Guided Learning Through Problems

I love new information. I read all the time (both fiction and non-fiction), and I like to learn new things about the world. I have interests in mathematics, science, and plenty of other niche topics like typography and running. As such, I’m frequently consuming information. I begin my day by reading, and I must spend at least an hour each day reading (particularly when I have time off between semesters, as is the case now). Put simply: I love learning new things.

To illustrate this, when I finished up with my exams for this semester (and felt pretty burnt out), I woke up the next day and started reading textbooks for classes I’ll be taking next semester and other topics I was interested in. I think that’s the best example that I can come up with to show just how much I enjoy the process of learning.

That being said, the observant reader may note that in the first paragraph, I used the word “consuming” instead of “absorbing”. That was not a mistake. The dirty little secret that I have is that I go through a bunch of information, but most of it isn’t retained as well as I would like. For some things, this doesn’t matter. But when I read a book about some scientific concept, it seems like a bit of a pointless journey if I don’t even recall the details a couple of weeks later. Likewise, I’ve found myself looking at textbooks with a cursory glance, interested in the new topics, but as soon as the problems and exercises came, I would stop reading. It was too much work to actually get into the book. Reading about a topic at a high level is fine, but digging into the details requires work that I was seldom giving.

This is a problem, because it means I’m only passively coming across new information. I’m not spending energy reflecting and really thinking about the content, which is where true learning and understanding comes from. That’s the difficult part of learning. Jumping into qualitative descriptions is relatively easy, but when is it time to sit down and really go through the details?

I bring this up because I’ve recently begun working through a textbook that is a bit different from the usual. Instead of being about theorems and proofs, the book is mainly a bunch of problems. The idea is to guide one through doing examples. This is much more difficult in the sense that working through a page goes from being a two-minute endeavour to a potential total-afternoon one. Progress is obviously much slower, since one has to think of the problems instead of simply reading through them. I’ve found that it’s very easy for me to skim through a worked example and say, “Oh, that makes sense.” Yet when it comes time to sit down and do a problem, it’s not so easy. You have to put pencil to paper and try things out. And in the end, that’s what helps us learn.

I think these kinds of books (or notes, or whatever you want to call them) are really the best idea if one wants to do some self-studying. It’s not that textbooks are bad. In fact, they contain a lot of useful information that actually does have a purpose. However, I think that this purpose is more of a reference than a tool to actively learn. For the learning part (within mathematics, at least), I think a lot of problem solving up front with the relevant details interspersed at the right moments can be much more effective than a traditional textbook. Do I think that this would work with every topic within mathematics? Of course not. For proof-based courses, one needs those definitions and previous theorems if there is ever a hope of doing more proofs. However, if I take as an example the particular topic I’m trying to learn at a more in-depth level (combinatorics), this is a great topic where working through problems can be more beneficial than reading through a textbook. In particular, I’m working through Kenneth P. Bogart’s Combinatorics Through Guided Discovery, which has been a great resource so far. The book forces me to think about the topics I’ve learned, as well as how an answer can be found in several ways. The writing is clear, and the problems flow together towards a total understanding of the material.

I really think that these kinds of resources are the next big wave within education. Instead of trying to learn from a textbook or video, the concepts will be illustrated directly through problems. Yes, it will take longer to get through the material, but if a student is motivated to learn, that shouldn’t be a large enough barrier. Sites like Brilliant do exactly this. They focus on problem solving over a bunch of details on the concepts, and I think this makes for an experience that stays with a student for longer.

Teasing Out One's Reasoning

I pride myself on being honest with the students I tutor. I don’t like the idea of telling a student that they’ve made a “good” attempt when really there answer is incorrect. My goal with them is to help them learn whatever material they are struggling with, not to give them what some people call “compliment sandwiches”. These are the result of giving someone a piece of criticism in between two compliments in order to make the criticism easier to digest. Perhaps that works for some people, but for myself, I find it more efficient to simply tell the student what’s right and what isn’t.

However, I’m not set in my ways yet, so I try to keep an open mind about different ways of working. As such, I’ve been thinking about the way my typical response would be received by the student I’m working with. For example, if we were working on a problem where variables needed to be set up given some constraints, and they asked me if a certain function would work (yet I know it does not), my initial response would usually be something along the lines of, “No, that doesn’t work since it doesn’t take into account this fact.” But I don’t think that the student would necessarily get this. Instead, I realize that there’s a good chance the student will simply think, “Alright, if Jeremy says you can’t, then I guess we can’t.”

This is a thought process I absolutely want to avoid. It’s just not conducive to learning. Is it productive? Maybe, but that’s not what I’m here for. I want to give the student a slightly more nuanced understanding of the world of mathematics, and telling them that what they’re doing just won’t work isn’t good. I might know it doesn’t work, but they don’t, and that’s the key difference. I’ve already absorbed the lessons that come from making the mistakes they have, but they haven’t, and so if I don’t give them a taste of why their idea does not work, they will always wonder why their idea didn’t work but mine did.

Therefore, I think the solution is fairly clear. I need to allow the students to pursue more “blind” alleys. I put the term in quotation marks because I’m not blind to the dead end, though they might be. Will this take more time? Of course, but I think it will also give the students a firmer understanding of what works and what doesn’t. It isn’t a magical set of rules. Additionally, they might even find new and creative ways to solve the problem, which in that case shows that mathematics is a creative endeavour, something else I want to cultivate.

I know that what I wrote above can sound a little strange. Isn’t the exact reason I’m being paid to work with the student is to make them avoid these kinds of issues? That is true, but I also know from experience that it is much easier to solve problems when one isn’t trying to remember a set algorithm to follow. Instead, my best problem solving sessions is when I deeply understand the concepts and can focus on the precise nature of the problem. I know this isn’t necessarily the goal for those that I tutor, but I do know that giving them a better foundation to work from can only benefit them.

Consequently, my new plan of response when a student tells me something that I know is wrong is not to tell them immediately. First, I’ll question them about their statement, and ask them to be more detailed. My goal is for them to see that their idea doesn’t completely work, and to then refine it. This will hopefully reinforce the good strategy in their mind, which can only help in the classroom.

I’m most proud when I see a student give an answer, reflect on it out loud, and then shake their head because they figured out something wasn’t quite right. If I can get a student to develop this one skill, I’m very confident that they should be capable of doing well on any new topic they study within mathematics. One does not need to know everything at once. However, by being able to think critically, it becomes so much easier to advance in a new topic.

Calculating $\pi$ Factorial

One of the things I like most about mathematics is its ability to generalize results to realms that one might not have previously thought of before. Historically, this is what happened with rational numbers, negative numbers, irrational numbers, complex numbers, and so on.

Most of you have probably heard of the factorial operation before, but here it is again explicitly. Essentially, if you have a natural number $n$, then it’s factorial is denoted as $n!$ and is defined as $n! = n(n-1)(n-2) \ldots (2)(1)$. That’s an easy enough definition. Start with your number, and just multiply it by all of the numbers that came before it (until you get to one). We also have the base cases of $1!=0!=1$.

The factorial is what is known as a recurrence relation. Instead of getting an explicit formula for how to calculate each term, you get what the $n^{th}$ term is in relation to the next term below it (for this specific case). Let’s do an example with $5!$. If we look at our definition above, we get $5! = 5 \cdot 4!$, since the rest of the multiplication is “hidden” within $4!$. As such, we don’t get a number for $5!$ until we figure out what $4!$ is, which we can only do if we find out what $3!$ is, and so on. Therefore, the factorial is really a recipe, and the only time we get a real answer is when we hit $1!$, which we set to $1$.

That’s great, but it doesn’t look very helpful for calculating $\pi!$. After all, $\pi$ is definitely not in the naturals, so we can’t make use of the definition above. However, let’s keep in mind the kind of relation that the factorial gives us. It tells us that $n! = n \cdot (n-1)!$. Let’s see if we can get something else to work like that.

Somewhat completely out of the blue, let’s take a look at the following function:

\begin{equation} \Gamma(a) = \int_0^\infty x^{a-1}e^{-x} dx, \,\,\,\,\, a \gt 0. \end{equation}

This function is called the Gamma function, and it’s used in probability distributions (which is where I came across it). Now, this might seem like a really weird function to throw at you. How in the world does this relate to anything about factorials? Well, let’s start by trying to calculate the value of $\Gamma(a+1)$.

\begin{equation} \Gamma(a+1) = \int_0^\infty x^{a+1-1}e^{-x} dx = \int_0^\infty x^{a}e^{-x} dx \end{equation}

Then, we can integrate by parts using $u=x^a$ and $dv = e^{-x}dx$ to get:

\begin{equation} \left[-x^a e^{-x} \right]_0^{\infty} +a \int_0^\infty x^{a-1}e^{-x} dx. \end{equation}

The first term evaluates to zero at both boundaries (which can be seen by taking the limit as $x \rightarrow \infty$). Therefore, we are only left with the second term. However, look at the form of the integrand. It’s simply $\Gamma(a)$. As such, we conclude with the following relation:

\begin{equation} \Gamma(a+1) = a \Gamma(a). \end{equation}

This is really neat, because it’s another recurrence relation that gives us an answer in terms of the previous (lower) one. We can also look at the result of $\Gamma(1)$, and confirm that is indeed equal to one. In fact, this means that we get the following result: $\Gamma(a) = (a-1)!$. It’s a recurrence relation exactly like the factorial, but formulated in a totally different way. The one important difference though is that the value of $a$ is not limited to natural numbers. Now, we simply need $a \gt 0$, which means we can easily calculate $\pi!$. This corresponds to a value of $a = \pi + 1$. Inserting this into the integral definition and evaluating the integral numerically gives a result of:

\begin{equation} \pi! = \Gamma(\pi + 1) = \int_0^\infty x^{\pi}e^{-x} dx \approx 7.18808. \end{equation}

How do we interpret this result? I don’t know! But roughly, I can say that it’s a bit more than $3! = 6$, so at least something is on the right track. However, calculating the result of $\pi!$ in particular isn’t of too much importance. It’s simply a neat extension of how one can think of factorials.

One thing I do want to note is that, just because we have a recurrence relation with this Gamma function, this doesn’t mean we technically have the same thing as a factorial if $a$ is not a natural number. Really, we then just have a recurrence relation. However, it’s still an interesting connection that’s worth sharing. Sometimes, seeing operations and concepts you were only used to seeing in one setting suddenly operating in another scenario can broaden one’s perspective on mathematics.

Easy Examples Miss the Point

I’ve been thinking recently about what it takes to make a concept “stick” in a student’s mind. When first looking at a topic, it’s tempting to show a student easy examples that get them familiar with the mechanics, before moving on to more difficult problems. However, when this new concept is a new way to see an old idea, it can be difficult to sell the concept to the students if the old idea seems to be just as effective as the new one. After all, why should the student have to learn a new method if the old one still works?

This is the topic that Dan Meyer talks about in his presentation on the needs of students. His main point is that mathematics education is too much like a collection of solutions, and not enough of giving students “headaches” in which mathematics can help. Throughout the talk, he gives several examples in which the audience is asked to solve an easy problem, and then to solve a similar but much more demanding problem which creates the “need” for new mathematical tools. In particular, I was struck by the example at around the 35:00 mark in which he asked two people from the audience to describe the location of a chosen point in a sea of other points. Crucially, there were many points, so it wasn’t a trivial matter to describe with words where the point one chose was located. The first person struggled with this, but when it was the other person’s turn, the dots were underlayed with a Cartesian grid. You can hear the audience at that moment laugh at how easy everything becomes. I highly encourage watching the whole thing, but if you don’t want to spend the full fifty or so minutes, watch that one part.

Related to Mr. Meyer’s point is the idea of showing easy examples. I’m reminded of the fact that many students are given easy examples only. You’ve seen this before in the fact that assignments have “nice numbers” as answers, and that nothing is too difficult. This is good to start, but it can often lull students into not really grasping why certain methods are used versus others. If there are five ways to answer the question, why does a student have to do it in a specific way? If this method really is important, I think one needs to create problems that show-off the merits of a particular method. As Mr. Meyer says, give the students a headache and show them a particular method as the aspirin. Don’t just give them easy example problems to work through when trying to motivate them on a new technique or concept. Skipping this step means students won’t have buy-in on the idea, and will simply wonder why they have to learn this esoteric method.

Now, I’m not saying that easy examples aren’t useful. They can help during the beginning of practice, where students are motivated by the concept, yet still need to work on getting the technique. Then, it makes sense to give easier examples. This can also help students check their work, since they can compare the new method to their old method. However, once the technique is mastered, don’t give them problems which are so easy that a different (and more familiar) method can be used. Make sure that students see the need for this new technique. There’s a reason it’s there.