# Tag Archives: Math

## MATLAB and critical thinking

My apologies for being a little behind the curve on the MATLAB-course-blogging. It’s been a very interesting last couple of weeks in the class, and there’s a lot to catch up on. The issues being brought up in this course that have to do with general thinking and learning are fascinating, deep, and complicated. It’s almost as if the course is becoming only secondarily a course on MATLAB and primarily a course on critical thinking and lifelong learning in a technological context.

This past week’s lab really brought that to the forefront. The lab was all about working with external data sets, and it involved students going to this web site and looking at this data set (XLS, 33 Kb) about electoral vote counts of the various states in the US (and the District of Columbia). One of the tasks asked students to make a scatterplot of the land area of the states versus their electoral vote counts. Once you make that scatterplot, it looks like this:

The reaction of most students to this plot was really surprising. Almost unanimously and without consulting each other, the reaction was: “That can’t be right.” When I’d ask them why not, they would say something like: It looks strange; or, it’s not like scatter plots I’ve done before; or, it just doesn’t look right.

The first instinct of those who felt like they had made a critical error in their plot was to ask me to verify whether or not they had gotten it right. That’s understandable, but it doesn’t go very far because I have a rule that I don’t answer “Is this right?” questions in the lab. (See the instructions in the lab assignment.) Student teams are responsible in the labs for determining by themselves the rightness or wrongness of their work. So it’s time for critical thinking to take center stage — which in this context would refer to using your brain and all available tools and information to self-verify your work. (I wrote about the idea of self-verification here using Wolfram|Alpha.)

Some of the suggestions I gave these teams were:

• Have you checked your plot against the actual data? For example, look at the outliers. Can you find them in the data set itself? And look at the main cluster of data; given a cursory glance through the data set, does it look like most states have a land area less than $10^6$ square miles and an electoral vote count of between 5 and 15?
• Have you tried to create the same scatterplot using different tools? For example, everybody in the class knows Excel (because we teach it in Calculus I); the data are in Excel already, so it would be virtually no work to make a scatterplot in Excel. Have you tried that? If so, does it look like what MATLAB is creating?
• Have you taken a moment just to think about the possible relationship between the variables, and does the shape of the data match your expectations? Probably we don’t really expect much of a relationship at all between the land area of a state and its electoral vote count, even with the outliers trimmed out, so a diffuse cloud of data markers is exactly what we want. If we got some sort of perfectly lined-up string of data points, we should be suspicious this time.

Once you phrase it like this, students pretty quickly gain confidence in their results. But, importantly, most of them have never been put into situations — at least in the classroom — where this sort of thing has been necessary. If critical thinking means anything, it means training yourself to ask questions like this and pursue their answers in an attempt to be your own judge of your work.

I was particularly surprised by the rejection of any scatter plot that doesn’t look like points on the graph of a function. “Authentic instruction” is a term without an operational definition, a lot like the term “critical thinking”, but here I think we may have a clue to what that term means. Students said their scatterplots didn’t “look right”, meaning they didn’t look like what their textbook examples had looked like, i.e. the points didn’t have an overwhelmingly strong correlation despite the existence of a few token outliers. In other words, students are trained by the use of made-up data that “right” means “strong correlation”. So when they encounter data that are very much not correlated, the scatter plot “looks wrong” rather than “looks like there’s not much correlation”. Students are somehow trained to place value judgements on scatter plots, with strong correlation = good and weak correlation = bad. I’m not sure where that perception comes from, but I bet if we gave students real data to work with, it would never take root.

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## Is Khan Academy the future of education?

Salman Khan is a former financial analyst who quit his day job so that he could form Khan Academy — a venture in which he makes instructional videos on mathematics topics and puts them on YouTube. And he has certainly done a prolific job of it — to the tune of over a thousand short videos on topics ranging from basic addition to differential equations and also physics, biology, and finance.  Amazingly, he does this all on his own time, in a remodeled closet in his house, for free:

I can attest to the quality of his linear algebra videos, some of which I’ve embedded on the Moodle site for my linear algebra course. They are simple without being dumbed down, and what he says about the 10-minute time span in the PBS story is exactly right — it’s just the right length for a single topic.

What do you think about this? What role do well-produced, short, simple, free video lectures like this have in the future of education? Will they eventually replace classrooms as we know them? If not, will they eventually force major changes in the way classroom instruction is done, and if so, what kinds of changes?

## Five reasons you should use LaTeX and five tips for teaching it

Over the weekend a minor smack-talk session opened up on Twitter between Maria Andersen and about half a dozen other math people about MathType versus $\LaTeX$. Maria is on record as being pro-MathType and yesterday she claimed that $\LaTeX$ is “not intuitive to learn”.  I warned her that a pro-$\LaTeX$  blog post was in the offing with those remarks, and so it comes to this. $\LaTeX$ is accessible enough that every math teacher and every student in a math class at or above Calculus can (and many should) learn $\LaTeX$ and use it for their work. I have been using $\LaTeX$ for 15 years now and have been teaching it to our sophomore math majors for five years. I can tell you that students can learn it, and learn to love it.

Why use $\LaTeX$ when MathType is already out there, bundled with MS Word and other office programs, tempting us with its pretty point-and-click interface? Five reasons.

1. $\LaTeX$ looks better. Seriously. MathType is getting better at visual appeal — it doesn’t look appalling any more — but nothing beats $\LaTeX$ for refinement and polish.
2. $\LaTeX$ is the mathematical typesetting standard in all technical disciplines and in many related fields. Most, if not all, major publications in math, computer science, engineering, and physics use $\LaTeX$ as the preferred typesetting system. arXiv prefers $\LaTeX$ over all other formats.
3. $\LaTeX$ is becoming a standard elsewhere, especially on the web. Last year, Google Documents added an equation editor that is basically a stripped-down $\LaTeX$ editor with a point-and-click interface. The wildly popular online presentation tool Prezi has said that $\LaTeX$ integration is coming. WordPress.com blogs like Casting Out Nines can do $\LaTeX$, and so can Wikispaces and several other web services. Online $\LaTeX$ typesetters abound, and more are popping up. The web likes open standards, and since MathML is all but impossible to use, $\LaTeX$ fills a gaping need for free, open-source mathematical typesetting. Which brings me to the next point:
4. $\LaTeX$ is free. Free as in beer and free as in freedom. You can download it right now for just about any operating system imaginable, and have the full strength of the system available to you at no cost. And this is a system that has been around for 40 years (if you count TeX) and has millions of users, many of whom actively contribute to the further development of the system by writing specialized packages and macros. This is in stark contrast to MathType, which is proprietary and closed, and although you get the “Lite” version bundled in with office software, the full version will set you back at least \$37.
5. $\LaTeX$ is what you make it. You can use $\LaTeX$ with a point-and-click IDE, or you can type everything out by hand with a text editor and compile from the command line, or anything in between. You can tinker with the low-level creation of fonts or just quickly type out a letter. It’s up to the user. Other proprietary programs force a menu-driven point-and-click approach upon you, which you may like but may not like.

Others may add to these in the comments. But if $\LaTeX$ is so great, how come nobody ever seems to learn it until graduate school? I’m not sure, but it’s not because $\LaTeX$ is counterintuitive. It’s not totally obvious, either, but with a little guidance, $\LaTeX$ can make perfect sense even to high school students. If you’re a math or science teacher, make it a project to learn $\LaTeX$ yourself and start using it in your classes, then teach it to your students. Here are five ways to make that a painless process.

1. Use an IDE or a user-friendly text editor rather than a plain, no-frills text editor or EMACS. For Windows machines, use the free TeXNicCenter IDE that gives point-and-click code insertion (or you can just type the code in) with syntax highlighting. On Macs, use TextMate if you have the money and Aquamacs if you don’t; both of these are text editors with tons of great $\LaTeX$ goodies built in. (In TextMate, for instance, typing begin and hitting the Tab key automatically creates an environment with the matching \end{}. ) On Linux, try Kile. These provide user-friendly interfaces and syntax highlighting that take the edge off some of the learning curve.
2. Have someone else do the installation and setup, or provide a total handholding guide for doing it. The only really hard thing about using $\LaTeX$ is simply getting it to work in the first place. This is one of the advantages MathType has over $\LaTeX$, but the payoff is worth it. New users will need to be walked through the whole process in high-definition detail. But once that’s over, the fun begins.
3. Start small and simple, and build gradually. When first getting students to use $\LaTeX$, restrict them to just a small, relatively simple document, one that’s mostly text with a little bit of math typsetting required. Small, early successes will convince them that learning $\LaTeX$ is worthwhile. I like to give out my training videos to students and have them learn the system on their own; then have a grace period where students get extra credit for doing their assignments in $\LaTeX$; and then start requiring it after the grace period expires.
4. Use it yourself. Students will learn from your example. Try writing your next syllabus in $\LaTeX$; and your class handouts; and your tests (perhaps using the excellent exam package). When you use it, and students begin to use it, they see that they are producing math that looks as good as what the pros do, and they get excited.
5. When you give a document made with $\LaTeX$, also give out the source code that generated it. Students can then look at what you created, ask “How’d s/he do that?”, and get the answer immediately from your code and do it themselves. I myself have learned about half the $\LaTeX$ I know from this method, and adapting/tweaking someone else’s code is a time-honored and very effective means of learning almost anything done on a computer.

Once they are over the initial learning curve and producing beautiful mathematical documents, my students look back on the dark days of MS Equation Editor and wonder, along with me, why anybody would put themselves through something like that. Happy $\LaTeX$-ing!

Filed under LaTeX, Math, Profhacks, Social software, Teaching, Technology, Twitter, Uncategorized

## How to memorize the value of e to 15 decimal places

I learned the following trick for memorizing the value of e from my colleague, Gene White. It never fails to impress calculus students (given a wide enough definition of “impress”).

Start by carefully looking at this picture:

That’s a 20 dollar bill, so memorize “2″ and put down the decimal point.

The picture on the bill is of Andrew Jackson. He was our seventh President, so put a “7″ after the decimal point to get 2.7.

Jackson was elected in 1828, so put down “1828″ next. Since there’s a 2 in front of the decimal place, put “1828″ a second time. We’re now up to 2.718281828.

Now look at the red square over Jackson’s face. The diagonal creates two congruent right triangles with angle measures 45, 90, and 45. So, add on 459045 to get 2.718281828459045. And that’s e to 15 places.

I’m open to suggestions on how to memorize more of the digits.

Filed under Calculus, Geekhood, Math, Teaching

## 12 videos for getting LaTeX into the hands of students

There seem to be two pieces of technology that all mathematicians and other technical professionals use, regardless of how technophobic they might be: email, and $\LaTeX$. There are ways to typeset mathematical expressions out there that have a more shallow learning curve, but when it comes to flexibility, extendability, and just the sheer aesthetic quality of the result, $\LaTeX$ has no rival. Plus, it’s free and runs on every computing platform in existence. It even runs on WordPress.com blogs (as you can see here) and just made its entry into Google Documents in miniature form as Google Docs’ equation editor. $\LaTeX$ is not going anywhere anytime soon, and in fact it seems to be showing up in more and more places as the typesetting system of choice.

But $\LaTeX$ gets a bad rap as too complicated for normal people to use. It seems to be something people learn only in graduate school, with few undergraduates — and even fewer high school students — ever seeing it, much less using it. There is a grain of truth there; $\LaTeX$ is not a WYSIWYG word processor, and the near-programming aspect of using $\LaTeX$ can overwhelm users used to pointing-and-clicking for everything.

But I think that the benefits of using $\LaTeX$ outweigh the costs, and undergraduates and high school students can, and ought to, learn how to use $\LaTeX$ as fluently as they use a word processor for other courses. A couple of years ago, I put together a series of twelve screencasts for use in our sophomore “transition-to-proof” class on learning $\LaTeX$. I put these screencasts online, but mainly they were only advertised to my students and colleagues. Now, however, I’d like to throw these out there for everyone to use.

All twelve of these are done on a Windows system running MiKTeX and the free $\LaTeX$ IDE known as TeXNicCenter.  This provides students with as close to a point/click interface to $\LaTeX$ as you could expect to get. Within that context, there are two basic intro videos:

These two videos are enough to learn how $\LaTeX$ works and will allow you to make a simple file with uncomplicated math and text in it. The remaining 10 videos follow from these two. Some are prerequisites for the others — and those prereqs are stated explicitly at the beginning of any video that has them — but if you watch them in the following order there will be no dependency problems:

Some of these are pretty long, but all totalled (including the two “basics” videos) this is less than two hours of viewing.

When I’ve used these in class, I give students some printed instructions on how to download and configure MiKTeX and TeXNicCenter, and then I have them watch these videos out of class. They are instructed to work along with the videos. I give them about a week to do so. Within that week, if there’s a problem set or something else in the class that could be done with $\LaTeX$, I’ll offer extra credit to students to do so, to incentivize their learning the system. After the end of that week, I will insist that all major assignments have to be done in $\LaTeX$, or else the assignment gets a grade of “0″.

Students have sometimes struggled to get up the learning curve, but if they’re allowed and encouraged to help each other, everyone eventually gets to the point where they are quite fluent writing up homework and so on. Students have even elected to use $\LaTeX$ on assignments in other courses, even non-math courses.

I’m going to use these videos in linear algebra this semester (our transition-to-proof course is now defunct) and I’ll be making up a new screencast on MATLAB and $\LaTeX$. Later, probably during the summer, I’ve been thinking about redoing the entire video series; I now have better screencasting tools than I used to have, and I’d like to keep all the videos under 10 minutes so they can go on YouTube.

So feel free to use these (attributing authorship to me is appreciated but not required), and if you have suggestions or comments, please email them or leave them below.

Filed under LaTeX, Linear algebra, Math, Problem Solving, Technology

## Fractal Doritos!

Students and faculty at University Preparatory School in Redding, CA have created the world’s largest Sierpinski triangle constructed entirely out of Doritos. (Well, it’s probably the only one, but still.) It is 64 feet long and made out of 12,000 Doritos. This was done as an entry to the Doritos Crash the Superbowl contest. Watch, and be awed:

Can a 128-, 256-, etc. foot long Dorito Sierpinski triangle be far behind? I bet the parent company for Doritos would seriously consider some corporate sponsorship.

Thanks to Cory Poole, math and physics teacher at U-Prep, who sent this in. That’s a great, creative way to get students interested in math. (And you can eat it when it’s done.) There’s more on the video here.

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Filed under Geekhood, High school, Math

## Girls inheriting math anxiety from female teachers?

The LA Times reports on a study suggesting that female elementary school teachers who are anxious about math transmit that anxiety to the girls in their classes:

Girls have long embraced the stereotype that they’re not supposed to be good at math. It seems they may be getting the idea from a surprising source — their female elementary school teachers.

First- and second-graders whose teachers were anxious about mathematics were more likely to believe that boys are hard-wired for math and that girls are better at reading, a new study has found. What’s more, the girls who bought into that notion scored significantly lower on math tests than their peers who didn’t.

The gap in test scores was not apparent in the fall when the kids were first tested, but emerged after spending a school year in the classrooms of teachers with math anxiety. That detail convinced researchers that the teachers — all of them women — were the culprits.

It’s no surprise that teachers who are weak in or nervous about a subject do not inspire confidence, or performance, in that subject among their students. What’s different here is the gender connection — female teachers having a pronounced effect upon girl students — and the subject area. It would be interesting to see just how many elementary school teachers view themselves as “anxious” about teaching math, and then to see how that self-description breaks down by gender. Do a lot of female elementary teachers feel anxious about math? Is it more than male elementary school teachers? I don’t know, but that is certainly the stereotype.

At any rate, the opposite seems to be implied by this study too — female teachers who are strong with math and comfortable with teaching it to kids will have an enhanced positive effect on girls’ perceptions of math and their performance with it.  And it seems like a no-brainer that elementary education curricula ought to stress a strong degree of math content mastery among all preservice teachers — of both genders — and demand a high level of fluency with doing and teaching math.Teaching math to little kids is hard, and you have to know a lot of math outside of what you are going to teach if you’re going to do it well. We need to have done with another stereotype: that you major in elementary education because “you just love kids” (you need more than sentimentality to be a good teacher) or because it’s supposedly an easy major (it isn’t, or at least shouldn’t be).

Filed under Early education, Education, Teaching

## Courses and “something extra”

Some of the most valuable courses I took while I was in school were so because, in addition to learning a specific body of content (and having it taught well), I picked up something extra along the way that turned out to be just as cool or valuable as the course material itself. Examples:

• I was a psychology major at the beginning of my undergraduate years and made it into the senior-level experiment design course as a sophomore. In that course I learned how to use SPSS (on an Apple IIe!). That was an “extra” that I really enjoyed, perhaps moreso than the experiment I designed. (I wish I still knew how to use it.)
• In my graduate school differential geometry class (I think that was in 1995), we used Mathematica to plot torus knots and study their curvature and torsion. Learning Mathematica and how to use it for mathematical investigations were the “something extra” that I took from the course. Sadly, the extras have outlived my knowledge of differential geometry. (Sorry, Dr. Ratcliffe.)
• In the second semester of my graduate school intro abstract algebra class, my prof gave us an assignment to write a computer program to calculate information about certain kinds of rings. This was a small assignment in a class full of big ideas, but I had to go back and re-learn my Pascal in order to write the program, and the idea of writing computer programs to do algebra was a great “extra” that again has stuck with me.

Today I really like to build in an “extra”, usually having something to do with technology, into every course I teach. In calculus, my students learn Winplot, Excel, and Wolfram|Alpha as part of the course. In linear algebra this year I am introducing just enough MATLAB to be dangerous. I use Geometers Sketchpad in my upper-level geometry class, and one former student became so enamored with the software that he started using it for everything, and is now considered the go-to technology person in the school where he teaches. In an independent study I am doing with one of my students on finite fields, I’m having him learn SAGE and do some programming with it. These “extras” often provide an element of fun and applicability to the material, which might be considered dry or monotonous if it’s the only thing you do in the class.

What kinds of “extras” were standouts for you in your coursework? If you’re a teacher, what kinds of “extras” are you using, or would you like to use, in your classes?

## Piecewise-linear calculus, part 3: Integration

This is probably the last of three articles on how piecewise-linear functions could be used as a helpful on-ramp to the big ideas in calculus. In the first article, we saw how it’s possible to develop some of the main conceptual ideas of the derivative, without much of the technical notation or jargon, by using piecewise-linear functions. In the second article, we saw how to use the piecewise-linear approach to develop an alternative limit-based definition of the derivative of a function at a point. To wrap things up, in this article I’ll discuss how this same sort of approach can help in students’ first contact with integration, again by way of a hypothetical classroom exercise.

When we took this approach with derivatives, we used the travels of three college students from their dorm rooms to the cafeteria. Each student had a different graph showing his position as a (piecewise-linear) function of time. From these we could get instantaneous velocities. Now let’s consider the reverse situation. A fourth student, Dominic, is traveling from his dorm room across campus, and we have this graph that shows his velocity (in meters per second) as a function of time (in seconds):

Question: How far did Dominic travel in the two-minute span shown here? This is easy, of course, and students get this right away: He traveled at 1.5 meters per second for 120 seconds, so that’s 120 x 1.5 = 180 meters. Distance equals rate times time.

Well, it turns out Dominic has a roommate, named Eric. Eric is leaving his dorm room for a walk too, and his velocity graph looks like this:

Same question: How far did Eric travel in two minutes? There’s a small amount of thinking to be done this time, but it’s still easy: He went 0.5 meters per second for 60 seconds, which is 30 meters; and then 1.5 m/s for 60 more seconds, which is 90 meters. Grand total: 120 meters.

A simple but very important question can be posed here: How come we couldn’t just use distance = rate x time to calculate Eric’s distance travelled? The answer is simply that Eric was not going the same velocity all the time. He had a “piecewise-constant” velocity, so we can use d = rt on either of the two time blocks we want to calculate distance; but we can’t use it globally because his speed changes. In other words: A nonconstant speed requires a kind of “local” d = rt calculation but we cannot use d = rt globally because the r isn’t a single number all the way through.

Now consider Frank, who is following both Dominic and Eric around but whose velocity graph is:

I’ve added the dashed vertical lines just to show where the graph breaks. How far did Frank go in two minutes? Still easy, but this time more work: Total distance = (0.5)(30) + (1.0)(30) + (1.5)(30) + (1.0)(30) = 120 meters.Related question: What does this calculation compute in terms of Frank’s velocity graph? With the dashed lines added in, students pretty quickly see that the sum they did is just an area sum, which we are using because we are doing four local d = rt calculations.

At this point students can stop and think about a few things they are learning:

• Calculating the distance traveled by a moving object cannot be done by calculating d = rt if the velocity changes.
• Instead, we have to “localize” the d = rt calculation by breaking up the time interval into chunks on which the r is constant. Do this on each chunk and then add up the resulting distances to get the total distance.
• This “chunk-wise” calculation is really just finding the areas of a bunch of rectangles.
• “Chunk-Wise” would be a very good name for a rock band. But we digress.
• This is really exactly the opposite sort of thing we did for derivatives. With derivatives, we were given a position function that was piecewise-”straight” and found velocity. Here we are given velocity graphs that are piecwwise-”straight” (actually constant) and finding positions (actually displacements).

Now comes the twist in the problem. We realized, when studying derivatives, that human beings cannot change velocity in an instant. So in the case of Eric above, he cannot possibly go from 0.5 meters per second to 1.5 meters per second without some kind of acceleration in between. His velocity graph is more likely to look like this:

Question: How far did Eric travel now?

Just like when the twist in the problem came for derivatives, I like just to throw this question out there to students and see what they come up with. Most will get the distance travelled on the 0-30 second and 90-120 second interval correct because those are the places where d = rt is in effect. But the 30-90 second interval in the middle doesn’t have constant velocity, so we can’t do that here. I find students do one of three things:

1. Transfer the idea that distance traveled = area under the velocity graph, then use geometry to calculate the area from t = 30 to t = 90.
2. Split the middle interval up into subintervals (usually two of them) and do some kind of rectangle approximation.
3. Average the heights of the endpoints of the middle line segment — that would be a height of 1 m/s — and do a d = rt calculation based on that average.

Each of these three approaches contains a lot of right ideas. The first and third will give them the exact results, and the second one might if they pick the approximations wisely. But any way they go at it, they acquire the right ideas: (1) Distance travelled = area under the velocity graph, and (2) when the velocity graph is not constant, we either approximate or use geometry to find the distance. Note also that if they get this far, they can do any displacement problem like it as long as the graph is piecewise-linear, because they have geometry on their side. For fun, throw in a graph where one of the pieces is below the t-axis and see what they do with it. It goes back to the idea from derivatives that the sign of velocity indicates direction — an idea they will carry with them if their intuition is sufficiently built up at first.

From here it’s an easy jump to start students thinking about non-piecewise linear velocity graphs. Give them one, and ask them to find the distance traveled. The natural thing to do based on their previous work is to try and approximate with piecewise-linear or piecewise-constant graphs. The latter approach is what we call a Riemann sum, and it’s very intuitive to students that more piecewise-constant “chunks” gives better results.

Some ways I think this approach is an improvement on the way calculus textbooks usually do integration:

• The usual approach starts students off with “the area problem” — find the area under the graph of a function, above the x-axis, and between x = a and x = b. There is no real reason given to the students to care about this problem, and the all-important connection between areas and displacement is relegated to the tail end of the section. Instead, here we are developing the notion of area as a necessary tool for calculating distances traveled by objects whose velocity isn’t constant.
• Because the usual approach buries the connection between areas and displacement, by implication it also buries the connection between derivatives and antiderivatives. By contrast, here we are making the connection between velocity and position via areas the focal point of the problem. There will be no surprises once we get to the Fundamental Theorem of Calculus.
• The usual approach presents Riemann sums as the solution to the area by fiat. It’s just “the way we do it”. Here, we build the idea of Riemann sums as a refinement of an intuitive idea, namely that of breaking up the non-constant parts of the velocity graph into constant chunks. Riemann sums are something that the students would have come up with themselves if they’d just been given the chance and the motivation to do so.

As always, I’m interested in your thoughts and criticisms of these three posts. Leave those in the comments.

Filed under Calculus, Math, Teaching

## Simplifying calculus by assuming linearity

Last semester I stumbled upon an approach for teaching the concept of the derivative, and later the integral, that worked surprisingly well with my students. It stems from a realization I had that much of what students see when they first learn about derivatives has very little to do with understanding what a derivative is. The typical approach to introducing the derivative throws students directly into the trickiest possible case: a smooth nonlinear curve, and we want to calculate the slope of a tangent line to this curve at a point. To do this, we have to bring in a lot of “stuff”: average rates of change, tables of sequences of average rates of change, and in a vague and non-rigorous sort of way the notion of a limit. It’s this “stuff” that confuses students — not because it’s hard, but because maybe it’s not suited for their first contact with the idea of the derivative. Maybe we need to build their intuition first.

In a nutshell, the approach is: Assume linearity. All too frequently, students do assume linearity, but in the algebraic sense; they tend to want to think that $\ln(x+y) = \ln(x) + \ln(y)$ and so on. But I mean, assume linearity in the graphical sense. More specifically, the pedagogical idea is use only piecewise-linear functions until students have a sufficiently solid grasp on the concept of the derivative. No smooth curves, no tangent lines, no average rates or limits, until students can explain what a derivative is and what it has to do with slopes.

Here’s how this approach might play out in a classroom.

Consider Alex, a student at our college. Let’s suppose Alex is leaving his dorm room for the cafeteria, which is 100 meters away. His distance $y$ from his dorm room is a function of time $t$ (measure distance in meters, time in seconds). Suppose the graph of this function looks like this:

Question: How fast was Alex going? It’s crucial for students to understand that his speed was the same at all points. If his speed changed, we’d see a difference in shape in the graph; going faster means a steeper graph since he covers more distance in the same amount of time, similarly for going slower. This is the essence of his distance being a linear function of time — his distance changes at the same rate all the time. That rate, or speed, is the slope. So the question is trivial to answer. Alex covered 100 meters in 120 seconds, so that’s a speed of $100/120 \approx 0.833$ meters per second. (That’s about 3/4 of a normal human walking pace.) Students learn at this point that the rate of change in a function has something to do with slope; for linear functions, the rate of change is equal to the slope of the line.

Now suppose Bob, Alex’s roommate, also leaves from the dorm room for the cafeteria, but his distance function looks like this:

Question: How fast was Bob going? It is extremely important for students at this stage to recognize that the answer is: It depends. Bob, unlike Alex, has two different speeds, one prior to the 60-second mark and another afterwards. So the question “How fast was Bob going?” is ambiguous. We have to ask instead: How fast was Bob going at a particular point in time? The answer to this question is what in calculus we call an instantaneous velocity and unless we have a function that is changing at the same rate at all times, any time we talk about a velocity we must be talking about an instantaneous velocity.

OK, so: How fast was Bob going at, say, 30 seconds? Well, at this point on the graph the function is linear, so we can calculate speed by calculating a slope. He is on pace to cover 20 meters in 60 seconds, so his speed at t = 30 is $20/60 \approx 0.33$ meters per second. And of course this is the same speed throughout the first minute. (The 60-second mark will need a little separate treatment.) And what about the second half of the trip? Well, Charlie covered 80 meters in 60 seconds, so the slope/speed is $80/60 \approx 1.33$ meters per second.

Now suppose Charlie, who lives next door to Alex and Bob, is also leaving the dorm room for the cafeteria. (Must be feeding time.), but his distance function looks like this:

First of all, what’s his story? How would you give a play-by-play announcement for Charlie’s short trip to the cafeteria? In particular, what’s different about his trip versus the other two? Students tend to be good at reconstructing stories like this, and they’d say that Charlie headed out the door and made it most of the way to the cafeteria, then had to turn around and go most of the way back, and then moved really quickly back to the cafeteria.

How fast was Charlie going? Again, it depends; but it’s easy to calculate. From 0 to 60 seconds he was going $80/60 \approx 1.33$ meters per second. From 60 to 90 seconds, the slope of the line is negative: $\frac{80 - 20}{60 - 90} = -60/30 = -2$ meters per second. (Implication: Negative velocities indicate an opposing direction.) Then in the final phase, he was going  $\frac{100-20}{120-90} = 80/30 \approx 2.67$ meters per second.

So now students have learned the following important concepts/facts about calculus:

1. Rates of change are calculated with slopes;
2. Functions that aren’t linear have different slopes in different places, so we must talk about the slope at a point; and
3. Rates of change can have different signs (positive or negative) and these signs indicate some notion of direction. (Essentially, we learn that rates of change are vector quantities, not scalar.)

Note well that we have developed all these fundamental concepts without introducing formulas (except the well-known slope formula), limits, epsilons, deltas, Δx’s, or any other technical jargon. This is because we are building students’ intuition and conceptual understanding first, using the simplest possible functions — piecewise-linear functions — before introducing the general case of a smooth curve. Once the students’ intuition and conceptual understanding is built up, then they’re ready to tackle the much trickier case of a smooth nonlinear curve and all the notational “stuff” that this important problem requires.

I have at least 2-3 more posts about this planned. The next one will discuss the crucial step of dealing with functions that are not piecewise linear; how do we use the piecewise-linear function approach to ramp up into the general case of differentiable functions? Then, I’ll talk about how this approach works for developing the idea of the integral. And in a later post, I’ll try to go in to some of the devils in the details of this approach, such as how to deal (pedagogically) with the junction points between the linear pieces and to what extent this assumption of piecewise linearity actually works in general — although some of you who are more knowledgeable in analysis than I am might beat me to it in the comments.