12

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.

Xander Henderson
  • 7,801
  • 1
  • 20
  • 57
Mahdi Majidi-Zolbanin
  • 1,102
  • 4
  • 17
  • 26
    Perhaps "mathematical maturity". – Sue VanHattum Oct 16 '22 at 02:34
  • 39
    "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. – Will Orrick Oct 16 '22 at 04:51
  • 7
    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? – Tommi Oct 16 '22 at 12:03
  • 1
    Under this specific formulation, with the use of the asterisk, I had to read this twice to figure out there was a redefinition of it from multiplication as usual into a formula for this question only. That might be an issue? – Malady Oct 16 '22 at 17:43
  • 5
    @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? – Him Oct 16 '22 at 22:54
  • 1
    As this question seems more to do with Algebra than Calculus, it might be worth considering the effect of studying Algera III. Perhaps the observation is related to how many Calculus III students have previously done Algebra II. – Peter Oct 17 '22 at 03:57
  • 6
    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. – Francis Davey Oct 17 '22 at 04:28
  • 2
    This kind of operation definition is also common in programming. – user10216038 Oct 17 '22 at 17:09
  • No. At the time I studied there was no national curriculum. they were set by examination boards. My board was one of the largest in the country and I suspect that most children 12/13 in roughly the North of England would have been exposed to some, or possibly many, problems like this. The curriculum was quite algebra heavy. By the time I got to university I was strong in algebra (including vector algebra) but weak in dynamics. – Francis Davey Oct 18 '22 at 15:00
  • 2
    This question can be rewritten to be asking about a pipe problem comparison between apprentice and journeyman plumbers and be just as valid. Math isn't a singular skill, it is a collection of skills. Like any skill, math skills require practice and interest. Intelligence, aptitude, maturity, or any other name will only modify the amount of practice and interest required to augment a skill. This appears to be a concept vs algorithm test, if they can identify related concepts and solve or rely on following learned steps to solve identifiable problems. – David S Oct 18 '22 at 16:54
  • 1
    Some people are just inherently better at math than others. Just as importantly, experience matters. That should be obvious. – RonJohn Oct 18 '22 at 17:38
  • People are manifestly NOT the same at different points of their lives and educational levels. That’s so obvious, this must be a troll question. – RonJohn Oct 19 '22 at 17:01
  • What happens to every young person? They mature physically and (hopefully) mentally while accumulating more knowledge and experience (even if that knowledge and experience are in video games or sports). – RonJohn Oct 19 '22 at 19:33
  • 1
    As the asterisk is often used in computers for describing a multiplication, I would opt for another operator, like $a \circ b$ or $a \square b$ or $a \oplus b$ or so. – Dominique Mar 20 '23 at 13:37

6 Answers6

35

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

Stev
  • 23
  • 4
zunojeef
  • 459
  • 2
  • 3
  • 1
    @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. – zunojeef Oct 17 '22 at 00:04
  • @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) – zunojeef Oct 17 '22 at 00:30
  • 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) – zunojeef Oct 17 '22 at 00:39
  • Let us continue this discussion in chat. – zunojeef Oct 17 '22 at 00:39
  • This is a much better answer than the accepted one. – Reid Oct 17 '22 at 15:06
  • The only thing I'd change about this answer is the number of "yellow balls" in Calc III. There would be zero IMNSHO. – RonJohn Oct 18 '22 at 17:35
  • @RonJohn, that might usually be the case. At my community college, we do get a few people who start out not good at math and improve enough over time to finish all our math courses. Maybe less than one a year, but I'd guess at least one every two years. – Sue VanHattum Mar 19 '23 at 16:51
  • @SueVanHattum does your CC teach Calc III? (The one near me tops out at Calc II.) – RonJohn Mar 19 '23 at 17:20
  • @RonJohn yes, and Differential Equations. Those are the last two. – Sue VanHattum Mar 19 '23 at 17:49
  • @SueVanHattum impressive. Of cvourse, my state, and it's schools, are near the bottom in all rankings... – RonJohn Mar 19 '23 at 20:16
28

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.

TomKern
  • 4,332
  • 13
  • 18
  • Could you specify what you mean by "accept new definitions without knowing what they mean"? To me, the only meaning of the definition in the question seems to be that one considers a mapping that maps two integers $a,b$ to the integer $ab-a+b$ - which is obvious from the definition. What other kind of meaning do you have in mind? – Jochen Glueck Oct 16 '22 at 06:20
  • 8
    @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. – Ben Oct 16 '22 at 14:27
  • 1
    @Ben: I of course agree with you that it is much easier to work with a definition of a concept if one has some intuition of the concept. Whenever I read about mathematical notion that is new to me, the first thing I ask is "What does this mean intuitively?" My point is just: There is no "notion" or "concept" behind the definition "$a\star b:=ab-a+b$". It's just an arbitrary example of a function. So would you, more generally, consider "being able to work with rather arbitrarily given examples of functions" also as an instance of "accept[ing] new definitions without knowing what they mean"? – Jochen Glueck Oct 16 '22 at 16:06
  • 5
    @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. – Jochen Glueck Oct 16 '22 at 16:10
  • @MahdiMajidi-Zolbanin: Good question. I'm note sure, since where I live there are no calculus courses in math programs; the programs rather start with proof based courses in linear algebra and analysis, and in those courses students are usually expected to pick up the general notion of a function very quickly. I've seen many students who have various difficulties with this; but working with an example of a function that is given by a concrete formula is, in my experience, quite ok for most of them (at least, it's much less abstract than many other things in those courses). – Jochen Glueck Oct 16 '22 at 16:25
  • @MahdiMajidi-Zolbanin: Regarding the second question in your comment: Why would one even speak of "solv[ing] an equation with respect to $\star$"? Solving an equation simply means to find the set of all solutions; it does not matter which symbols are used to denote any functions that occur in the equation. I think most beginning math students I'm familiar with would, right after learning what the notion "function" means, be able to solve $4 \star x = 36$: they would simply substitute $4x - 4 + x$ for $4 \star x$ and then solve for $x$ (as one would expect them to do). – Jochen Glueck Oct 16 '22 at 16:33
  • 4
    @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.) – Jochen Glueck Oct 16 '22 at 16:37
  • 6
    “Lots of practice performing symbolic manipulation” was the explicit justification a professor gave me for the continuing requirement for calculus 2. – KRyan Oct 16 '22 at 17:22
  • 11
    @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!) – Kevin Oct 16 '22 at 18:30
  • 1
    @JochenGlueck "Abstract" possibly isn't the best word for what I mean, but Kevin seems to have a similar understanding. My point is that I believe a lot of people have the instinct to approach mathematical definitions by finding the "notion" or "concept" behind the maths, so they can lean on their intuitive understanding of that notion when working with the maths. So when you say there is no notion or concept behind the definition, that's exactly what would make this exercise uncomfortable (even if not actually hard) for people who think that way. I had to train myself out of that habit. – Ben Oct 17 '22 at 00:29
  • (Maybe it would be possible for a moderator to move the comments to chat, so that we don't keep spamming TomKern with our ongoing discussion?) – Jochen Glueck Oct 17 '22 at 14:13
  • @Kevin: Hmm, I see that it can make sense to call an unmotivated definition abstract (although I would probably describe it differently), but it still feels strange to me to apply this to things like a very easy function. Maybe the following will help me to better understand your point: In the exercise "Define $f: \mathbb{R} \to \mathbb{R}$ as $f(x) = x^3-x$ for all $x\in\mathbb{R}$. Find all local maximal points of $f$.", would you also consider the definition of $f$ as purely abstract? – Jochen Glueck Oct 17 '22 at 14:39
  • @Kevin: Regarding the comparison to groups, I somehwat disagree: for the function $\star$ in the OP, the domain and the co-domain are explicitly known, and an explicit formula for the function is known. In the definition of a group $(G, \circ)$ on the other hand, the set $G$ is just a general non-empty set, and no concrete formula is given for the mapping $\circ: G \times G \to G$; one merely has a number of axioms. To me, this appears to be much more abstract than $\star$ (even if one thoroughly discusses the motivation regarding symmetry and many examples for groups). – Jochen Glueck Oct 17 '22 at 14:41
  • @Ben: Thanks for your reply, too! To make sure that I understand your point correctly: Let's say we rephrase the sample question in the following way: "Consider the mapping $f: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ that is given by $f(a,b) = ab-a+b$ for all $a,b \in \mathbb{Z}$. Find all $x \in \mathbb{Z}$ that satisfy the equation $f(4,x) = 36$." Would you also say that this version of the exercise will probably be uncomfortable for many students since there does not seem to be an intuitive concept behind the maths? – Jochen Glueck Oct 17 '22 at 14:54
  • I suppose I should have anticipated that the word "meaning" would mean different things to people at different levels of mathematical maturity. Meaning is just connections back to what someone already is familiar with. For mathematicians, this is being able to connect something new to their existing practice of symbolic math, but for precalc students I imagine this is more being able to connect something to their real-world experience. – TomKern Oct 17 '22 at 15:43
  • 3
    This seems disappointingly simple, but based on the comments here it may just be a familiarity with defining new binary operations (that look like but aren't multiplication), a skill practiced in calc 3 with the cross and dot products. This is a bit hard to test, since switching the question to Jochen Glueck's f(a,b) would measure familiarity with defining binary functions, a skill also practiced in calc 3. – TomKern Oct 17 '22 at 15:49
  • 2
    @JochenGlueck: I would regard it as abstract in the sense that it doesn't relate to any physical object or otherwise appear to be directly "useful" in its own right. – Kevin Oct 17 '22 at 17:10
20

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.)

Tanner Swett
  • 631
  • 4
  • 7
12

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.

Mike
  • 181
  • 5
  • 1
    +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). – WaterMolecule Oct 17 '22 at 14:37
8

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.

Todd Wilcox
  • 635
  • 5
  • 9
-3

Intelligence. With familiarity with math being an additional factor.

Acccumulation
  • 1,440
  • 7
  • 6