of students. Successfully completing worksheets within the SCT demonstrates that the student has mastered the material and is ready to move on. [source]

In practice, "near perfect" usually means "at least 90%."

• This is the "East Asian" concept of "mastery," which is quite different from the "Western" concept. – Joel Reyes Noche Jul 9 at 14:22
• Thanks for the links. My primary question is which concept of "mastery" is more likely to show expertise of a topic. I tend to criticism from the [Wikipedia page][2]: "This format of learning may only be beneficial to learners who are interested in surface rather than deep processing of information. This contradicts many of today's modern learning approaches which focus less on direct assessment of knowledge, and more on creating meaningful applications and interpretations of the obtained knowledge." [2]: en.wikipedia.org/wiki/… – Robbie_P Jul 9 at 14:46
• Thanks for the comment. You might want to read Jon Star's "Reconceptualizing Procedural Knowledge," in particular, Table 1 and page 408: "Deep procedural knowledge would be knowledge of procedures that is associated with comprehension, flexibility, and critical judgment and that is distinct from (but possibly related to) knowledge of concepts. Separating these independent characteristics of knowledge (type versus quality) allows for the reconceptualization of procedural knowledge as potentially deep." – Joel Reyes Noche Jul 9 at 15:33