I get a bit tired of the trope that students today are subjected to boring math with boring procedures and boring problems. (Although I must say, I find the real-world problems that are supposedly interesting to be quite tedious and boring). Essays abound with links to something called Lockhart’s Lament which was written by a mathematician named Lockhart and is a lamentable whine about how he found math in K-12 boring.

In his essay, he laments about why students can’t be made to be curious over the relationship between perimeter and area, and the question of what perimeters of various polygons yield the greatest area.

The problem has become a poster child of what math classes are supposed to be about, and of course Dan Meyer (of dy/Dan fame) is no exception to this. Yes, it is a good question, but one that can be attacked with the tools of mathematics rather than by guess and check or other more sophisticated iterative and inductive procedures augmented with TI calculators and Desmos-generated tools.

The principle involved is the subject of something called the Isoperimetric Theorem which can be proven with Euclidean-based geometric theorems and leads to the relationship between perimeter and area of regular n-gons. If you really wanted to teach it with “understanding” and mathematical principles, teach it in a proof-based high school geometry class. “Geometric Inequalities” by Nicholas D. Kazarinoff (from whom I had the pleasure of taking a differential equations course at U of Michigan). You can find the book on the internet. It was published by Random House as part of the New Mathematical Library and is one of the better products of the 60’s new math era.

If you insist on guess and check and other inefficient ways, then you can stick with dy/Dan, but it doesn’t take a mathematical genius to guess how I feel about this particular “lament”.

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Guess-and-check was the introduction to that problem and not its conclusion. Can I make that clearer?

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No, it was pretty clear that the methods were not limited to guess and check. Using the tools mentioned in the post is fine as an intro, but my point is that there are mathematical approaches to the problem as I mentioned other than the iterative graphics and applets described. I would like to see these addressed for those students who are so inclined to study them, in addition to the graphical approaches. Nothing against Desmos or other applications–I use Desmos in class myself.

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I think Barry’s point was that this does not solve the “boring math” (supposed) problem — while one can (but few do) talk about the importance of isoperimetric properties in nature, few students will see this problem as intrinsically interesting or credibly related to their likely future careers. Adding candy or digressive activities and explorations deemed by current educational dogma to be “engaging” — such as experimentation with paper models, guess-and-check, computer graphics and so on, also does not make the subject matter less boring or more interesting. A lot of what I see purporting to make mathematics engaging consists of doing non-mathematical activities, such as building houses, playing with computer graphics or playing a guessing game. Nothing wrong with such things, but simply adding those things does not make the underlying material more engaging, it’s still what it is.

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> Adding candy or digressive activities and explorations deemed by current educational dogma to be “engaging” — such as experimentation with paper models, guess-and-check, computer graphics and so on, also does not make the subject matter less boring or more interesting.

I wouldn’t dream of defending that list in part or in whole. (Indeed I have no idea what thread connects all of its elements except you don’t like them.) But psychologists have studied curiosity and interest empirically. Those studies transcend dogma and people who want to be taken seriously in discussion of engagement ought to take them seriously. Berlyne on curiosity. Silvia on interest. The Johnsons on controversy. Kasmer has studied the role of prediction specifically in mathematics classes. That’s the foundation for my Desmos activity.

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All teachers have to directly teach something or else they will never cover all of the material in the time allotted. Discovery takes too much time and is often done in groups where a few excited students cover the lack of excitement, engagement, and discovery of others. The light bulb students directly teach (or try to) the others. Also, many of the traditional teachers I’ve had made their classes very interesting. They would pose questions in class to lead us down a path towards discovery. Sometimes it worked and sometimes not, and of course, many didn’t get a chance to discover anything before the others piped up. Then we had individual homework sets where the real work and learning took place. It’s a place that’s little affected by excitement and curiosity. It’s a place facilitated by mastery of prior content and skills – i.e. previous homework assignments.

Curiosity, interest, and prediction do transcend dogma, but are those tools presented that way and do they work? What’s the main goal, to encourage these attributes and skills or to get students to go beyond excitement to actually be able to do all sorts of problem variations automatically? Pedagogy must concern itself with individual mastery of content knowledge and skills and not vague class goals of curiosity and excitement.

Often, however, these ideas are presented outside of the overall requirements and restrictions of a course and curriculum (and what will best help students for their further education), and some courses are re-designed to cover less material just to provide more time for curiosity, excitement, and other vague concepts of understanding. It’s easy to cover less material and then claim better excitement or whatever. There is also the complication that schools pass along students of much wider ability range into the one classroom. The goal is to fix this problem and not find some sort of curiosity, excitement, and discovery concepts (however nice) to hide behind as if that’s the solution.

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Agreed, and I think your latter point is especially important. I often say, over the objections of educationists that what some call “differentiation” is a necessary evil. Necessary, of course, because of those differing ability ranges — which is often not a variation in intrinsic ability, but a consequence of differing standards and content in earlier-years instruction.

Educationists don’t like my phrase because of the second word, “evil”, as they regard differentiation, per se, to be a positive good in instruction, as if it were malign to design lessons around the idea that students would all master more-or-less the same material at the same time in a more-or-less uniform instructional environment.

But I say that it is an evil, because differentiation is intrinsically inequitable — by providing different instruction for different individuals the most likely outcome, that intrinsically encoded into this approach, is different levels of learning, mastery, exposure and (learned) “ability”.

I say that it is necessary because differentiation can be used by a discerning and capable teacher to narrow the learning gap for those falling behind or with identifiable deficiencies. But to use it willy-nilly out of a feeling that it represents an aspect of instruction without which learning is shortchanged will lead to unnecessary inequities.

Like many evils, differentiation begets more differentiation. Differentiation in Grade 1 leads to further differentiation in Grade 2. If this is done indiscriminately it leads to further widening of the backgrounds, knowledge and mastery of students and so necessitates even more in Grade 4. And the snowball grows. At some point a teacher may neglect to differentiate for the increasing disparity between achievement levels … and still the gap widens, because of the Matthew effect.

Finally, as Steve points out here, differentiation is often done at the expense of content, and it can lead to thinning out of the substance of instruction, harming both the high achievers and those falling behind.

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Looking at just this one problem, it still has to be examined in a mathematical context and not just as part of some sort of inspiring or vague thinking process. That’s Barry’s point.

This is a problem that wants a student to find a maximum. That has a specific mathematical meaning and refers to a class of problems. Instead of using just some sort of vague thinking process, students need to learn basic mathematical skills and processes, like drawing a picture and labelling unknowns. The they need to learn how to create equations and combine them to find out whether they have MN. Once that is done, they need to understand what can be done in each case – do they need more equations (make up equations? boundary conditions?) or do they need to delete equations or apply some sort of fit to the data. Perhaps they need to define a merit function that they can optimize. All problems have one solution and math provides you with a proper way to do that. It may be based on your preferences, but there are proper mathematical ways to quantify one solution.

Having a student who is well below calculus solve an optimization problem may or may not accomplish anything. It’s not wrong out of context, but I don’t see how it defines any sort of ideal engaging, creative, or thinking learning process. There are more concrete and mathematical tools for that, and they can be creative and inspiring. They can also be taught directly or done individually for homework, thereby wasting a lot less class time.

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MN – sorry, it doesn’t like less than or greater than symbols. I’m referring to how many equations and unknowns you have and what mathematical tools apply to each case. Students have to understand what maximum or minimum means mathematically and when it applies. What does maiximum mean for a system of equations needing one more equation to get M equal to N? How about missing two equations? What are global and local maxima and minima and when do they occur? Understanding comes in layers and it’s based on mastery of concrete mathematical concepts and skills, not a vague process driven by guess and check.

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Or even more to the point, it may represent a perfectly fine instructional piece all by itself in isolation, but not contribute meaningfully to coverage of the curriculum for a given class. That is what is wrong with this stress on digressive activities of the sort I collect under the rubric “candy” above. In a piece we wrote some time ago, Barry and I made a point about the distinction between “main dish” and “side dish” instruction. Side dishes can indeed make the experience of classes interesting and engaging, but they may or may not actually represent progress toward the main dishes in the course, as laid out in the curriculum. The problem with these miscellaneous “bright ideas” for one-off lessons you find on the internet for free or offered by these mercenary educational software groups is that they tend to be side dishes. Too much of that and your students may not get to through the main dish, and all these sparkly bright ideas will turn out to be a poor idea in the big picture.

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“Traditional” also implies traditional criteria of success, which means the AP calc track in high school. That’s what my son just got from his high school and what most high schools still directly teach. Some students might benefit from another track and emphasis, but the emphasis has to be on concrete mathematical goals and not vague thinking or excitement goals. Excitement is no help if the student does poorly on the Accuplacer test when they get to the community college or vocational school. They also won’t help the college bound student who needs to pass college level statistics. I saw too many students who had to change their majors because they did not have the basic skills to pass statistics.

Discovery, curiosity, excitement, and whatever else sound great, but do they specifically help students for what they need in the future, and how, exactly are they defined and used for different pedagogies?

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