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Consider this elementary problem:

Define an operation $*$ between integers as follows: $a*b=ab-a+b$. Solve the equation $4*x=36$.

If we give this problem to Pre-Calculus and Calculus III students (assuming that none of them have seen this type of problems before) I think Calculus III students are more likely to solve it (you may disagree, but that can be tested). If we accept that Calculus III students are more likely to solve this problem, it means Calculus III students have developed "certain ability" as a result of taking two semesters of Calculus. What would you call this "certain ability"?

As I created this question for a certain purpose, I do have a name for this ability, but I am curious what others would call it.

Edit 1 (added later). A number of responses have questioned my statement that the said "ability" of Calculus III students was developed as a result of taking two semesters of Calculus. That's fine, let's not assume that it was a result of taking two semesters of Calculus. But still, what would you call this "certain ability", regardless of how it was developed?

Furthermore, one can perform this experiment slightly differently. One way would be to perform it every semester, varying the pedagogy used to teach the Calculus sequence, and then observe the change (if any) in results, over various semesters.

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    $\begingroup$ Perhaps "mathematical maturity". $\endgroup$
    – Sue VanHattum
    Oct 16, 2022 at 2:34
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    $\begingroup$ "it means Calculus III students have developed 'certain ability' as a result of taking two semesters of Calculus": How would you demonstrate causality? Only a fraction of Pre-Calculus students make it all the way to Calculus III. As a result, the Calculus III group is heavily selected. This includes both self selection--students enrolled in Calculus III will tend to have interest and affinity for STEM subjects--and also the selection effect of having succeeded in the prerequisite courses. $\endgroup$ Oct 16, 2022 at 4:51
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    $\begingroup$ Is Calculus 3 a course one takes at a specific year of a specific study program and with specific contents that we should be aware of? $\endgroup$
    – Tommi
    Oct 16, 2022 at 12:03
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    $\begingroup$ @WillOrrick do we need to demonstrate causality? Perhaps OP would be satisfied with correlation: i.e. that students selected for Cal3 have said ability (and Precal not) regardless of whether or not the two preceding semesters of Calculus "caused" this ability? $\endgroup$
    – Him
    Oct 16, 2022 at 22:54
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    $\begingroup$ It is worth noting that in some educational systems (I have no idea which this is) definitions like * will be routine in earlier years. When I was 12/13 we had numerous invented algebraic operation type questions. So you might want to check that. $\endgroup$ Oct 17, 2022 at 4:28

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Great question! I suspect "the ability to accept new definitions and work with them" although perhaps it's more specifically "the ability to accept new definitions without knowing what they mean and work with them symbolically". Students get lots of practice of both in the calculus sequence.

That's not to speak ill of the practice of working with definitions symbolically without knowing what they mean: this was a vital part of breakthroughs on the part of (among others) Hilbert and Gödel.

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    $\begingroup$ @JochenGlueck I think a lot of people find it difficult (or at least uncomfortable) to work with pure abstract definitions like that. Rather than doing so they use the definition to build an intuition about what they are doing, and reason more with the intuitions than the mathematics. Speaking for myself at least, I think it was quite a while into my education before I really "clicked' with the idea of using arbitrary definitions rigidly and actively trying to keep intuitions about what is "behind the maths" out of my reasoning. $\endgroup$
    – Ben
    Oct 16, 2022 at 14:27
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    $\begingroup$ @Ben: Hmm, now that I read your comment once again, I would also be interested to better understand what about the definition "$a \star b := ab - a + b$" you consider as "pure[ly] abstract". From my point of view, this is as concrete as it can possibly get: a function that is defined by an explicit (and simple) formula. $\endgroup$ Oct 16, 2022 at 16:10
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    $\begingroup$ @MahdiMajidi-Zolbanin: But that's actually an interesting point. Next time I'll teach a first semester course, I'll try this (assuming that I remember to do so) and see whether what I said here is correct. (If it's not, then the problems in math education are even worse than I perceive them to be right now.) $\endgroup$ Oct 16, 2022 at 16:37
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    $\begingroup$ “Lots of practice performing symbolic manipulation” was the explicit justification a professor gave me for the continuing requirement for calculus 2. $\endgroup$
    – KRyan
    Oct 16, 2022 at 17:22
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    $\begingroup$ @JochenGlueck: Well, I'm not Ben, but I similarly view this definition as "purely abstract," because it is not obvious (to me) why you would define the symbol in that way, or what the symbol is "trying" to accomplish. This is the same reason that students struggle with group theory: All of the definitions are perfectly clear, but unless you actually tell them about symmetry and group actions, it all just looks like pure abstract nonsense. (I managed to make it all the way to a bachelor's without a single person ever telling me that matrices represent linear transformations of the plane!) $\endgroup$
    – Kevin
    Oct 16, 2022 at 18:30
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In your proposed scenario of testing pre-calculus students as well as calculus III students, you seem to be assuming that the calculus III students are, overall, equivalent in their pre-existing level of mathematical aptitude/interest to the pre-calculus students, but having received additional education and being a couple years older. However, in real life, it would almost never be the case that the two groups are equivalent in their overall level of pre-existing math aptitude. Here are a couple reasons why:

  • Culling Effect: Many students struggle with calculus I and or II, and will therefore not be able to pass the courses that are pre-requisites for calculus III, and/or may elect to drop out of the sequence if it's not necessary for their degree. These will tend to be studentss with lower math aptitude, meaning that the students remaining in the sequence (who do make it to calculus III) are, on average, a higher-aptitude subset of the original co-hort of pre-calculus students.

  • Self-Selection: Pre-calculus is often a requirement for students to take, even for non-STEM fields of study. Calculus I and II are sometimes (but less often) a requirement. It is even less common for calculus III to be required, especially for a non-STEM degree. The students taking calculus III therefore will tend to be students who are required to take it as part of their STEM degree, and not students majoring in, say, English or history. These students will usually have a higher math aptitude than students from the original pre-calculus cohort, as the calculus III students chose a STEM field of study partially because they were good at math and found their success in math courses to be rewarding. Therefore, many students who were in the original pre-calculus cohort who were stronger or more interested in another area (compared to math), may have been likely to choose a different field of study and therefore stopped taking math classes when they fulfilled their requirements. If calculus III was not required for them, then they never ended up taking it.

Trying to draw conclusions by comparing these two groups would be a form of selection bias, as you are comparing two groups that have not been selected in the same way. The calculus III students are a subset of the original pre-calculus cohort that are enriched in math aptitude and interest. This may be comparable to what's known as survivorship bias, in that you are considering the calculus III students without including their peers from pre-calculus who didn't "survive" within the sequence up to the level of calculus III.

Here's a pictorial representation of what I'm talking about (each colored circle represents one student):

Hypothetical Scenario

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    $\begingroup$ @MahdiMajidi-Zolbanin When I say "mathematical aptitude," I mean an inherent mathemtical ability. Part of this may be inborn (i.e., present at the time of birth), and part of it may be modifiable in the early years of life (often called a "critical period") based on education and educational resources available to the child. But there comes a point when a person's general ability to understand mathemetical concepts is somewhat fixed, and even with the same education/training, some people will be better at math than others. Also, see the image I added to the post. $\endgroup$
    – zunojeef
    Oct 17, 2022 at 0:04
  • $\begingroup$ @MahdiMajidi-Zolbanin I should add that I don't think the difference in students' ability to solve the given problem is 100% due to differences in inherent ability or aptitude, but aptitude is a major factor. There is also an effect of training/education, and of time "practicing" math and allowing concepts to "incubate" in the mind and become more fully integrated with one's existing knowledge. If I were designing a study to try to tease apart these two factors (inherent aptitude versus the effect of training/time/experience), I would do so as follows: (continued) $\endgroup$
    – zunojeef
    Oct 17, 2022 at 0:30
  • $\begingroup$ Study Design: First, create 20 questions roughly similar difficulty that test mathematical ability from a non-standard perspective or in some sort of creative way, like the question you included in your post. Randomly divide these questions into two groups of 10 quetsions each, called Questions Set A and Questions Set B. Select a group of pre-calculus students as study subjects (ideally, students in different schools who don't know each other). Half of the students receive Question Set A and half of them receive Question Set B (randomly assigned). (continued) $\endgroup$
    – zunojeef
    Oct 17, 2022 at 0:39
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$
    – zunojeef
    Oct 17, 2022 at 0:39
  • $\begingroup$ This is a much better answer than the accepted one. $\endgroup$
    – Reid
    Oct 17, 2022 at 15:06
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I'm not sure if there's one single "certain ability" that makes students able to solve this problem. In order to solve this problem, a student should:

  • Know what it means to "define an operation between integers."
  • Know what an equation like $a * b = a b - a + b$ means when it's interpreted as a definition.
  • Recognize that it is both permissible and useful to apply the substitution property to this situation.
  • Know how to apply the substitution property to this situation.
  • Recognize that, after the substitution property has been applied, the answer to the new problem is the answer to the original problem.

These are all distinct pieces of knowledge, and it's possible for a student to possess any four of these but not possess the fifth.

That said, these pieces of knowledge are all fundamental facts about elementary algebra. (Of course, "fundamental" doesn't mean "easy to learn"!) So I think I'd summarize these as something like "a solid understanding of the fundamentals of elementary algebra."


In response to your comment:

If you were to put all the 5 bullet points in a box and give it a label, what would you write on that label? "A solid understanding of the fundamentals of elementary algebra" doesn't seem to capture the concept, because as far as I know, in elementary algebra they don't teach equations like $4∗x=36$.

That's a great question, and I'm not really sure.

Let me start by clarifying that when I say "elementary algebra," I mean "manipulating expressions and equations involving numbers, and especially solving equations"—in other words, the area of math that's usually just called "algebra" in the context of secondary education.

Next, let me pretend for a moment that the problem that you wrote was:

Define a function $f$ on integers as follows: $f(a, b) = a b − a + b$. Solve the equation $f(4, x) = 36$.

Now that I think about it, I think that there is one particular skill that a student needs in order to solve this problem. If I had to give a name to that skill, I think I would call it familiarity with functions. This skill consists essentially of the pieces of knowledge I listed above: knowing what it means to define a function; knowing what an equation means when it's interpreted as a function definition; and knowing how to use the substitution property with a function definition.

In order to solve the problem you originally described, a student needs to have familiarity with functions, and also needs to know that "define an operation $a * b$" means exactly the same thing as "define a function $f(a, b)$" (aside from the difference in notation).

(If I may ramble for a few moments, I think that there are two important "levels" of familiarity with functions. The first level is what I described above—it's the level where a student is capable of reading "$f(4, x) = 36$" as meaning "$a = 4$, $b = x$, $y = 36$." The second level is where a student is capable of treating a function as an independent object in its own right, and solving problems that involve multiple values of the same function without getting confused.)

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I hypothesize this is due to notation, particularly formula notation: Calculus III students have been exposed to much more sophisticated notation than students in the novice courses. Calculus III students have seen and used notations like multiple integrals, higher order derivative notations, vector notation, cross product notation, gradients, etc. This work develops a certain comfort level with notation that earlier students are less likely to have. Most importantly, Calculus III students are used to dealing with formula notation, like cross product formula, chain rule, divergence theorem etc. and know how to apply them. This skill of application of formula is the key cognition involved in solving this. Also, I have seen questions like this while tutoring for standardized tests like the SATs, and it is mostly the students who are uncomfortable with formula notation that struggle with these types of 'fictious operator' problems.

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    $\begingroup$ +1 If you ask the average person what math is, they will say something about numbers. The average Pre-calc student will as well. I think Calculus III or maybe some college level physics classes are the first place that you do algebra with objects that are explicitly not numbers and don't follow the rules of numbers (cross products don't commute). $\endgroup$ Oct 17, 2022 at 14:37
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To me this seems much more straightforward than the question and other answers suggest. I would describe the problem itself as "abstract algebra", with some overlap with linear algebra. In any case, I don't see knowledge of calculus itself as relevant to the problem.

However, for students to progress to calculus III, they must have either had a solid foundation in algebra before starting calculus, or they must have been able to fill in their knowledge of algebra to succeed in calculus I and II. Personally, I understood algebra a lot better after taking a year of calculus.

In short, my answer to the question is:

The calculus III students are better at algebra than the pre-calculus students.

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Intelligence. With familiarity with math being an additional factor.

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