Getting Ready for Day One

A few years ago, I stopped going over my syllabus in class on the first day. For the lower level classes, I created a Syllabus Quiz, asking them when my office hours were (and addressing all the most important points on my syllabus), due on the 2nd day of class. Both semesters last year I began with the Put Your Group on the Axes exercise. I like that, but I want to think more about my first day.

Chris Shore just posted about his first day plans. What I liked most was this idea:

... an American martial artist who toured fighting schools in China ... said that for all the good teachers, he learned everything they were going to teach him in the first lesson, and the rest of his time with them was spent mastering those first-day lessons.

Reading that made me want to clarify for myself what my goals are for my classes, and to plan my Day One activities to point students toward those goals.


My goals for each class:

• help students form a learning community, where they feel safe taking risks
• provide tasks that are deeply engaging
• help students learn to problem-solve
• help students become more persistent (teach them about how useful a flexible mindset is)
• help students learn the 'material' of the course, along with a broader understanding of what math is

My first question to myself, then, is whether my axes activity helps build a community. It definitely helps students bond in their groups of 4, and they use math in an unusual way. But I have noticed that they struggle with it. (And many give up the struggle too quickly.) If I have a group of 4 of us up front model it, then they'll have a better sense of how to do it (and maybe I can get a student to model persistence, even). I'll be working for a few minutes with the whole class; I think that might be a better way to start this activity. In Pre-calculus, we'll begin with this. In both my calculus classes, I'm starting with an activity that gets at the heart of the course. I've figured that out below for Pre-calculus, too. So the axes activity will come after the more course-specific activity.

In Calculus, we'll be starting with graphing y=x², and drawing a tangent line at x=2, freehand. They'll do that in their groups of 4, and estimate the slope of their tangent line. Together, we'll talk about the various meanings of the word 'tangent'.  This task is at the heart of the course. Then they'll do the axes exercise.

In Calculus II, they'll start with finding the area of a circle. I want them to begin to address the difference between definitions (pi is defined to be the ratio of circumference to diameter) and proved theorems (area can be shown, by ancient methods that lead toward calculus, to be half the circumference times the radius). They'll also do the axes exercise.

I'm working now on creating a task for the Pre-Calculus class. It won't be as deep as the other two, unless I come up with something better in the next week. I like to think the students' job in that course is to learn to identify families of functions. So I'm writing up some dialogues that should go with some graphs. One each for linear, quadratic, exponential, and periodic.

...

Here's what I've got for now:

Stories and Graphs

Alicia: “The days are getting shorter now. The sun’s still up until past 8pm, but I’ve noticed that it’s a bit earlier each week.”

Bo: “I wonder what the longest and shortest days are.”

Cristina: “If you go back in my family, I have my two parents, 4 grandparents, 8 great-grandparents – did you know my great-grandma lives with us?”

Diego: “Whoa, mine too! No wait, she’s my great, great. She’s 93.”

Cristina: “We’d have 16 of those. I wonder how far back until everyone’s related.”

Diego: “Hmm, there’s 7 billion people on Earth but it would be less the further you go back…”

Erlino: “I pay $30 a month for this phone, plus 1 cent a minute. Sure glad I don’t pay for data too.”

Foday: “I use about an hour a day. My plan is $40 a month with no per-minute charges.”

Greta: “We visited the Grand Canyon this summer. At one of the view points, I stood at the edge of an incredible cliff. My brother dropped a little rock, and we heard it hit 3 seconds later.”

Hazel: “I wonder how far that is.”

Please draw a graph (with axes labeled) to go with each story. Identify each one as linear, quadratic, exponential, or periodic.

What are you planning for your first day?

First Day of Class: Calculus II and Linear Algebra

I gave myself the gift this semester of no lower-level classes. I think I'll have more mature students, and I hope to have a great time with them.


Calculus II
I've taught this the previous two semesters, so it should be a breeze. We've switched to a new text, but I don't think that will have much effect on my teaching. I have a big class this time (36 or more), unlike the past two semesters. But putting students in groups of 4 makes it so much easier to learn their names.

We did the axes exercise (mentioned here), and I once again loved how it got them talking to one another and reminded me how weak they are on things I think of as pretty basic. (How would you label the axes?)

We also worked on the Calculus Review sheet I've used each term. Part of their homework is to make a list of 5 Calculus I problems that they can't do, and share it with their group tomorrow.


Linear Algebra
It's been over a decade since I've taught Linear, and I knew I needed lots of work on the material over the holidays to be well-prepared to teach it. That task is done.

We're using the text by David Lay (Linear Algebra and Its Applications, 4th edition), and it starts out tougher than many texts. He says, "I think that students' opinions of the course are set somewhere in the first two weeks, and they need to feel the conceptual emphasis early." (page xv, Notes to the Instructor) My colleagues have told me that our students have done especially well with this text. One thing that threw me at first was (from page 35, in section 1.4):

"If A is an mxn matrix, with columns a1, ..., an, and if x is in Rⁿ, then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights; that is,

I don't think I had ever seen that before. I've been talking with Owen Thomas (aka vlorbik) about this course, and he grabbed right onto that when I mentioned it - he loved it. I think I will too, once I get used to thinking this way.

After doing the axes exercise, I gave my students a warmup sheet which was a review of what they've seen before regarding systems of equations.  Of course it wasn't the breeze I thought it would be, so we're already 'behind'. That's ok. Class went well. I had fun showing them three axes on the edges of my desk and in the air.


Pre-Calculus
We did the axes exercise, and compared the different forms the equation of a line can take.

I'm tired - headed home...

First Day of Class

Mostly fabulous. Definitely exhausting. Must write it down so I won't forget the details.

My first class was Pre-Calc at 10, but I had to get to campus before 9 so I could stop by another teacher's 8am calc II class, which had too many students, and recruit for my 1pm calc II class, which had too few students. (Many of the science students are in labs at that time.)

Pre-Calc
I finally got a chance to look at my Pre-Calc classroom and was disappointed to find out it's not a 'smart' classroom - no internet to screen capacity. I showed it to a colleague who prefers chalkboards. He said he'd think about switching rooms with me.

Here are my notes for Pre-Calc:

Before class: Push desks into pods of 4, number the groups

Stand at door with handouts for exercise, and seat cards.

On side board:

·      “Look here every day,

·      make a name tent (diagram) 

·      make sure to sign the attendance sheet neatly (groups diagram)

·      adds at end of week

Axes exercise:  (10:10-10:25)

·      Groups do it

·      Share out

Do ‘good at math?’ exercise  (10:25-10:35)

Talk about brain function and learning: neurons, synapses, myelin sheath, confusion  (10:35-10:45)

http://vv.carleton.ca/~neil/neural/neuron-a.html

Crossing Tiles problem: (10:45-10:58)

Do in groups

Announce: (10:58-11:00)

·      Must have a textbook, can use older edition, see me to learn how to one for under $10, you will  be dropped if you don’t get one (I want to give my effort to those who will care enough about succeeding to take care of their end.)

·      Photos on Tuesday

 

Hand out syllabus pack at door at end

I wasn't able to get into the room before the students did, so my seat cards were useless. But the students helped me move the desks, and the grouping gave us more space between the groups - very nice in our crowded classrooms.

It was so different from my usual first day. I talked way less, and they got to play with math more. The axes exercise went well, and I used it in all 3 classes. I wonder if any problems will arise from not going over my policies. I handed out a 'syllabus search' as part of their homework in 2 of the 3 classes, so they'll be nudged to read the syllabus. The times I used on my plan were no use at all.

[Please let me know whether the links in this post work. They're to documents I've shared in google documents. It's my first time doing this. I am so slow to pick up each new techie bit - that's why I had to go to Maria's workshop.]


Axes Exercise

Work in the middle of the table with your team. Label the x and y axis each with an attribute (no physical
attributes please) such that each dot represents one person in the team.

This came from the Complex Instruction workshop. I loved how much mathematical thinking they had to do while being prompted to get to know each other. For next year, I think I'll change the instructions on the bottom. (People thought the dots had to represent them in the same order they were sitting. Question for CI (Complex Instruction) experts: Is it better to have less instruction on a sheet like this, so they have to figure it out?)

If I were good at CI, I would have been writing myself notes about good things students were doing, so I could let them know how their actions helped their group. Mostly I had no time for that. When I did have a moment, I had trouble hearing enough to know what was going on.

In the Pre-Calc class, I got one volunteer for the share out, and rolled dice for a second 'volunteer'. I had 10 groups but no 10-sided die. I rolled 2 dice, and thought about how the odds were stacked against group 7. I didn't bring that up, but I might another time. We clapped for the brave souls who came up and explained their group's work.

We discussed conventional ideas about what qualities someone who's good at math would have, versus what's really true. That's about all we got to.

I want to remember to use the side board for instructions. I want to keep trying to respond to their questions with questions, instead of answering them. I'm not good at that.


Calc II
I had two hours between class, but I still had to print out my syllabi. I had purposely not prepped much for this class, since I wasn't sure it would go. We got 16 students, so it's good to go. And I know from last semester that more people may join us over the coming week.

This group was slower to get started with the axes exercise than my lower level classes. Once they got into it, they did it well, and enjoyed talking to each other, but they seemed less comfortable with jumping into something that was strange to them.

As they finished, they picked up a Calc I review sheet.

I had no homework sheet for them, so I told them their homework was to find 5 Calc I problems they could do and do them, and to find 5 they couldn't do and write them down to share with their group tomorrow.


Intermediate Algebra
Terrible classroom. One whiteboard covered in information I thought might need to stay, one chalkboard on wheels. But we worked with what we had. I'll try to change rooms, but it might be impossible.

This class is part of a program funded by First Five money, and has a smaller enrollment cap than our usual 40. Unfortunately our computer system can't handle the quirks of this class, so there were lots of people who came, hoping to get in, even though it was full. And everyone who is in still has to register online for this one, but I couldn't get their add codes, because WebAdvisor was down.

It took a while to get all the registration details taken care of, but this is a 2 1/2 hour class (5:40-8pm), so we had plenty of time to do some math. Their homework included putting some fractions in order, and the suggestion to play Flower Power at Manga High if they had any trouble with that.

After our break, I use Energizing Brain Breaks, a little book I bought online; I find a silly physical exercise in it for us to do together, to wake us up. I also talk about how cross-lateral motion is supposed to help brain development.

At 8:30pm, after 12 hours at work, I dragged myself home, mostly content. And now I'd better get to work prepping for today!

When Math Tells a Story

On the Living Math Forum group, I claimed that Algebra 1 tells a story more than Algebra 2 does. N asked me to explain what I had in mind. Here's my reply (with a few revisions):

Lately I've been saying this sort of thing in my pre-caclulus class, not about the course, but about individual equations. We say math is a language. If it is, then we should be able to tell stories in it. Each equation makes a statement, and sometimes those statements tell stories.

The equation for a circle is (x-h)² + (y-k)² = r² . Many students see each equation like this as separate from any other equations/formulas they know. I try to get them to look at this deeply. From the structure of it (square plus square equals square), I see that it's really the Pythagorean Theorem. Why would something for right triangles show up in the equation of a circle?! (That blew me away a few years back. I've been teaching math for 30 years, but that question seemed deep.)

It's because our coordinate system has the two axes perpendicular to each other. So the distance from the x-coordinate of a point on the circle to the x-coordinate of the center is measured horizontally and the similar y distance is measure vertically. You can build a right triangle from the center to (almost) any point on the circle. The constant radius is the hypotenuse of that right triangle.

So this equation tells a little story.

How does a whole course tell a story?

Algebra is about solving equations and about graphing. We want to see how real life situations (anything with data that has two components, like time and height) can be represented with equations and with graphs. In Algebra 1 students learn to solve simple equations and to graph. And hopefully they learn how the two skills are connected. There's a bit more. Systems of equations allow us to model slightly more complex situations, using more variables. And in my (community college) Beginning Algebra course (which is pretty much equivalent to a high school Algebra 1), our grand finale (after factoring) is graphing and recognizing equations of parabolas.

Near the beginning of the course, I introduce my favorite problem. I bought a tree and planted it. It was one foot tall at first. It grows two feet a year. Let's make a data table for height versus time, and a graph, and an equation. What does the input variable (let's use t instead of x) mean? What does the output variable (let's use h instead of y) mean? What does the slope mean? What does the y-intercept (or h-intercept) mean? The graph and the equation both tell the story of the tree. Linear growth is modeled with lines, which have equations of the form y = mx + b (or, in our tree story, h = 2t + 1). You can come at that from so many angles.

The course can tell the tree's story, or any story that can be told through data, graphs, and/or equations. It tells the story of using math to help us think quantitatively about problems we care about. (And of course there are plenty of things we care about that cannot be quantified.)

Today in Class: A Good Day to Do Without the Textbook

My son had a doctor's appointment, so my morning classes didn't meet today. Maybe I came into my afternoon class with more energy, I don't know, but it felt really great. I gave a short mastery test on solving equations. Five questions; last one was a story problem. I think I'll break it up into two grades - one for solving equations, and one for the story problem. That way lots of them can know they've passed the equations portion. (Because I don't think many will have the story problem right.) After the test, I started our new unit on graphing, and it seemed to go really well. (Three of the students with difficult behavior had left after the test; that might have something to do with it.)

I started out with the function game. We had played it before, so this wasn't new. I chose a harder relationship than the ones I'd done before, and not many people were seeing it. Here's how it goes: I have one student up front as the scribe. I ask students for a number, and then I say "Casey said 4; I say 11." The scribe writes it down in a two-column table, and I call on someone else. Each time we play I've told them: "The most important rule of the game is - don't say the rule! If you think you know what's going on, say the number I'm going to say." After a few numbers are up, some of the students will call out the number I'm supposed to say. This time only a few people were getting it.

I decided to use this function game as my entree into graphing. I had meant for it to be one example among many, but when I saw that people weren't seeing the rule in my head, I thought maybe the graphing would help some people see it. And that would help them see the power of graphing! So I told them I wanted them to plot all the numbers we had on the board so far. Their first point would have an x-coordinate of 4, and a y-coordinate of 11. I walked around as they were doing that, and helped a few people with the typical difficulties (0 to 1 is often about twice as big as 1 to 2 and all the rest, or they put tick marks between the blue lines instead of on them so it's hard to be accurate, ...). After that I got to mention axes, quadrants, and all that. We got to see the linear relationship, and we got to talk about the rule (multiply by 3 and take away one) and what it would look like as an equation, y (or output) = 3 * x (or input) - 1. This was so much more fun than doing section 3.1 in the book, where all they do is plot points, identify coordinates, and plot data for (number of years since 1970, number of Walmart stores)!

We did another function game and its graph, and then we did a worksheet from Maria Andersen's Algebra Activities workbook (from the free teasers pack pdf).

Students kept asking me how to determine the equation of a line from points. (They didn't ask it that clearly.) And I kept asking them to wait until we had built up a bit more background. I answered that question just before class ended, by doing the function game a third time, and stopping after we had just two number pairs.

I think this is the best intro to graphing I have ever done. I hope it goes as well in my morning classes!

Tutoring and Conics

My tutoring sessions with Artemis* continue to go well. I never prep, except for fleeting thoughts about topics and problems he might enjoy. Today, I showed him the books that came a few days ago from James Tanton. (I will blog about them soon.) We started to look at Thinking Mathematics, Volume 3: Lines, Circles, Trigonometry and Conics, and before we'd looked at two pages, I asked if he'd like to figure out the equation for a circle. Sure.

I drew a circle, and asked him for the definition. He said, "All the points are the same distance from the center." I said that what we were going to do is called analytic geometry - the marriage of geometry and algebra that Rene Descartes helped found. I drew x and y-axes and asked him what we should call the center. He said (x,y). I felt bad that I'd asked the question, since I didn't want to use (x,y) for the center. So I talked about how we have a tradition of using x and y for the points that would move around (said while tracing over the circle), and so the center would traditionally get another name. One tradition would be just to use (a,b), another would be to use (h,k). I have no idea why we use h and k... He chose a and b.

I pointed to his definition and asked how we'd think about the distance. He said, "Use the Pythagorean Theorem." My algebra students tend to think the 'distance formula' is something separate, so of course it tickles me that he thinks of it in a way that feels more basic to me. I drew a triangle with a bad circle around it...


...and had him tell me what to do next. He worked it out to and I told him the tradition is to write it as .

Then we looked at and completed the squares together. He told me the center and the radius.

We decided to move on to other conics, and started with the definition which states that a parabola is all the points equal distance from a focus and a directrix (a line). But we also were talking about how you cut the cone to get the conics, and I said I had never figured out how we know that a plane cutting the cone parallel to the side gives us the same thing as this definition that uses focus and directrix.

We worked it out for an example, using x,y,z coordinates. After a bit of fiddling around, we called our cone and our plane . We used a 3D graphing calculator to check whether it was right. A bit of algebra gives us , which is a parabola opening to the left. That was nice, but has nothing to say about focus and directrix, and of course it's not a proof.

We decided to look it up. The explanation I found is for an ellipse, not a parabola, but we decided to work our way through it. It's titled Dandelin's Spheres, after the French/Belgian mathematician Germinal Dandelin (1794 – 1847) who came up with this proof - and it's dazzling! I've never seen this before, and I want to know why - it's so elegant, and pretty simple. At the end, it says the hyperbola and parabola can be thought through following almost the same steps. I'm going to do it!

This was the first time I did some stretching mathematically while tutoring him. It's going to happen more and more. I'm very curious when he'll "outgrow" me.


______
*Artemis (not his real name) is 8, and is very smart.

Note: The equations were done at CodeCogs. I had to redo the drawing because the website I used to put it up is gone now.

Math is Like Mountain Climbing

It must be true; I've read it in 3 different places! :^)

I like the analogy: they're both very hard work, and both give their enthusiasts lots of pleasure in what they achieve, along with a view of the world that most people don't get.

~~~

In Out of the Labyrinth: Setting Mathematics Free, a book full of wisdom gleaned from their experiences conducting math circles, Robert and Ellen Kaplan write:

Those who love to climb mountains have a very different view of them, and it may be no accident that so many mathematicians are also mountain walkers and climbers. It isn't just the exhilaration of solving the rock face, but the fresher air along the way and the long views from the top that draw them on. ...

We aim to take acrophobia away by having our students do the climbing however they will, with us as their Sherpas. We bring up the supplies and peg down the base camp; we point out an attractive col or a dangerous crevasse; but they do the exploring on a terrain we've brought them to. (page 11)

In an earlier description about why math might be particularly difficult to teach well, they write:

Fall from a ledge and the odds are slim that you'll climb back up to and past it. (page 8)

The handholds seem to grow fewer the higher you climb. Mathematics is all ledges. You no sooner acclimate yourself to breathing the thin air at this new height than the way opens up to one still higher... (page 10)

~~~

In The Art and Craft of Problem-Solving, one of the first treatments of problem-solving I've found that doesn't just regurgitate Polya's lovely four step process*, Paul Zeitz writes:

You are standing at the base of a mountain, hoping to climb to the summit. You first strategy may be to take several small trips to various easier peaks nearby, so as to observe the target mountain from different angles After this, you may consider a more focused strategy, perhaps to try climbing the mountain via a particular ridge. Now the tactical considerations begin: how to actually achieve the chosen strategy. For example, suppose that strategy suggests climbing the south ridge of the peak, but there are snowfields and rivers in our path. Different tactics are needed to negotiate each of these obstacles. For the snowfield, our tactic may be to travel early in the morning, while the snow is hard. For the river, our tactic may be scouting the banks for the safest crossing. Finally, we move onto the most tightly focused level, that of tools: specific techniques to accomplish specialized tasks. For example, to cross the snowfield we may set up a particular system of ropes for safety and walk with ice axes. The river crossing may require the party to strip from the waist down and hold hands for balance. There are all tools. They are very specific. ... (page 3)

As we climb a mountain, we may encounter obstacles. Some of these obstacles are easy to negotiate, for they are mere exercises (of course this depends on the climber's ability and experience). But one obstacle may present a difficult miniature problem, whose solution clears the way for the entire climb.For example, the path to the summit may be easy walking, except for one 10-foot section of steep ice. Climbers call negotiating the key obstacle the crux move. We shall use this term for mathematical problems as well. A crux move may take place at the strategic, tactical, or tool level; some problems have several crux moves; many have none. (page 4)

After so richly developing his metaphor, Zeitz uses it to explain mathematical problem-solving. For example:

Let us look back and analyze this problem in terms of the three levels. Our first strategy was orientation, reading the problem carefully and classifying it in a preliminary way.Then we decided on a strategy to look at the penultimate step that did not work at first, but the strategy of numerical experimentation led to a conjecture. Successfully proving this involved the tactic of factoring, coupled with a use of symmetry and the tool of recognizing a common factorization. (page 6)

I haven't finished this book. It's full of hard mathematical problems that I can come back to over and over - a whole mountain range I can carry with me!

~~~

Each of these 3 authors has used the metaphor to a slightly different purpose. Mike South's piece (here) reminds me of Lockhart's Lament. They're both about how destructive 'teaching' can be in math. As a teacher, I continue to struggle with this conundrum.

I'll attempt to tantalize you with the beginning of his essay:

On the distant planet of Lanogy, all of the population centers are in sight of ... mountains. It's not just because there are lots of mountains on the planet, although that is also true. There are certain resources which are only available in the mountains. You need them to build cities, hence the proximity. But not only that, the mountains are the source of resources needed to facilitate trade in Lanogian economies, so all trade that takes place is near mountainous areas out of convenience. In addition to that, many other technologies turn out (some times unexpectedly) to benefit dramatically from the resources the mountains have to offer.
Now, interestingly enough, despite how useful the mountains are, almost no one on Lanogy likes them. Spending time in the mountains voluntarily is, to almost anyone you talk to, such a laughably improbable concept that it would only occur to them in jest. Everyone knows that builders and commerce agents have to do their share of mountaineering as part of their jobs (in fact, that very fact encourages a lot of people to eschew those professions), but the only people that would ever spend most of their time there would be the mountaineers. These very rare and very peculiar people (those whose only job is to climb mountains) might, possibly, do it voluntarily. But what they would do, why they would do it, and, indeed, what they do professionally is a complete mystery to the rest of the Lanogians.
Now, the reason for this general dislike of climbing in and retrieving resources from the mountains could be due to the simple fact that most people are really not very good at it. Now why, in a society that can obviously see the value of the resources obtained from the mountains, people still aren't good at climbing them, is widely disputed.

I am not into mountain climbing myself. Too scary. But I can wish I were braver, and I can understand better so many people's fears of math by reading these pieces. I can also get better at math myself by using Zeitz's strategies, tactics, and tools.

---
Polya's work, written back in 1944, was so helpful most of us mere mortals haven't discovered anything much more to say. I wrote a problem-solving handout for my classes, in which I updated Polya's language, and added a few ideas too basic for him to have included, like: Write 'Let x =' the quantity you're trying to find. Maybe I can write a separate post on Polya, and include it there.

Calculus: Reviewing Functions with Desmos

I like to dive into the calculus ideas in my calc I course, so I do not start with review. (I use a just-in-time approach, reviewing what we need when we need it.)

But I know that the students' understanding of functions is weak and needs to be brought to mind. So I was excited about having them outline their own faces in Desmos as a homework assignment, which I learned about at Twitter Math Camp from Deb Boden (@debboden).





Here are my instructions:

Desmos Graph of Yourself

1. Set up an account on desmos.com. (It’s free.)
2. Upload a selfie into desmos. (Click the + in the upper left corner of the desmos calculator screen to add your image. Photos with you facing front are easiest to use.)
3. Use various functions to outline features of your face. (At least: lines, arcs of circles and ellipses, parabolas, and trig functions. Try including exponential and log functions, hyperbolas, and cubics.)
4. When you’re done, you can hide your photo to display the icon you’ve created. You can also hide the axes by clicking on the wrench in the upper right corner.
5. Add a link to your completed desmos work on our class google doc: [Link removed, for student privacy. I didn't need google, actually. They could have submitted directly to Canvas. But we may need the google doc for our viewable collection.] (My icon is linked there. You can check it out to figure out how to do this.)
6. We’ll share these in class and see how many classmates everyone can recognize.

Every time I got stuck, I googled my question. For example, "Desmos function restrictions" helped me make short pieces of the curves I used. If you are still stuck, start a discussion item here.


Rubric
50% Required function types  (lines, arcs of circles and ellipses, parabolas, and trig functions) 10% each (Extras can bring this score up to 60%)
15%   Good Match with Photo
15% Visually Engaging
20% for using Desmos and Canvas (our "learning management system")

(I didn't post the rubric until after they turned in their work, but I will next time.)

31 out of 44 students turned it in. I am loving Canvas, which our college just started using. (We used d2l before and I hated it. Yes, I have strong feelings about things.) I took hours grading this, but once I get good at it, I think I could do this in about an hour. Canvas made it easy. And now I know that a third of my class is having trouble. So I know I need to intervene somehow. Good information to have.

Here are some of my favorites...

Math Poems

I was in a mood one day, and this was my response to a math question...
(Do you know any math poems?)



Imaginary Numbers Do the Trick
by Sue VanHattum
(written on December 15, 2008)


In an email group of 4,000 homeschoolers,
a member wrote:
   My son asks, “The square root of 1 is 1,
   So what's the square root of -1 ?”
This was my reply to her…

What we call the real numbers
is everything on a number line,
positive, negative, zero.

If you're thinking about those real numbers,
on that number line,
none of the negative numbers can have a square root,
because anything times itself will come up positive
(or zero).

But, once upon a time (for real),
mathematicians dueled
by giving each other lists of thirty hard problems.
The winner got recognition
and perhaps a job.

All this dueling led to
a solution for cubic equations:
these mathematicians
created a formula
that would find the numbers
that would solve a thing
like 2x3-3x2+4x-5 = 0.

But that formula was a problem!
It came up with square roots of negative numbers,
which drove the mathematicians wild.
No, no, no. There is no such thing!
Well, maybe there could be…
and if there is,
what would it look like?

With a wave of the magic wand of imagination,
These mathematicians
made up a new number,
which later got the name i.
(Imagine it written in fancy script.)
i is the square root of -1,
so i squared must equal -1.
i is the first step in creating …
the imaginary numbers.

Picture, if you will, a new number line
of imaginaries
crossing the line of real numbers at 0,
with the real number line horizontal,
and this new one vertical.
(It looks just like x and y axes,
but now it's all one number system,
a bit more complex.)

i sits one step above zero.
Another step up this imaginary number line,
we see 2i,
2i is the square root of -4.
(It is?!
Why yes, 2i times 2i equals 4 times i squared,
and i squared equals -1,
so we get -4.
Cool, huh?)
And on it goes.

Now all of this wouldn't really solve much
if there were no square root of i,
and that seems too weird to think about.
But, once you study trigonometry
(how'd that get in here?!),
the solution to that little problem
is actually quite elegant.

Day Two

8am
Calculus. I talked about what we had done yesterday with finding a line tangent to y=x² at x=3. In algebra, we find the slope when we are given two points. We know one point, (3,9), and there is no other point that we know. [Last semester, at least one person used the y-intercept of the tangent line they had graphed as their second point. I liked that, but forgot to mention it today.]

I asked them to give their definitions of the word tangent.
First student definition of tangent: A line that touches the curve in one place only.
Sue's counter-example: I drew y=x³ and drew at tangent line at about x=1. They agreed that I had drawn a tangent. Then I extended the curve and the line. They cross at x = -2. I suggested that we could add the word nearby, and maybe this would work.

Second student definition of tangent: A line that touches the curve but doesn't cross it.
Sue's counter-example: I asked them what the tangent to y=x³ at x=0 would look like. They told me it would be horizontal. I drew it in. Hmm. (I told them that later we'll talk about concavity, and showed it with my hand curved. I said that I think the only time the tangent line crosses the curve is when it's tangent at an inflection point. Is that true? I should try to prove it.)

Third student definition of tangent: A line that determines the direction of the curve.
I think this one is about as good as we can get at this point, although it's hard to turn it into something precise. I talked about thinking of the curve as a road, and your point being a car driving along the curve. Its headlights make half the tangent line, and its taillights make the other half.

Talked just a bit about history of calculus, and gravity. Got some volunteers who will drop a heavy and a light object, and see what happens.

Then we started our circle activity. I had a picture of a circle of radius 10cm on the back of the handout. I asked for the radius, a rough estimate of the area, and a more careful estimate of the area. (I asked them to pretend they knew no formulas. Next I had them fold a round coffee filters in half through the middle over and over, then cut it into wedges, and play with them. Tomorrow we'll do the area formula from that. Today I gave the definition of pi (C/D), and talked about how C=2*pi*r comes easily from this definition. I got a few volunteers who will measure around a circle and across it, using string, and will bring in their string tomorrow. Area is different...



10am
Linear Algebra. I used a desk corner as the origin, drew the x and y axes with my finger along its edges, and the z axis coming up from the corner. I asked them to figure out (in groups of four) what the equation x+y+z=1 would look like. I heard someone say circle. It is not at all obvious to most of them yet that it will be a plane. But we got there.

Was that before or after we worked on the definition of a linear equation? Yesterday I had asked for their definitions from their heads. I got four volunteers today (yay!) to give me their definitions to put on the board. They were all different, and none matched the official definition. So, after I went over the official definition from our textbook, I asked them to use that to prove or disprove each of the statements given by students. I think this will help them with proofs and with what a linear equation is.

Next I continued with the problem we had done, algebra style (no matrix), yesterday. I talked about computers, and representing it with just the coefficients, and wrote the matrix. I showed them the matrix that would represent the solution, and said our steps will be similar to those we used yesterday, but our order will be different. We did our same problem matrix-style, and I identified the three elementary row operations as we used them. (We never used the swap rows operation, but I talked about when it would be needed, and how you'd never do that with the algebra-style method.)

I finished up with one book problem.


11am
Pre-Calc. Stamped their homework. Had them share with their group the list of 5 problems they couldn't do. Had them each pick a problem from their partner's list, that they would later explain to their partner. Some people working hard; others feeling unsure what to do. (Everyone willing to participate.)

Showed them y=mx+b on desmos, but got caught up in another problem. We'll come back to this tomorrow.

They worked on finding a_n, with the hint that it might be good to find a₁₀₀ first, for the sequence 12, 17, 22, 27. (I got starting value and jump size from students. Good it was five - some people struggle with arithmetic.) We worked on that a while, and then did a problem from 12.1 (Stewart) that turned out to be geometric. It was good to see the similarities.

I loved my day. Now I'm off to the chiropractor.

What are our intuitions about temperature?

I'm teaching exponential functions and logarithms in pre-calc right now. That means it's time to pull out my murder mystery, in which they will use logarithms to solve an important problem - which of their classmates killed John Doe? Since the murder mystery uses temperature to find the killer, I want to lead in with some thinking about how temperature changes over time.

On Wednesday and Thursday, I told my classes a story, and asked them to draw a graph. I said I was mixing some cake batter up to make a Halloween cake. I asked what temperature it should cook at. We decided to set our oven at 350 degrees. (In one class, I talked about how silly the Fahrenheit temperature scale is, but how, even with Centigrade, zero is just attached to water freezing. It's not the same as zero length, volume, or weight. Temperature is different...)

I also asked what temperature the batter was now. They told me room temperature, and we decided that was about 70 degrees. Then I drew axes on the board, labelled them, and asked the students to graph the temperature of the batter over time. Only one person (out of over 50 in the two classes) came close to the right shape. No one seems to have much intuition about how temperature changes. I did this once before, with the cooling coffee we always think about, and got slightly better results.

Here are my approximations of what students thought:

The green one may have been influenced by our attention in the past week to exponential growth, while the purple one seems to have taken the exponential growth we were studying and limited it by the temperature of the oven. I have often seen students give a linear graph like the blue one, and a logistic-like graph like the orange one. No one wants stuff to heat up fast at first, and then slower.

What makes their intuition bad here? Is there a physical experiment / demonstration we could do to improve their intuition? What would make exponential decay feel like the natural choice to them? Maybe cake is the wrong object to be heating?

Please help me think about this.