# Start class with the good stuff

Not much is going very well in this most cursed school year, but I like the way I rewrote this lesson today.

Here is the original activity, from the Illustrative Mathematics Algebra 1 course:

I like the idea, but I saw room for improvement. Here were my issues:

• The problems were somewhat repetitive.
• Students could answer the questions without using the function expressions.
• This might make it so they’d miss the main ideas until the last two questions.
• I like creating situations where students have a principle that they try to apply to a lot of different situations. The way this is designed, they wouldn’t be applying any sort of general mathematical principle until the very last question.

For my redesign, I wanted to get a generalization out in front. I started by showing a parabola and asking students to identify the x-intercepts and the y-intercepts.

Then, I displayed a worked example showing how someone could use the standard form formula to find the y-intercept. I asked students to read all the text that was in green and to put their thumbs up when they were done reading. Then, knowing I had some deeper questions coming up, I explained each line of the example. “Because the y-intercept is always on the y-axis, we know that x will equal … say it aloud,” I gestured for everyone to chime in. “That’s right, x will equal 0. If you input that into the function, your output will be the y-coordinate of the y-intercept.”

After briefly explaining, I revealed those deeper questions:

First, a “what if” question that swapped the numbers from the original function around. Instead of “-5x + 4” I asked about “+4x – 5.” What would the y-intercept be if the function looked like that? This question is asking students to take a step towards a generalization, and that second question goes all the way towards generalizing.

The third question, “What about the factored form formula?”, is a bit of an extension question. It was a more challenging question than I could have asked at this stage if I hadn’t lead off with the example. I didn’t expect every student would necessarily arrive at a correct answer to this question, but every student could benefit from trying to apply what they’d learned to factored form.

As it happened, there was a nice little debate about whether the y-intercept could be recovered from factored form. One student said, only if you turn factored form back into standard form by multiplying. Another chimed in to say that they noticed that -1 x -4 = 4, which connects to the y-intercept, but they weren’t sure if it were a coincidence. Then a third kid pointed out that -1 x -4 would always be part of the binomial expansion, so it wasn’t a coincidence. And then I pointed out that the same logic as before — input x = 0, your output is the rest of the y-intercept — applies to factored form, and (0 – 1)(0 – 4) = 4. Good talk.

Then, it was time to apply this knowledge. I started with a question that was probably a bit too easy, but I wanted a chance to connect factored form to the graph one more time. I asked students to put a thumb up when they’d found the x-intercepts of this function:

Good, good, I asked them to shout out the x-intercepts and I pointed out that they could have gotten this just from the equations. Ditto for the y-intercepts.

Anyway, on to the good stuff. I assigned partners and asked students to talk together about what the x- and y-intercepts were of this function:

Surprisingly challenging for the kids, even given all that we’d talked about! Most got the x-intercepts, the y-intercept required a bit more discussion. All were interested in the reveal, where I removed the grey box.

Moving along, I asked students to talk to their partners about the x- and y-intercepts of this graph:

And then this one:

This one threw kids for a loop! There was much discussion among pairs as to what the “other” x-intercept was going to be. And then there audible “ooohs” from a few kids when I revealed the graph, and there was only one x-intercept.

I feel as if it’s important to mention at this point that this was a class of pretty strong students. Their reaction to this graph says as much about them as it does about me. I want to be real, I don’t often elicit “ooohs” from my teaching. I don’t even know if I want to be a teacher that gets a lot of “ooohs.” But I want to tell you the truth, and the truth is that there were “ooohs.” I could be making that up, but I’m not.

Finally came the “Are you ready for more?” problem from Illustrative Math:

The lesson as written by Illustrative Math was pretty good, but it definitely fell into a paradigm that we might call “Notice, Notice, Generalize, Apply.” To give students enough mathematical data to make the generalization the lesson asks kids to do something a bit repetitive — to find a lot of x-intercepts and y-intercepts. Because students don’t yet know how to use formulas to do this, the graphs need to be provided. Because the graphs are provided, kids will probably just use them. Only at the end, when they’re prompted to generalize, will kids go back and try to connect their answers with the formulas. Kids then are asked to apply that generalization to a single new case where a graph wasn’t provided.

My rewrite follows a different paradigm, something I’ve sometimes called “Analyze, Explain, Apply,” but might also be called “Generalize, Apply, Apply, Apply.” The generalization is front-loaded in the lesson, which means that I was able to turn some of those “notice” problems into chances to apply the big idea about using the formulas to find the intercepts. I hope it’s clear that this hardly eliminated the challenge for students. If anything, it deepened our mathematical discourse.

The generalization is the thing we want to help students think about. Sometimes it can be nice to give students a chance to form that generalization on their own. But the tradeoff is that they then have less time to try out that new mathematical idea with new problems. If you put that big idea towards the top of the lesson, they get many more chances to apply the idea to problems. Of course, that means that you have to provide that new idea to students, either in the form of an example or an explanation. That’s a trade that I’m increasingly willing to make.

1. Who should be allowed to go to college?
2. Should everyone?
3. Who benefits from attending college?
4. Suppose there were a test that could predict perfectly whether you would successfully graduate from a particular college or not. Would that be the fairest way to determine whether you should be allowed to attend a particular school?
5. Suppose you had a number that could perfectly summarize a student’s prior achievement. Colleges could admit students above a certain number X of prior achievement. Would that be fair?
6. If every student who graduated high school could choose to enroll in free public college, what percentage of students would? How many students now seek to attend college but aren’t able to?
7. What percentage of students would like to attend a more selective college but are not able to, either because of money or rejected applications?
8. Why does it benefit students to attend more selective colleges?
9. Should there be elite private colleges?
10. Who should be allowed to elite private colleges?
11. Who benefits from attending elite private colleges? And how?
12. Are there benefits to attending elite public colleges?
13. Should there be elite public colleges?
14. Who should decide who goes to elite private colleges?
15. Do the “official admissions criteria” impact who private colleges admit? Or does it just force them to use different criteria to create demographically identical classes?

I find all of this confusing. I found it more confusing after reading this and this, both of which I recommend reading. Feel free to answer these questions. Are these even the right questions? Feel free to ask better questions.

# Good explanations connect particulars to principles

What should count as an explanation in math? Even though most people in education think that explanations are important, they often struggle to clearly define what an explanation is. This has led to a situation where it’s relatively easy to make fun of mathematical explanations: You want kids to explain why 2 + 3 is 5? Solve the problem, what other explanation do you need!

In support of the pro-explanation camp, there is a large body of evidence that finds that it helps people when you prompt them to explain things to themselves during their learning. And this indeed does seem to settle the case, until we ask the exact same question as before: what does it mean to explain something to yourself?

Making it even more confusing? The use of “self-explanation” by researchers is a moving target. Sometimes it is supposed to be self-talk, a coherent explanation to yourself; other times it’s just any old inference you’ve made, even if it doesn’t add up to an “explanation.” Sometimes it counts as self-explanation if you say it aloud. Other times self-explanation is supposed to be an exclusively internal phenomenon.

I indeed found this extremely confusing, but felt much clearer after I read a fantastic piece by Alexander Renkl and Alexander Eitel that lays the whole situation out in a coherent way. First, they note that the situation is indeed very confusing and the meaning of “self-explanation” has changed over time:

As use of this construct has become so widespread, its meaning has changed. For example, Chi (2000) now sees (at least some) self-explanations as the process by which learners revise and improve their yet imperfect mental models. Presumably due to the widely varying use of this construct, its recent characterizations are rather general such as “inferences by the learner that go beyond the given information” (Rittle-Johnson et al., 2017) or explaining “the content of a lesson to themselves by elaborating upon the instructional material presented” (Fiorella & Mayer, 2015, p. 125). If one relies on such definitions, it is hard to draw the boundaries between other established constructs in research on learning such as “elaborative inferences” (in text learning; Singer, 1994) or “elaboration strategies” (Weinstein & Mayer, 1986). It is also questionable whether all the phenomena labeled self-explanations in previous research can be justifiably called explanation (for discussions about the concept of explanation, see, e.g., Keil, 2006; Kiel, 1999; Lombrozo, 2012).

Then they say something very sensible, which is that self-explanation is probably not one but rather many phenomena, related but distinct:

Given that even the initial characterization of self-explanation already included relatively heterogeneous phenomena (see the four findings reported by Van Lehn et al., 1992) and given that the subsequent extensions led to very general characterizations, there would appear to be little justification for discussing the learning effects of self-explanations and their use in (classroom) instruction on a general level. It is highly probable that different types of self-explanations have different functions, lead to better learning via different mechanisms, and should not be regarded as a unitary construct when providing practice recommendations.

They then announce that they are going to focus in on just one particular form of self-explanation. They call it “principle-based self-explanations” but that isn’t terribly evocative. What they’re describing is connecting the particular details of whatever it is we’re looking at to generally applicable patterns or rules. This form of self-explanation is a form of explanation; it’s explanation by way of connecting these specifics to that general rule:

We focus on principle-based self-explanations (Renkl, 1997): Learners “self-explain” a step in a problem solution (e.g., physics problem) or a feature of an object (e.g., the appearance of an animal) in reference to an underlying principle (e.g., one of Newton’s laws or mimicry). Such self-explanations were part of the initial self-explanation concept (e.g., VanLehn et al., 1992). They can also reasonably be called explanations. Although the concept of explanation can have different meanings (e.g., Keil, 2006; Lombrozo, 2012), explaining a case in terms of underlying principles is a quite prototypical case of explanation. This type of explanation fits the subsumption and unification accounts of explanation (e.g., Lombrozo, 2012). These accounts regard a case as explained when it is subsumed under a more general pattern (e.g., Newton’s laws or the general strategy of mimicry).

Let’s come back to 2 + 3 = 5. How could you connect this to some general principles? It wouldn’t just be taking two things and three more things and counting them. That’s of course valuable, but to get the benefits of what Renkl and Eitel are talking about it would have to be something more abstract and general. It probably shouldn’t involve the words “2” or “3,” or at least that’s not the important part. Maybe you’d say, “Whenever you are adding you can count on from one number by the other number.” Or maybe “You can always put both numbers on your fingers and count them together.” The point is the always. You are seeking to connect the particulars of this to a general pattern that repeats a lot.

The benefits of this kind of self-explanation, Renkl and Eitel write, are to be better prepared to apply that generalization to future cases. This makes sense since you are articulating a generalization, in the process abstracting some example or solution and turning it into a generally useful principle, procedure, strategy, structure.

They even have a nifty diagram:

I think this is the simplest way to think about what a mathematical explanation is, or what it should be aiming for. It’s simply asking students to make a generalization. Articulating a generalization, if one is able to, is pretty clearly going to help students handle future cases that fall under that general pattern.

Just as equally, if students aren’t yet able to articulate a generalization then they probably won’t get much out of being asked to explain. A lot of the cases that are easiest to make fun of are situations where students just don’t yet know how to articulate those generalizations. If students don’t know how to explain 2 + 3 = 5, they can be taught what those general principles are explicitly and then asked to apply that to other problems such as 4 + 5 = 9 or whatever.

One of my favorite techniques I’ve been playing with this year is giving students two generally applicable principles and asking which better explains a given solution. This removes the burden of articulation from the kids, and instead asks them to focus on deciding which abstraction best matches the particulars. And if it’s sometimes ambiguous (as arguably the choices below are) that can be OK too, since it’s all still about connecting particulars to general principles.

Practically, this means that teachers can be clearer to students about what we’re looking for. We can ask students to tell us something that is always true, or that will always work. We can teach that great explanations aren’t just about this problem, but that they show how a particular instance is part of a larger pattern. In this way we can teach students how to give good explanations, at first to others. But my guess is with time this habit is internalized, and you start giving better explanations to yourself.