7
$\begingroup$

How can students learn to discover results themselves, if textbooks just spoonfeed them right in the exercise, before asking them to prove it? Why don't these textbook problems educate students on proofs' thought process? I quote four popular undergraduate textbooks.

Charles Pinter, A Book of Abstract Algebra (1982 2 edn), page 158.

Let G and H be groups. Suppose J is a normal subgroup of G and K is a normal subgroup of H.

1 Show that the function $f(x, y) = (Jx, Ky)$ is a homomorphism from $G × H$ onto $(G/J) × (H/K)$.

I would change the last sentence to either —

From what set — onto what other set — is the function $f(x, y) = (Jx, Ky)$ a homomorphism? Prove your claim.

or —

Onto what set is the the function $f(x, y) = (Jx, Ky)$ a homomorphism from $G × H$? Prove your claim.

Page 236

9 For any prime $p > 2, x^2 \equiv -1 \pmod p$ has a solution $\iff p \not \equiv 3 \pmod 4$. (HINT: Use part 8 and E3.)

I would change the last independent clause to —

  1. for what $p$ does $x^2 \equiv -1 \pmod p$ have a solution?
  2. if $p \not \equiv 3 \pmod 4$, then when does $x^2 \pmod p$ have a solution?

David Bressoud, Radical Approach to Real Analysis (2006 2 edn), p 263.

6.4.1. Prove that if $f(a - 0) \neq f(a) \neq f(a + 0)$, then $F(x) =\int^x_0 f(t) dt$ cannot be differentiable at $x = a$.

I would ask —

If $f(a - 0) \neq f(a) \neq f(a + 0)$, then is $\; F(x) = \int^x_0 f(t) dt \;$ differentiable at $x = a$? Prove or disprove.

Terence Tao, Analysis I (2016 3 edn), p 300.

Exercise 11.9.3. Let a < b be real numbers, and let $ f : [a, b] \rightarrow \mathbb{R}$ be a monotone increasing function. Let $F : [a, b] \rightarrow \mathbb{R}$ be the function $F(x) := \int_{[a,x]} f$. Let $x_0$ be an element of $[a, b]$. Show that $F$ is differentiable at $x_0$ if and only if f is continuous at $x_0$. (Hint: one direction is taken care of by one of the fundamental theorems of calculus. For the other, consider left and right limits of f and argue by contradiction.)

I would change the second last sentence to —

When is $F$ differentiable at $x_0$? When is $f$ continuous at $x_0$? Prove your claims.

Terence Tao, Analysis II (2016 3 edn), p 205.

Exercise 8.3.3. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ be absolutely integrable, measurable functions such that $f(x) \le g(x)$ for all $x \in \mathbb{R}$, and that $\int_\mathbb{R} f = \int_\mathbb{R} g.$ Show that $f(x) = g(x)$ for almost every $x \in \mathbb{R}$ (i.e., that $f(x) = g(x)$ for all $x \in \mathbb{R}$ except possibly for a set of measure zero).

I would change the last sentence to —

How does $f(x)$ relate to $g(x)$, for all $x \in \mathbb{R}$ except possibly for a set of measure zero? Prove your claim.

$\endgroup$
13
  • 10
    $\begingroup$ And would your students be able to answer the harder versions? I don't teach classes at this level, but in Linear Algebra, I get great problems for teaching proof from the True/False sets of questions (in Lay's text). $\endgroup$
    – Sue VanHattum
    Commented Nov 4, 2022 at 4:58
  • 10
    $\begingroup$ Given that 3 out of 4 texts of problems in analysis you quoted are garbled, and that you are suggesting in good faith that calculus students should figure out the sum of inverse squares by themselves, I suggest that you start with trying to solve some simple analysis exercise given in your form yourself before trying it on the students. So let's start with "Describe all permutations $\sigma$ of positive integers such that if $\sum_k a_k$ converges (conditionally), then $\sum_k a_{\sigma(k)}$ converges as well. What can you say about the relation between these two sums for such $\sigma$?" $\endgroup$
    – fedja
    Commented Nov 4, 2022 at 9:04
  • 5
    $\begingroup$ See also "Moore Method" en.wikipedia.org/wiki/Moore_method $\endgroup$ Commented Nov 4, 2022 at 9:15
  • 7
    $\begingroup$ @user131533 The question you have raised is rather interesting one, but I want to make sure that you understand fully what you are talking about here before trying to share any thoughts on it. The way you presented the analysis problems and your suggestions for their modifications just have raised some uneasy doubts in my head. That's all :-). $\endgroup$
    – fedja
    Commented Nov 5, 2022 at 22:46
  • 6
    $\begingroup$ @user131533 1) Look at your conditions on $f$ and $g$. One of them is $f(x)=g(x)$ for all $x$, which makes the problem nonsensical. It should be $f(x)\le g(x)$ or something else like that :-). 2) What you are asking about is replacing a clearly focused problem (with a good reason to believe that the proof should be based on the ideas of the previous section) by a freestyle "figure it out" exercise with "no hands barred". I'm trying to demonstrate to you that it is a totally different game both in the resulting question difficulty and in the required background mathematical culture. $\endgroup$
    – fedja
    Commented Nov 8, 2022 at 0:37

6 Answers 6

16
$\begingroup$

I largely agree with you that your approach often makes sense. Since you are asking "why people don't (always) do that", I will only describe the downsides.

One framework to think about teaching tasks and their goals is the Bloom's taxonomy, in which there are 6 levels of knowledge: Remember, Understand, Apply, Analyze, Evaluate, and Create. In every teaching situation, it is good to mind what level you are teaching on, and choose the tasks accordingly. What you are suggesting is, essentially, to add a bit of create to each task, but it may simply be out of place.

For instance, in your first example, once the students remember the definition of homomorphisms and factor-groups, and understand them - i.e., know what they need to check, why $Jx$ is an element of the factor-group, etc. - the exercise is solved. So, this particular exercise is aimed at reaching these two levels. It goal is simply not to "teach the students to discover results by themselves", it is to let the students who were just shown some abstract definitions for the first time in their life, and are utterly confused, to get some hands-on working experience. If you want to teach them to create (i.e., re-discover by themselves) results in group theory, that's great, but it should be done when they have way different level of knowledge and understanding. And - arguably - with more meaningful exercises.

On a more technical side, it is not easy to ask an open-ended question with a well-defined correct answer. Many of your suggested formulations have a perfectly correct answer which is different from the implied one. E.g., in the first one, I answer "from $J\times K$ to the one-element set $\{J\times K\}$, equipped with its unique group structure, since any map from a group to a one-element group is a homomorphism", and demand full credit. It's not just nitpicking, it's really what will happen: on top of struggling with the material, the students will get confused about what you are asking, what constitutes a right answer etc. More harm than good.

$\endgroup$
3
  • $\begingroup$ Thanks! Can you please suggest improvements to my rewrites of these exercises? $\endgroup$
    – user131533
    Commented Dec 4, 2022 at 9:03
  • 1
    $\begingroup$ Some of them are poorly suited for such a rewrite at all. But you can never do it mindlessly. In your third example, the student might answer "not necessarily, here's an example where it's not differentiable". And the fourth is even worse - are you giving full grade to an answer simply reciting the definition of differentiability of $F$ and continuity of $f$? If not, why? Better formulations may be "Can $F$ be differentiable at $a$? Give an example or prove that it is impossible" and "how is differentiability of $F$ at $x_0$ related to continuity of $f$ at $x_0$?" $\endgroup$
    – Kostya_I
    Commented Dec 4, 2022 at 13:20
  • 1
    $\begingroup$ It's a bit of a general problem, if you take a question "Prove that if $A$, then $B$" and reformulate it as "When does $B$ happen?", it is inherently ill-posed, as it is not clear what kind sort of conditions are allowed to go into the answer. And making an unambiguous formulation may be awkward. $\endgroup$
    – Kostya_I
    Commented Dec 4, 2022 at 13:25
7
$\begingroup$

Good point! That said, disclosing the result in the question could be to make the question more doable / less intimidating for the student, so they know what to aim for.

Note that popular undergraduate analysis textbook Understanding Analysis by Abbott does include more challenging questions:

The exercises (even more so now than in the first edition) are superb: many call for proofs, but there are also quite a lot that ask for examples (or an explanation why no examples exist; the reader isn’t always told which is which) or ask the reader to explore the consequences if a definition is modified slightly. Few if any constitute “busy work”; some are quite challenging.

(See Hunacek's review at https://www.maa.org/press/maa-reviews/understanding-analysis.)

Maybe a mix of question types would be a good idea? Of course, we cannot rule out that the lack of questions of the type you suggest could also be because of mere oversight or force of habit.

$\endgroup$
7
$\begingroup$

One reason that this is the case is that, as other members have stated, questions may intentionally be not testing these different aspects of knowledge, they might just be introductory problems made to ensure people have a good grasp of material. Another closely related reason might be to moderate for difficulty- problems might be too hard otherwise.

Another important reason is that sometimes, it simply doesn't make sense to ask the question in such a way. For example, in your first question, $Jx$ is an element of $(G/J)$- the way the question is stated already makes it clear that the homomorphism is onto $(G/J) \times (H/K)$ and removing that from the question doesn't make it any better or more interesting of a question, it's just clunkier.
In your two examples from Tao, the answer space is far too broad for it to be a very good question- "When is $f$ continuous at $x_0$" could be answered via an epsilon-delta definition of continuity, for example.
The other two examples do have a possible way to ask in such a manner, although for your number theory one only the first question you asked really makes sense- the second question is somewhat nonsensical (what does it mean for $x^2 \pmod{p}$ to have/not have a solution?)
For those two examples, if an author wanted to ask the question in such a way, they certainly could (in the number theory example, they could even give less information and ask for what primes $p$ instead of letting on that there is a mod 4 condition). However, this would make the problem much harder in the number theory case, which might be undesirable, and would barely change the problem, and therefore is unnecessary, in the Bressoud case.

In general, textbook authors ask questions that they think are pedagogically appropriate for the readers answering them- if they have chosen to give information, it often is for a good reason.

$\endgroup$
2
  • $\begingroup$ Thanks! Can you please suggest improvements to my rewrites of these exercises? How would you rewrite Pinter's p. 159 exercise on homomorphisms, with my mindset? I edited rewrite #1 on p. 236 as you suggest. But how can I improve my rewrite of #2 on p. 236 on modulus? $\endgroup$
    – user131533
    Commented Dec 4, 2022 at 9:04
  • $\begingroup$ The Pinter one I think the best you can do is "Is the function f(x,y)=(Jx,Ky) a homomorphism from G×H onto (G/J)×(H/K)? Why or why not?" For the modulus one, you don't need a second question- to fully answer the first question, they'll already have specified what p values will and won't work. $\endgroup$ Commented Dec 6, 2022 at 15:25
6
$\begingroup$

I think there are two parts to the answer:

  1. The purpose of a textbook – as a reference or a learning tool.
  2. How experienced mathematicians read books.

Addressing 1: A textbook may serve as a reference or as a learning tool. These two uses are very different (though many textbooks try to do both at the same time). If a textbook is designed exclusively for learning, it makes a lot of sense to have theorems stated in terms of exercises. This is the idea behind Inquiry Based Learning, and you can find many such question sequences in the Journal of Inquiry Based Learning. As a reference, of course, it doesn't make sense to leave core content as exercises; there are times when you're justified in just wanting to know what is true.

Addressing 2: When an experienced mathematician picks up a book to learn something (i.e., is not looking for a reference), they stop, put the book down, and try to work out parts along the way. Essentially, they turn theorem statements into questions, work out the questions, and then continue reading to see if the author had some cleaver tricks that they missed. In this way, any reference book can also serve as a good learning tool if you're disciplined enough.

I try to teach my students to read math this way, but it takes a long time to develop the skill.

$\endgroup$
5
$\begingroup$

Late to join this party, sorry about that. Posting this largely to get some feedback and criticism myself.


A problem with including a discovery phase in a homework problem is simply that those take a lot of time. And the students usually are unwilling unable to invest much (yours is not the only course they are enrolled in), unless you are lucky to find a particularly enticing question! Alas, those are rare. Please share, if you have a surplus!

A possibility is to split the exercise into steps:

  • experimenting/testing,
  • forming a "conjecture",
  • proving the conjecture.

I have experimented with the following two part exercise. Basically the two first bullets formed one homework problem, and the last bullet was then an extra credit problem. This was for a first course in abstract algebra. The students have just been introduced to the cycle notation of permutations. My first and most obvious goal is to make them familiar with the process of rewriting a composition of permutations as a product of disjoint cycles. There are the obligatory rote examples to be worked out serving that end. And then this...

A property of permutations we have not yet discussed at all is the number of disjoint cycles in it. Here we do a bit of research! Unlike in the text, here we also count the 1-cycles, so for example the permutation $\sigma=(12)(456)=(12)(3)(456)\in S_6$ has three cycles of respective lengths 2,1 and 3. The problem we investigate here is what happens to the number of cycles in a permutation, when we compose it (from the left) with the 2-cycle $(12)$. So compare the number of cycles in the permutations $\alpha$ and $(12)\alpha$ with the following choices of $\alpha$ $$\begin{aligned} &(12)(34), (13)(24)\in S_4,\\ &(13)(245), (1324)(5)\in S_5,\ \text{and}\\ &(135)(246), (123)(456), (16)(235)(4), (135264)\in S_6 \end{aligned}$$ You should see that the number of cycles in $\alpha$ always differs from those in $(12)\alpha$ by the same amount, but in different directions. By how much? Can you formulate/guess a general result as to when the number of cycles increases and when it decreases? Do more experiments to test your guess! You will only be asked to prove that your guess works in a follow up problem.

(apologies for my unpolished choice of phrase in the formulation)

The student response is, unsurprisingly, very mixed:

  • "TL;DR;" (I'm sure you have your share of such students, too)
  • "We calculated all those permutations, but what exactly are we supposed to do with the answers?" This may (or may not) be accompanied by a "WTF!" This from students who take notes in class and can do basic variants of examples from lectures, but feel lost without a model example.
  • Exalted students (future researchers and a few others willing to try their hand at figuring things out for themselves).

In a typical problem session, this does generate a lot of discussion. Two of the three groups of students have correctly figured out that the number of cycles always changes by one. I have about 20 students per session, and usually 1-3 suggest a conjecture. If I'm lucky, the conjecture is wrong, and I can quickly refute it with a different example. Even if the conjecture is the correct one (= it depends on whether $1$ and $2$ belong to the same cycle or to distinct cycles), I still test it with at least one more example on the chalkboard.

Then in the extra credit follow up exercise I ask a subset of the following:

  • Prove that $(12)\alpha$ has one cycle more than $\alpha$, if $1$ and $2$ appear in the same cycle of $\alpha$, and one cycle less than $\alpha$, when $1$ and $2$ appear in disjoint cycles of $\alpha$.
  • What changes, if we use another 2-cycle instead of $(12)$?
  • What happens, if we compose by $(12)$ from the right? Compare the number of cycles in $\alpha$ and $\alpha(12)$!
  • We saw that if $C(\alpha)$ denotes the number of cycles in the permutation $\alpha$, then $C((12)\alpha))\equiv C(\alpha)+1\pmod2.$ Recall the definition of parity of a permutation (=odd/even). Prove that if $\alpha\in S_n$, $n$ even, then $C(\alpha)$ is even if and only if $\alpha$ is even. But when $n$ is odd, then $C(\alpha)$ is even if and only if $\alpha$ is odd.

So the idea is to give the students an easy enough conjecture to form, and then later prove. The result could (should ?!) veer a bit off the baseline material. I think it is necessary to split the problem into many parts, because such an exercise requires quite a bit more time from the students. My example does not take much (I have had bright students who turn in a proof for the conjecture at that same problem session), but overall I'm happy with the student response (I may be wearing rose-tinted specs when interpreting it).

I'm afraid I cannot cite any piece of research based on testing similar ideas. It seems to fit into the Bloom taxonomy Kostya described.

$\endgroup$
1
  • $\begingroup$ A point is that the result to be discovered will usually not be covered later in the course material. It is mostly a curiosity that does enhance the students understanding of the cycle structute of permutations. That's what I meant with "veering a bit off the baseline material". The last bullet in the extra exercise is useful, but the upshot (parity is flipped when we compose with a 2-cycle) is often proved by other methods. $\endgroup$ Commented Feb 25, 2023 at 5:47
4
$\begingroup$

I would like to add my 2 cents to this question, which already has many great answers. This is rather long, sorry about that!

Discovering results is hard. It's why most discoveries are labeled theorems. It requires experimentation, trial and error, perseverance over long stretches of time. Your proposed examples attempt to address some of the aspects of research in a sandbox environment: you want students to learn the ropes of discovery but with well established, routine proofs. You hope that they will develop grit that will serve them well in their careers. However, it is likely that these will have the opposite effect: it will diminish their willingness to engage meaningfully with the material, end up with a spotty understanding of the fundamentals and ultimately unmotivated to go further in their studies.

To help explain why, I have broke down the answer in multiple parts.

Cognitive load and memory

Mathematics is cognitively demanding. It is strenuous to do mathematics at the edge of your skill level, i.e. stretching the limits of your current abilities. Working at this level implies high cognitive load, requires full attention and complete focus. When someone is learning new mathematics, they already are exercising grit. They are already discovering the results for themselves, because it is new to them and their mental schemas.

Cognitive load is the mental workload you have to deal with when learning. It is split into two categories: intrinsic and extrinsic cognitive load. Intrinsic cognitive load is the work necessary to understand the material and directly contributes to your learning. Extrinsic cognitive load is useless work that does not contribute and instead actively hinders learning. The objective is to maximize intrinsic and minimize extrinsic cognitive load. It is tempting to think that this approach will do that, but I believe it does the opposite.

Research shows that we have a very small working memory (WM), which holds information about what we are doing. Usually this means 3 to 4 unrelated items at a time. It is no surprise that mathematics easily fills one's working memory and becomes taxing to learn. There's a catch: you can bypass the limits of working memory by using knowledge and schemas stored in your long-term memory (LTM).

Someone learning new mathematics does not have knowledge and schemas stored in their LTM related to the new area. Therefore, they will struggle to incorporate important details in long-term memory by themselves, let alone think creatively and experiment to discover the established results you want them to.

Flow state and success rate

As Jason Siefken has pointed out, it's important to make a distinction between a reference and a textbook. In a textbook you want novices, with respect to the subject, to be able to confidently apply definitions, lemmas, theorems, know established examples and counter-examples, reason in terms of the new structure and understand how it fits into an overall big picture. It does not mean working at the knowledge frontier of that particular area.

With this in mind, I feel it's important to help learners achieve a flow state, a concept created by Mihaly Csikszentmihalyi where their perceived challenge is high but so is their relative skill level, in a way that they have relatively high success rate but not perfect.

Scott H. Young has a good article discussing what he calls the 85% Rule for Learning. From his article:

If you’re always successful, it’s hard to know what to improve. If you constantly fail, you won’t learn what works. Only when we have a mixture of success and failure can we draw a contrast between good and bad strategies. [...] The research also suggests that the optimal success rate for fostering student achievement appears to be about 85 percent. A success rate of 85 percent shows that students are learning the material, and it also shows that the students are challenged.

Your proposed changes will turn out to have much lower success rates I assume, drawing from both personal experience learning mathematics and as a teacher.

Thinking mathematically and problem solving

In the book Thinking Mathematically, the authors distinguish three phases into the work of dealing with a problem:

  1. Entry phase,
  2. Attack phase,
  3. Review phase.

The entry phase is for answering questions like "what do I know", "what do I want" and "what can I introduce". As Kostya_I mentioned, these are in line with Bloom's taxonomy first two tiers (remember, understand). This is where the learner should be occupied understanding what is being asked, what are the nuances of the problem, etc.

The attack phase is when you start to conjecture possible ways to solve your problem, together with justifying and convincing yourself that the steps you followed are correct beyond doubt. This is where you construct examples, specialize cases, think of lemmas, try to generalize from concrete forms, etc. This phase should correspond to the next two tiers in Bloom's taxonomy (apply, analyze).

Last but not least, the review phase is where you check your resolution, reflect on key ideas and key moments, and try to extend these to wider contexts. These are the last two tiers in Bloom's taxonomy (evaluate, create), where you can create new problems, solidify understandings, build bridges between concepts.

My point in elaborating all of this is that this is difficult enough when you have clear start and endpoints, let alone when you have no idea what should be of interest and what's not. Naturally, research begins like that: you have muddled understanding of something and you want to make headway in a challenging environment. But this is not the point when you simply want students to learn and internalize well-known mathematics that should be foundational.

I also recommend Alan Schoenfeld's 1985 book, Mathematical Problem Solving. Amongst other things, he discusses how knowledge impacts problem solving by analyzing verbal data together with sketches from novices, semi-novices, more experienced problem solvers and experts at fairly elementary tasks. He tracks time spent in six episodes, or stages, labeled read, analyze, explore, plan, implement, verify. The data is very interesting and shows how differently they handle uncertainty in problem solving.

Suggestions

I believe the best way I've seen to introduce this sort of "result discovery for themselves" is done in both volumes of Multidimensional Real Analysis by authors J. J. Duistermaat and J. A. C. Kolk. They tend to break rather long problems into many subparts (I think I've seen up to twelve items!), each involving different concepts and fitting into a sort of chain of exercises throughout the two-volume set. They serve to both apply concepts, develop interesting results, and get your hands dirty in meaningful ways. This is one example:

enter image description here enter image description here enter image description here enter image description here enter image description here

I think you can more easily switch to this type of problems, which will enrich your problem sets and your students' abilities. As a bonus, they get to broaden their mathematical horizons!

References

John Mason, Leone Burton, Kaye Stacey - Thinking Mathematically

Alan Schoenfeld - Mathematical Problem Solving

J. J. Duistermaat, J. A. C. Kolk - Multidimensional Real Analysis, Volume 1 and Volume 2

Scott H. Young - The 85% Rule for Learning

Jan Plass, Roxana Moreno, et. al - Cognitive Load Theory (somewhat outdated)

John Sweller, Jeroen J. G. van Merriënboer and Fred Paas - Cognitive Architecture and Instructional Design: 20 Years Later - This article is the most recent I know regarding cognitive load theory from a broad perspective

Mihaly Csikszentmihalyi - Flow: The Psychology of Optimal Experience

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.