Integrated Math, Part 1 – Don’t Grant the Premise

As I’ve mentioned before, a couple years ago we made a decision as a department and school to replace our Algebra 1 through Precalculus courses with a sequence of integrated mathematics. Since September, a colleague and I have had a one-course teaching reduction so that we could develop these courses. In addition to developing the curriculum itself, we’ve been working on administrative details, including a clear, concise justification of why we’re making this curricular move. We feel pretty good about the justification at this point; in one concise, well-written page (for which I deserve none of the credit), it describes the reasons for moving to integrated math in a way that we can easily convey to prospective students, their parents, and other interested constituencies.

What we don’t have is a similarly strong and concise justification for colleagues in the math department, teaching candidates, and math teachers from other schools who might be interested in what we’re doing. This hasn’t been a pressing concern, but it is something I think we need. We don’t yet have complete buy-in from our department, which isn’t surprising since we’re talking about something that hasn’t been taught before and is still in development. As we continue to talk about it, it’s time to put something in writing that will serve our needs now and for the next few years. I was having trouble figuring out how to structure the argument until recently, when the  College Board’s AP Calculus listserve took care of it for me.

A few weeks ago, there was a question on the listserve about course sequencing. The question, which is frequently posed in math departments around the country, is which is a better sequence of high school mathematics courses: Algebra 1 – Geometry – Algebra 2 – Precalculus, or Algebra 1 – Algebra 2 – Geometry – Precalculus. There are benefits and drawbacks of each option, the most signficant of which revolve around when students forget the algebra they’ve learned and how much of that algebra will have to be reviewed, or retaught, later.

The debate over when in the sequence to teach geometry makes an implicit assumption: high school mathematics should be taught using these four courses to lead to calculus. While mathematicians are taught to look for fallacies in arguments, we seem to miss this one quite often. To argue over when to teach geometry is to grant the premise that these four courses are the best way to prepare students for calculus (and, more broadly, to be mathematically literate adults). If we refuse to grant this premise, the answers to the question suddenly become very different.

So let’s not grant the premise, but instead start with the questions of how do we best prepare students to study calculus, understand what it means to think mathematically, be mathematically literate, and see that there is much more to mathematics than solving equations. These were the questions that we started with and have kept in mind. In addition, we  ignored “the way things have always been done,” what textbooks are available, or what we’re most comfortable with (but, because we’re a college prep school,  we did pay attention to the SAT, ACT, and the college process). By not granting the premise, we came up with what we believe is a strong sequence of courses that pursues these goals. Over the next few weeks I’ll start to share some details of how we’re building these courses, what content we’re including (or excluding), and why we made the decisions we made. I hope people will tell us that they think and (especially) what hidden assumptions exist in our own work that we might have missed.

The Day My Students (Almost) Invented Calculus

One of my favorite things to do in Precalculus Honors is ask my students, “How do polynomials behave?” They started working on this deceptively simple question in early November and are still plugging away at it. In the process they’ve had to come up with a definition of polynomial, decide what we mean by “behave,” figure out a way to tackle the question, and then look for patterns from which they can develop conjectures that they can make convincing arguments for. In other words, they’re being mathematicians, not just doing math. Some people just stare at me in disbelief when I tell them that my honors class is in its seventh week of working with polynomials, but if didn’t spend this much time on it or do it this way, then we wouldn’t get classes like the one we had today.

A couple of groups in the class had been working on the patterns they were seeing for turning points on a graph, which are things like those labeled “A” and “B” in the graph below:

Cubic1

They had looked at a lot of graphs and were pretty convinced that for a polynomial of degree n, there should be n-1 turning points. (The truth is that there will be at most n-1 turning points, but I wanted them to explore their conjecture themselves, not just wait for the right answer from me.)

While this conjecture made sense to them, it led to some challenges in addressing curves like this:

Cubic2

This curve has no apparent turning points, but they knew that it needed to have some in order for their conjecture to hold, so they spent a lot of time trying to figure out where the turning points were. They come up with something like this:

Cubic3

When they looked at points “A” and “B” they knew that they were not maximum or minimum y-values, which they had figured out happened at turning points, but they felt like there was a distinct change in the shape of the graph at these points and so these must be the turning points.

As they were presenting their work to the class, there was a lot of discussion about these non-extreme-value turning points. Everyone was having trouble figuring out exactly where they were or how to define them. One member of the group said that she wanted to think of them as the places where the rate of change changes, but she knew that wasn’t correct because she knew the rate of change (slope) was continuously changing. This led a classmate to suggest that maybe they could pick a couple of points that were close together, find the slope between those points, and use that to get a sense of the rate of change. Prompted by another classmate, he went on to say that by doing that in a couple different places they’d be able to compare the rates of change, and by keeping the points really close together they should get at least a decent idea of what the rate of change was.

In other words, he was suggesting approximating the rate of change by calculating the slope of the secant line between points that were close together.

In other words, he was about 20 minutes from developing an intuitive definition of the derivative.

I do a lot of student-driven inquiry work with my classes because to me it’s like a road trip – we have a general sense of where we’re headed, but sometimes we take a detour because something interesting catches our eye. This was one of those times. What I really wanted to do was let three or four students starting playing around with this question of approximating the rate of change. To do so would have been an amazing, fun experience that would have let them continue to develop as mathematicians. Unfortunately, it also would have meant that I wouldn’t have gotten to something like sequences and series, plus I would have stolen the thunder of the AP Calculus teacher. Still, it does make me wonder – how flexible could our courses be so that we can follow a line of inquiry without worrying about going past what’s on our syllabus?

My Christmas Present from my Students (and My PLN)

I have really come to love Twitter for the way I can use it as an on-going source of professional development, and Wednesday’s AP Physics 1 with Precalculus Honors course is my latest, greatest example of why.

It all started Tuesday night when Frank Noschese (@fnoschese), a physics teacher in New York, retweeted something from Monica Owens (@MonicaKonrath), a physics teacher in Indiana. First of all, I’ve never met either of these people, but without them Wednesday’s class wouldn’t have happened. Anyway, here’s her original tweet:

Now, our plan for Wednesday, which was our class’s last meeting of the term, was to give a couple quizzes, work for a while on momentum problems, and then continue our open inquiry on polynomial equations. This activity sounded way more fun, and I really liked the idea of looking at texting while driving right before the students leave for break, so we decided on the fly to do this instead. The results were even more amazing than we’d hoped.

We started with a simple question: “If you sent a text message while you were driving, how far would you travel while sending the message?” They looked at us for a couple seconds, then turned to their groups, and what followed was music to our ears:

  • “We’re going to need to make some assumptions.”
  • “If the velocity is constant then we can ignore acceleration.”
  • “Let’s run some trials to see how long it takes to send a text.”

It really is quite a nice gift to hear that the students have internalized at least some of the things we’ve been working on all year. Because things were going well, my teaching partner suggested we add a second question: “Suppose you collide with a stopped car while you’re sending that text. How much force will the seat belt exert on you?”

This is a nice little momentum/impulse question that extends the life lesson we’re trying to get at with this problem, and it was once again a joy to see that they immediately recognized it as such and went to work on it. When all was said and done, we had a nice collection of whiteboards to share. Here are a couple of them:

2015-12-17 09.47.32 2015-12-17 09.47.43

We finished up by trying to give the students some context. First, we did some quick unit conversions so they could see that the force of the seat belt is the equivalent of having 1000 pounds or more sitting on your chest. Second, we took a little field trip to the street that borders the school, where the speed limit happens to be the same as the one assumed by the white board above. We measured out 80 meters so they could see how far that really was. We then noticed that in this distance there were seven cars parked by the side of the road, any of which a driver might hit while looking down. We also saw two pedestrians, one jay walker, and an illegal U-turn, which were just additional opportunities for a terrible accident.

It really was a lot of fun, and we think pretty powerful, to tie the physics we’ve been doing to a real world issue that teenagers (should) hear a lot about, and it was a great way to close out the month. Thank you, PLN, for bringing me just the right activity at just the right time. Happy holidays to all.

An Unintended Benefit of an Interdisciplinary Course

I co-teach an interdisciplinary course in AP Physics 1 and Honors Precalculus. Combining these two courses, which we did for the first time last year, was a no brainer: much of the math we do in the precalculus course ties directly to the math needed in physics; we tend to do some physics applications in a precal course, like projectile motion and simple harmonic motion; and there’s a big emphasis in setting up and solving word problems in both courses (although in physics they’re just called “problems”).

There are some other ways to connect the courses, too. Physicists and mathematicians work with vectors with different levels of rigor and formality, for example, which allows us to highlight the difference between “doing math” and “using math to do science.” Similary, what we can know or prove in math is very different than what we can know or prove in science, yet there are similarities in the rigorous structures the two disciplines use to collect evidence and draw conclusions. Seeing and understanding these parallels seems to help the students understand their work in both disciplines better than they otherwise would.

Today, we discovered a new benefit of having this interdisciplinary course and using team teaching for it. Our students recently took a quiz on solving non-right triangles, and even though they’ve been quizzed on this before, many of them are still making the same mistake: failing to check for multiple solutions in the SSA case. They also took a physics quiz that had an object moving down an incline plane before launching off the end of the ramp and falling to the ground. They did the first part of the problem just fine, rotating their coordinate axes to be parallel and perpendicular to the incline plane. When they got to the second part of the problem, which dealt with projectile motion, many of them made the same mistake: they left their axes rotated but solved the problem as if they weren’t.

Were we teaching these classes separately, we each would have addressed these mistakes in our own ways and moved on with class. Because we’re co-teaching and looking at student work together, we realized that the mistakes on both quizzes really stem from the same issue. In both cases, the students needed to recognize what kind of problem they were dealing with and then proceed based on that recognition. We were able to talk this through with our class, helping them see that this was less an issue of knowing specific content and more an issue of how to approach and solve problems. Had this not been an interdisciplinary course, it’s likely that no one, including the teachers, would have made this connection. Score one for interdisciplinary learning.

Can We Make Proofs Less Tedious?

My goal for the school year was to write one blog post each week. I made it two weeks before failing to reach this goal. I originally started this post in late September, and then school happened. I’ll try to be better, starting with the completion of this post.

One of my colleagues came into the office a few weeks ago and told me a story about his statistics class. They were working on samples and populations, and his class was struggling to understand the difference between the population and the population size. (For the record: the population is a group of interest, such as students at school, and the population size is the number of objects in the group, which in this example would be 502.) Frustrated, he related this struggle to his wife, who immediately identified the conflict. She noted that if a student is asked in history class what the population of France is, the student is going to respond, “66 million,” not “all the French people.”

It occurred to me that this is similar to something I happened to be thinking about this morning. I was thinking about geometry and proofs, and thinking about some of the hoops we ask students to jump through in completing two-column proofs (for the record: I do not like two-column proofs). A common occurrence in proofs is to see two steps like this:

  1. Angle A is congruent to Angle B | Given
  2. The measure of Angle A is equal to the measure of Angle B | Definition of congruent angles

To a mathematician, the difference between angle congruence and angle measure equivalance is an important and substantive distinction. To a high school student just learning geometry and proofs, this seems like a distinction without a difference – a lot like population versus population size. To them, if the angles are congruent then of course they have the same measure, and vice versa, so why does it matter which term they use?

Students are routinely put off by proofs in geoemtry, and one reason for this is the “necessity” of these steps and others like them. I’m not sure there’s any value in emphasizing such a distinction , or requiring it in proofs. What would be the harm in cutting out some of the rigorous detail in these proofs and focusing instead on the big ideas? More importantly, what would be the benefit? In calculus, we prove things without resorting to epsilons and deltas, so why can’t we mke a similar decision about geometric proofs? At some point, of course, students do need to learn that more detail can be given, that more subtle distinctions exist (and are important). But to a student new to the idea of formal proofs, this level of detail only serves to get in the way.

As we work toward our integrated math curriculum, we’ve been working with two important questions: what should we prove, and when should we prove it? To these, I’ll now add a third: how much “proof” is sufficient? Getting this right seems to me to be one of the keys to enticing students to dive deeper into math instead of scaring them away from it.

Why Are Word Problems So Hard?

One of the courses I teach is an interdisciplinary course in AP Physics 1 and Precalculus Honors. In a future post, I’ll describe the course, the things we’ve tried to do (successfully and unsuccessfully) in it, and what we’ve learned in the first two years it’s been offered. For now, suffice it to say that the course is a lot of fun to teach, and it’s also an incredible challenge. Wednesday was a good example of both those things.

One of the reasons to create the course was the natural overlap that exists in content between typical physics and precalculus courses. Among other things, both courses cover vectors and parametric equations; rely heavily on trigonometry; require students to solve equations; and spend time on word problems. It seems redundant to cover the same material in both classes, and it implies that the math they use in physics is somehow different than the math they learn in precalculus. Why not just talk about these topcs once, and use the time saved to go into more depth about something else? This made perfect sense when we created the course, and for the most part it’s worked pretty well. But…it hasn’t always worked well, and when it hasn’t it’s told us a lot about where our students are relative to where we think they are.

Our students have been working on one-dimensional motion for a few days now, and they’re having trouble with it. The ideas are hard, which we expected, but earlier this week we realized that their struggles were not really with the physics, they were with the problem-solving process itself. Despite the fact that they’ve been drawing pictures, stating knowns and unknowns, determining equations, and solving for two or three years at this point, they couldn’t seem to do it well enough to allow them to solve the problems. So, we came up with a way to diagnose their issues and help them through their struggles.

The problem-solving process we’re using in class follows the one laid out in our textbook, Eugenia Etkina’s College Physics:

  1. Sketch and Translate – draw a picture of the situation and put all the information from the problem on the picture
  2. Simplify and Diagram – model the objects in the problem as points (if appropriate), and draw the appropriate vector diagrams and/or graphs of the situation to help visualize the relationships. To this step we’ve also asked the students to explicitly state what they Want To Find (WTF).
  3. Represent Mathematically –  state the equation(s) that relate the knowns and unknowns in the situation
  4. Solve and Evaluate – solve the equations and decide if the answers make sense

Our students were getting bogged down in the third step – they couldn’t seem to figure out how to create equations. As we (my co-teacher and I) thought it about, we realized that wasn’t really where their issue was, though. Their real issue was in setting up the problem well enough to clearly understand the WTF, which in turn enables one to identify the appropriate equation. To help them understand the importance of the first two steps of the process, we threw out what we had planned to do in class and did the following instead.

We have groups of students in class, with three of four students in each group. Each group was given a different problem. Together, on a whiteboard, they had to complete the first two steps of the problem solving process, ending by stating the WTF. The whiteboards were then switched, but the problems were not. Each group had to try to find the WTF using only what was on the board. If the problem had been sketched, translated, simplified, and diagrammed well, then the second group should be able to finish the problem off. Our students quickly realized this because, well, the problems were not all sketched, translated, simplified, and diagrammed well.

When a group got stuck, we invited them to give feedback to the group that had set up the problem. For example, one group stated that they wanted to find the time when two objects’ paths intersected, but their diagram showed the objects moving away from each other. Another group correctly showed two objects as moving towards each other, but they stated both velocities as positive instead of changing the sign on one to indicate movement in the negative direction. With these and other mistakes, the students were able to see the importance of setting a problem up well. At the same time, we were able to see what concepts they had misconceptions about, and we were able to address those misconceptions on the spot.

This exercise went very well, and in theory it helped resolve a lot of the issues the students were having. Did it actually help? We’ll find out this afternoon when we ask them how their homework went.

Thinking about Coding

As I’ve written several times, we are spending the next year developing an integrated math curriculum, the first year of which will be implemented in 2016-2017. One of the things we’ve decided we want to do is include coding in the curriculum. The move to include coding is nothing new of course; in fact, some would say we’re behind the curve since many schools began adding coding to courses a couple years ago. I actually think we’re better off doing it now, because we can learn from the experiences of others and we can benefit from the historical perspective of looking back at what people were doing two or three years ago and what they’re doing now.

coding-future

I will say right up front that I’m not a huge fan of adding coding to our curriculum. It felt like a fad two years ago, and it still feels like one now. Rereading Peter Gow’s insightful piece, “Coding is Just the New Surveying” only reinforced that sense, but it also gave me a good way to think about what the role of coding should, and should not, be. To try to get a sense of what coding the in curriculum might look like, I read some articles on the other side of the issue and also spent some time with the newly published book, Doing Math with PythonBetween this reading a great conversation with colleagues, I’ve come up with an initial framework for thinking about how to bring coding into integrated math.

  1. The role of coding in our curriculum should not be, to borrow from Gow, “pre-vocational”. Our goal should not be to teach coding so everyone knows something about it because they’ll need it in the future.
  2. More to the point, coding should be integrated into the curriculum, not feel like an add-on. If we are working to tie geometry and algebra together, then we should make an equal effort to tie coding to other parts of the curriculum in a natural way.
  3. Whenever possible, the coding work students do should have long-term usefulness, not be a one-off project or assignment. This might mean writing a program you’ll use over and over during the year, or it might mean writing a program you’ll return to each year to modify and enhance.
  4. The purpose of being able to write a program is to get technology to do exactly what you want or need. There is no need to write a program to find the mean of a data set because spreadsheets and calculators can already do this for us. Whatever role coding plays in the curriculum, it should focus on enabling students to do things they can’t otherwise do.
  5. The corollary to this is that in order for students to know what technology can’t do for them, they need to learn what it can do for them. Therefore, familiarity with calculators, spreadsheets, and even websites like Desmos and Wolfram Alpha is also necessary, and should be a prerequisite before learning how to code.

Our first big milestone in the curriculum development is in about three weeks, when we hope to have a broad outline of what topics we think should go in each year of the course, how many levels of each course we should have, etc. It’s been hard to think about what to include in the courses since I’ve never been clear on the role of coding in the course, and this framework gives us a way forward. Next up: so what should we include in each course?