Why Calculus still belongs at the top

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AP Calculus is often seen as the pinnacle of the high school mathematics curriculum*–or the “summit” of the mountain as Professor Arthur Benjamin calls it. Benjamin gave a compelling TED talk in 2009 making the case that this is the wrong summit and the correct summit should be AP Statistics. The talk is less than 3 minutes, so if you haven’t yet seen it, I encourage you to check it out here and my first blog post about it here.

I love Arthur Benjamin and he makes a lot of good points, but I’d like to supply some counter-points in this post, which I’ve titled “Why Calculus still belongs at the top.”

Full disclosure: I teach AP Calculus and I’ve never taught AP Statistics. However I DO know and love statistics–I just took a grad class in Stat and thoroughly enjoyed it. But I wouldn’t want to teach it to high school students. Here’s why: For high school students, non-Calculus based Statistics seems more like magic than mathematics.

When I teach math I try, to the extent that it’s possible, to never provide unjustified statements or unproven claims. (Of course this is not always possible, but I try.) For example, in my Algebra 2 class I derive the quadratic formula. In my Precalculus class, I derive all the trig identities we ask the students to know. And in my Calculus class, I “derive” the various rules for differentiation or integration. I often tell the students that copying down the proof is completely optional and the proof will not be tested–“just sit back and relax and enjoy the show!”

But such an approach to mathematical thinking can rarely be applied in a high school Statistics course because statistics rests SO heavily on calculus and so the ‘proofs’ are inaccessible. I’d like to make a startling claim: I claim that 99.99% of AP Statistics students and 99% of AP Statistics teachers cannot even give the function-rule for the normal distribution.

Image used by permission from Interactive Mathematics. Click the image to go there and learn all about the normal distribution!

Image used by permission from Interactive Mathematics. Click the image to go there and learn all about the normal distribution!

In what other math class would you talk about a function ALL YEAR and never give its rule? The normal distribution is the centerpiece (literally!) of the Statistics curriculum. And yet we never even tell them its equation nor where it comes from. That should be some kind of mathematical crime. We might as well call the normal distribution the “magic curve.”

Furthermore, a kid can go through all of AP Statistics and never think about integration, even though that’s what their doing every single time they look up values in those stat tables in the back of the book.

I agree that statistics is more applicable to the ‘real world’ of most of these kids’ lives, and on that point, I agree with Arthur Benjamin. But I would argue that application is not the most important reason we teach mathematics. The most important thing we teach kids is mathematical thinking.

The same thing is true of every other high school subject area. Will most students ever need to know particular historical facts? No. We aim to train them in historical thinking. What about balancing an equation in Chemistry? Or dissecting a frog? They’ll likely never do that again, but they’re getting a taste of what scientists do and how they think. In general, two of our aims as secondary educators are to (1) provide a liberal education for students so they can engage in intelligent conversations with all people in all subject areas in the adult world and (2) to open doors for a future career in a more narrow field of study.

So where does statistics fit into all of this? I think it’s still worth teaching, of course. It’s very important and has real world meaning. But the value I find in teaching statistics feels VERY different than the value I find in teaching every other math class. Like I said before, it feels a bit more like magic than mathematics.**

I argue that Calculus does a better job of training students to think mathematically.

But maybe that’s just how I feel. Maybe we can get Art Benjamin to stop by and weigh in!

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*In our school, and in many other schools, we actually have many more class options beyond Calculus for those students who take Calculus in their Sophomore or Junior year and want to be exposed to even more math.

** Many parts of basic Probability and Statistics can be taught with explanations and proof, namely the discrete portions–and this should be done. But working with continuous distributions can only be justified using Calculus.

Product Failure

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I’ve been taking a grad course in statistics this semester and so I’ve been thinking about all sorts of real world examples of math, including the classic product-failure example that’s a mainstay of most stat classes.

One of the simplest continuous distributions is the exponential distribution which is a pretty decent way to model product failures. The probability of failure f(t) after time t is given by

f(t)=\frac{1}{\lambda}e^{-t/\lambda}.

I read this great article about product failure and testing in Wired this week. I encourage you to check it out. Read the last page of the article especially, where it talks about how cutting-edge companies are modeling minute variations in materials using an electron microscope and some statistics. Instead of actually testing the product over and over again using a fatigue machine, they can create surprisingly accurate models of the materials using computers. Prior to this, the behavior of materials was somewhat unpredictable.

Of course I was excited to see this figure in the article, which shows the Weibull distribution modeling failures of steel bars in a fatigue machine.

The Weibull distribution, unlike the exponential distribution, takes the age of a product into account. If the parameter k is greater than zero, than the rate of product failure increases with time. The probability of failure f(t) after time t is

f(t)=\frac{k}{\lambda}\left(\frac{t}{\lambda}\right)^{k-1}e^{-(t/\lambda)^k}.

The first obvious thing to note is that the exponential distribution is just a special case of the Weibull distribution, with k=1. The next thing to say is that this distribution is single-peaked. So how is the above a Weibull distribution? The article says it is, but I think it might be a linear combination of two Weibull distributions, don’t you? Whatever–normalize, and you’ve got yourself a probability distribution.

[pun warning!]

The real question is, if this is TWO Weibulls, would you settle for the lesser of two Weibulls?

Sorry. I had to.

Ten Questions about Flipping a Mathematics Classroom

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Dr. Gene Chase, guest blogger.

“Flipping the classroom” is doing problems in groups during class time, while listening to lectures and reading books during homework time. See previous blog post. It is a sufficiently vague and controversial method of teaching that I have questions to ponder rather than answers to push. What do you think? I’m focusing on the mathematics classroom especially in the light of the popularity of Khan Academy’s mathematics modules and because this is a math blog.

1. Will students learn more because they are discussing content together in groups in class? Or will they have trouble staying on task because the teacher can’t attend to all groups at once? We have no control over the schedules of students outside of school to get them to interact about content outside of class. Do your students interact about mathematics outside of class now? (If you are reading this as a student, do you interact with other students about mathematics outside of class?)

2. Will “just-in-time” teaching of content when in the middle of solving a problem be more meaningful and more motivational than lectures that “front load” a student with lots of answers that don’t yet have questions? Or will “just-in-time” teaching encourage the kind of thinking that says the answers are only one problem-solving step away? In Japan, students expect to struggle with a problem; in the US students expect a problem to have a ready answer.

3. If the outside readings or videos are in smaller segments than a class lecture would be (say 5 minutes) will this make the material more digestible? Will modularizing lessons help students especially with attention deficit disorder, or will these modules instead promote more scattered attention to the content.

4. Reading mathematics textbooks is a special skill. Mathematics texts need to be read with a pencil and paper at the rate of a line a minute; in contrast, fiction can be read relaxing on the couch reading at a page a minute. So students are tempted to start a mathematics homework assignment without reading the text, and then go back to the text on a problem-by-problem basis to find a problem like the one that they are working on. Will a flipped classroom help students to engage with their textbooks more actively?

5. Could the flipped classroom be a fad because videos are “hotter” than books? In Marshall McCluhan’s terms books are a “cooler” medium than videos because books demand more effort on the part of the reader. Showing videos in class, if they take up the whole class period, are a waste of time. Do teachers do that because students don’t have access to the media outside of class? Because it’s easy? Print is more effective than video in delivering content unless the video has interactive features. For example, a study showed that news from the printed New York Times was remembered better than news from the on-line New York Times.

6. Problem Based Learning (PBL) worksheets for use in class are very time-consuming to develop because they need to address multiple levels of student preparation, and time-consuming to evaluate. Unless you are using modules for outside of class prepared by others like Khan Academy, preparing the content for use outside of class is time-consuming as well. Could you flip part of a class? Perhaps assign listening to a narrated Powerpoint about a single topic, a Powerpoint that you used with your lecture in a previous semester? (Thanks to Dr. Jennifer Fisler of Messiah College for this suggestion.) Mathematics is skill-based. Could you “flip” a single skill?

7. At the college level, students are supposed to spend two hours outside of class for every hour in class. How can two hours be flipped with one hour? Outside material would have to be lecture plus half of the homework —the homework not covered in class the previous day. So we’re back to the traditional model. Laboratory sciences already recognize that PBL requires twice as much time as lecture, and they recognize that students will finish labs at different rates. Could mathematics be taught as as laboratory science? In the experimental sciences, concepts are exact, but the lab part can be messy. (Thanks to Dr. Richard Schaeffer of Messiah College for that observation in this context.)

8. Flipping a small class is easier than flipping a large class. Students who are home-schooled typically experience a small flipped class. Thirty students using PBL in an hour only allow a teacher to give individual help at the average rate of two minutes per student. Should the students be grouped heterogeneously so quicker students can help slower students?

9. Do I lecture because it’s energizing for me, whereas helping students at their desks is draining? Can I remain non-threatened by questions to which I don’t know the answer if I lose the control of the class that lectures afford?

10. This will only work for college, since secondary school teachers don’t have the option to ask students to leave the class because they didn’t do their homework. Would a “ticket to ride” be an extrinsic incentive to prepare for class by doing the reading or watching the videos in advance of the class? A “ticket to ride” is a little one-question pre-quiz at the start of class that gives students the opportunity to earn the right to attend the class.

 

 

 

Don’t flip out!

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Well, not yet at least. Everyone’s flipping the classroom, but is it really worth it? Yes and no, as NCTM president Linda Gojak explains in her column this week. I don’t always highlight her column, but I especially appreciated the nuanced way in which she approached this trendy subject. There’s something more fundamental that we need to aim for: engaging our students in mathematics and problem solving. Whether we flip or not may be immaterial, as Linda points out.

Here are a few excerpts from her article, which you should check out in full here.

To Flip or Not to Flip: That Is NOT the Question!

By NCTM President Linda M. Gojak NCTM Summing Up, October 3, 2012

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A recent strategy receiving much attention is the “flipped classroom.” Innovative use of technology to enhance student learning makes flipping possible and motivating for students and teachers.

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I believe that we need to go further. As we consider effective instruction that leads to student learning, we must remind ourselves of the characteristics of mathematically proficient students.

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Rich mathematical tasks provide students with opportunities to engage deeply in mathematics as opposed to a lesson in which the teacher demonstrates and explains a procedure and the student attempts make sense of the teacher’s thinking. Communication includes good questions from both teacher and students and discussions that develop in students a deep understanding by wrestling with the mathematical ideas.

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Although the flipped classroom may be promising, the question is not whether to flip, but rather how to apply the elements of effective instruction to teach students both deep conceptual understanding and procedural fluency.

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(more)

All that being said, I still DO want to try flipping my classroom on a small scale, one-lesson at a time basis. I promise I’ll try it someday.

Happy Mean Girls day

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In my Calculus class today I showed just a short clip from this video (“the limit does not exist!”).

Apparently, completely and totally without my knowledge, I showed this video today, on October 3rd, which just happens to be Mean Girls Day. So..happy Mean Girls Day! (happy/mean…that sounds a little funny)

Kudos to the folks who put these videos together!