Category Archives: Math Ed

The Dead Puppy Theorem and Its Corollaries

To preface, I normally celebrate mistakes in my classroom as a part of the learning process. But there are some things that really speed the progress of my receding hairline, a large of percentage of which involve bad algebra. I saw this from @lustomatical and thought YES. This is what I need to get my kids to stop distributing powers over terms that are added, and “canceling” things willy nilly, and not respecting the trig functions as operations. Let’s concentrate on the Calculus! So I made these posters for my classroom:

Enjoy. I know my students will, and it will actually give us a funny and memorable way to talk about and avoid these common algebra mistakes.

The other thing that I showed them today to get them to stop just playing around with letters while doing Algebra is the following, which I believe I picked up at a summer workshop:

They literally laughed out loud at this. I said (in a funny, not mean and not sarcastic way), “You think that’s funny?!?!? This is the kind of stuff you guys do on quizzes. When I am correcting your work I sit and laugh and laugh and laugh at the crazy things that you do! No more crazy algebra!”

How do we stop/prevent crazy algebra mistakes besides carefully and repeatedly addressing them when they happen? Any ideas?

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How Much Should Students Retain from High School Math?

So, after a wild foray into differentiation and applications of differentiation, we just started to learn how to deal with exponential and logarithmic functions, before starting our equally wild foray into integration. I gave a pre-test to see what they remembered about exponential and logarithmic functions from the past.

It was less than encouraging when I think about the quality of math education we give to high school students.

I gave them a little more than 10 minutes to fill in what they knew on a two page pretest. Most were just about as blank as the one below:

Most students could not sketch anything that even looked like an exponential function (these are four different student’s answers):

Very few knew how to evaluate a logarithm, or even knew what ln(x) meant:

Which might have also been because not even had any idea of what e was:

 

Okay, so I have fully embraced the whole idea of “teach the where they are.” I enjoy teaching this class because I have the liberty to slow down and fill in gaps in their math backgrounds and then can teach the Calculus material with the depth in conceptual understanding but without the depth in mechanical skills (if that makes any sense). Sure it’s harder to explain the idea of a limit as x goes to negative infinity for e^x when they don’t fully even understand what a negative exponent is, but I wouldn’t be in teaching if it was an easy job. So, I’m not complaining that I have to change my sacrosanct yearly plan and “lose time” – I’m just wondering if this is all they can remember about a major family of functions that I’m sure they learned about in both Algebra II and Precalculus, what did we accomplish in teaching them this in the first place? What do we expect our students to know coming out of high school? How big is that gap between what we teach and what they actually learn? 

I’m not blaming their past teachers, and I’m especially not blaming my students (though my non-AP Calculus students are labeled “weak,” many of them are really quite talented but have yet to find that spark of loving mathematics or need a bit more time). Worse, I have no ideas for solutions, just those questions. I guess I can find comfort in knowing that other people are asking the same thing, and even better, that those are the questions that are driving (some of the) reform in mathematics education right now.

Wannabe Math

There are a lot of times in Calculus where stuff that you want to be true is (like the limit of a sum is the sum of the limits) but then there are a lot of other times where sadly your intuition just doesn’t work (like the derivative of a product is NOT the product of the derivatives). This is a random little thing, but I’ve been doing “Wannabe Math” in thought bubbles on the board to distinguish between rules we know that work and ones that we don’t know are solid:

I want to encourage students to try things (so encourage them to use their intuition to explore) but give them tools to be able to test to see if they are correct. I did this for the product/quotient rules and then used it to derive the chain rule with the class – with the Wannabe Math on one side and the actual math on one side, we could easily see what was “missing” and come up with the whole derivative of the inside part of the rule.

Experiments with Math Whiteboarding

Getting five mini-whiteboards was a real game changer for my classroom. It has completely changed the way that I go about skills-based instruction (which is what a large chunk of the first term of Calculus ends up being) and has added so many new tools to my instruction toolbox. I think that too often in the past I relied on variations of the learn-practice-apply model, and the “practice” part not only always seemed to make class drag, but never really felt effective. Well, doing practice on the whiteboards with some sort of extra little component to make it interesting has proven to be not only interesting (I’ve never seen students more engaged while practicing skill-based math), but far more effective at teaching skills. The only downside to the whiteboards is that they don’t have anything in their notebooks to study at home, but I’m brainstorming ways to fix that (mostly posting pictures of the whiteboards on our course website or not caring because they can always find examples of solved problems in our textbook – it’s the practicing part that matters).

Here have been my favorite things to do so far with the whiteboards and a few ideas for experiments that I want to try in the future. I would love any and all comments about different non-topic specific modes of instruction that you use with the whiteboards to expand my repertoire.

1. UTILIZING DIFFERENT COLOR MARKERS

One of the great parts about whiteboards is that you can get get students to use different colors like you do on the board up front to get them to focus on different things. For example, above is a whiteboarding exercise I did with the Chain Rule. Students were in groups of threes – for each problem, one person had to rewrite the problem in different colors to indicate which was the outside and which was the inside function, the next person had to differentiate it still using the colors to point out where each part of the new expression came from, and then the last person had to rewrite the expression in a simplified form. This was perfect because the hardest parts of the chain rule are recognizing when you need, seeing inside vs. outside and then seeing where the parts of the new expression come from.

EXPERIMENT I WANT TO TRY: Mistake Marker. I want to have the students solve problems and then whenever they make a mistake, instead of just erasing part and fixing it, they write over their mistake with the mistake marker color (or make a note with the mistake marker if it’s not possible to write something in that color) and then continue the problem in a new line below with their original color. Then, at the end we can collect as a class the most common mistakes that are made when doing a complicated problem like, say, the Quotient Rule (not complicated you say? Then you have never taught students Calculus who have a terribly weak Algebra background).

2. THE MISTAKE GAME

I know I have mentioned this like twelve times already, but I absolutely love it. This is stolen from Kelly (read her description here), but the basic idea is that groups present solutions to semi-complicated/involved problems on whiteboards, but while presenting their solution, purposely make a mistake (and not an silly arithmetic mistake like 3+4=8, real hardcore-misconception-style mistake). Then, they present their work to the other students in the class, trying to sell their mistake as having been made for real. Then other students ask thoughtful questions about the presenting group’s solution to try to help everyone find the mistake. This is always great with a quick class followup at the end collecting the most common mistakes.

3. ROTATE

When I have wanted to show how a topic from my AP class could be applied to many different situations, I have done some sort of rotation so that students could be exposed to a wide variety of problems (without taking the time in class to do ALL of them). The first exercise with this is a simple gallery walk – each group of 2-3 students solves a problem individually. Then, when everyone is done, the groups rotate around to each station, taking a few minutes at each one. This is far more successful if you give them specific tasks, like “First, check if their answer makes sense, then see how they set up the limit definition of the derivative” or something concrete like that. The second exercise is when each group starts doing a problem, and then the groups rotate after about 5 minutes and they pick up where the last group left off (someone on Twitter gave me this idea, sorry, I forget who!). Then, they rotate every 4 to 5 minutes until all of the problems are completed. I tried to emphasize that while doing this, you must show your work neatly and clearly (an important skill for all math, but especially an AP test) so that the next group can quickly see what has been done to solve the problem and what still needs to be done. The thing that I liked about this was the meta cognitive mapping out of problem solving, though I don’t think I left enough time for students to really think about each one.

EXPERIMENT I WANT TO TRY: Rotate Marker. I didn’t love the rotating problems mid-problem solving, mostly because it stopped students in the midst of great problem solving, but I think one thing I am going to try is having the students rotate within their group who is writing. So a group would be solving a problem and every 2 minutes, the next person in the group would become the writer. This would ensure that all the students in a group are engaged in the problem solving process and that they are all talking math with each other.

4. SORTING

One of my many goals this year is to step back and focus more getting students to figure out how to go about problem solving. One thing I did that I really liked was I photocopied a few pages from the book and cut out like 50 functions for each group that all required a variety of differentiation rules. Then the students made categories on their whiteboards and sorted the functions based on which differentiation rule they needed. It was a really interesting process, especially because many of the functions needed more than one rule. I really enjoyed seeing how students solved this – most just made a bunch of different categories (like Quotient & Chain, Product & Chain, Product & Quotient & Chain), but one group made a crazy complicated Venn Diagram and another made a table kind of like one of those that shows the distances between cities (so like Quotient, Product and Chain both across the top and down the side) then placing the functions at the intersections of the rules they needed. It only took 15 minutes, but after learning so many differentiation rules, I think it was great to give them a chance to step back and figure out what types of rules they needed to use and where. The next day on the quiz, I saw tons of students circle parts of functions and write “Product” and “Chain,” which is something I have never seen them do before. To me, this is a wonderful problem solving strategy that was explicitly identified and strengthened by a quick activity.

 

Overall, I just think the added presence of the whiteboards has given my classroom a much more dynamic feel. When I asked for feedback from my students about how class was going these were my two favorites:

I like how we change up the routine. We do not sit and do the same thing over and over again, its changes up and keeps me interested.

 

One positive thing is the different types of work you give us because it is not all the same thing it is diverse so it keeps things interesting.

I guess it was a good indicator to me that switching up the routine for switching-up-the-routine’s sake is not a bad thing. Knowing some basics about how the human brain works, keeping the kids from sinking into a comfortably numb routine will certainly make everything a little bit stickier.

And in case you forgot in all of my blathering, I’ll repeat my plea from above…  I would love any and all comments about different non-topic specific modes of instruction that you use with the whiteboards to expand my repertoire!

Growth Mindset – Normalizing Mistakes

My first year teaching, I remember one of the elder, wiser, experienced teachers at my school looking at my first week plan and telling me that I think more deeply about setting routines in the class and creating a good class atmosphere. I kind of brushed this off as a sort of silly – I was there to teach Physics, and that’s what I would do. The other stuff would happen automatically. Well, luckily, I wasn’t totally wrong – I think that I inadvertently did a decent job of setting good routines, though I don’t think I did a great job of creating an atmosphere where mistakes were not only encouraged but celebrated. I realized by the end of the year that the hardest part of teaching Physics was not Physics at all, and tried to focus a bit more on all the “other stuff.”

This year, my third year teaching, one of my main goals is to really get my students to buy into the idea of a “Growth Mindset,” especially in my non-AP Calculus class. I started well with an awesome discussion, which was based on Dweck’s original mindset survey (which John Burk over at Quantum Progress turned into a cool data driven exploration of his students’ mindsets, which then he turned into a collaborative mindset data collection experiment in which you can participate). As my beginning of the year review rolled on though, I kind of ruined what I had started through my frustration with my non-AP Calc students. For some reason, they are incredibly weak, far weaker than the students I had going into the same class last year. Many don’t know the basic shapes of parent function graphs, don’t know how to correctly simplify rational or radical expressions
(\sqrt{x^2+y^2}=x+y    right??), have never seen a piecewise function, can’t find the domain of a rational function, can’t recognize a basic vertical shift etc. Sigh. I guess my surprise and confusion that they were at this level was pretty obvious, and both of my classes seemed embarrassed by not knowing things that I thought they “should have” and, yeah, worried that they were “dumb.” I sort of realized that I hadn’t bought into the growth mindset as much as I had thought – They’re weak? No. I was comparing them to the students from last year instead of just assessing their level of math and working from there.

The worst side effect of our really rough week of review is that the class started to get really, really quiet. I could only get a few students to respond to questions and take risks. I couldn’t tell when they didn’t understand something, even instructions, because they would just be silent – I have never had that happen in the classroom before. I decided to take some action and remind them (and remind myself) that we are a classroom committed to the Growth Mindset. Using PollEverywhere.com, a wonderful interactive polling website where students can vote and immediately see the results at the front of the classroom, I carved out 10 minutes from mathematics and took them through a series of questions that I designed to help normalize mistakes. We looked at the results of each question before moving on to the next. The results…

Observations: Though some of the questions were certainly leading, the students seem to really buy into the ideas and remember our growth mindset conversation. The questions were ordered perfectly, because after everyone realized that no one else judges other people for making mistakes, they were forced to think about really why they were having a hard time participating in class. We went through each of the statements for the last question and talked about if we believed that statement, how the results from the previous two polls might help us participate more. It was a really nice conversation and seemed effective. I saw that look that the students get when their gears are turning and stuff is clicking. Side note: I was a little surprised that students voted for the “Mr. B, you are intimidating option” but I used that as a spring-board to remind them that I buy into the growth mindset idea too. (Also, sra7a means “honestly” in Arabic).

We wrapped up with a PollEverywhere open-ended question, where they type things into the poll and they show up on the screen. I thought this might be a nice, low pressure way to share some thoughts with the class so that we could all be supportive of each other:

How do I know this was a wonderful use of 10 minutes? The first response to the question above was “Thank You” which was surprising and actually pretty touching. And theeeen, it quickly devolved into things like “Bring lasagna to class” and “apple juice breaks.” Really senior-in-high-school? Apple juice? Thanks for ruining a rare sentimental math moment.

Next step: Now that I have them a bit more prepped to be okay with mistakes, I want to find ways to go one step further and celebrate mistakes. I really love the Mistake Game , from Kelly over at Physics! Blog!, to use with Whiteboarding in Physics. Basically, students work in groups and present the solution to a problem that contains a mistake hidden in it. Students are encouraged to find the mistake through asking thoughtful questions instead of just saying “HA! I FOUND THE MISTAKE.” I love this because it is not only instructional, but teaches students how to constructively criticize each other’s work. The math department plans on getting mini-whiteboards any day now, so I am excited to experiment with this. Also, Kate over at f(t) has some great tips from the Virtual Conference on Core Values from this past summer where she describes the center of her classroom as being “We Make Mistakes.”

Moral of the Story? Growth Mindset takes more than a description and a survey to create buy-in. I will remember that teachers can unintentionally send subtle signals through their behavior. I’ve learned from my mistakes with this, which, paradoxically, will lead me to encourage lots more mistakes. I’ll certainly be coming back to this throughout the rest of the year.

September Review: Is it Actually a Necessary Evil?

It seems like math is the only subject in which teachers feel like they need to review for the first week or two of the school year. Teachers of other subjects seem to review as they go along, going back to old skills and ideas as needed, as motivated by their curriculum. To me, this makes much more sense… and I know because I say this a bit disheartened having plowed my students through a review of algebra for the first week and a half of school. After starting of my class with a bang, with some great metacognition and a good introduction to Standards Based Grading, I had a lot of trouble getting the groove, mostly because I knew that I wanted to push through the review to get to the good stuff, which means I ended up having an uninspiring week of a hugely teacher-centric classroom.

So that of course brings the question to my mind of “Why am I wasting time on not-good stuff?” I know some stuff is unavoidable (especially when a vast majority of my non-AP Calc students claim they have never seen or heard of piecewise-defined functions before), but I really believe, like a bunch of other math teachers I have talked to, that most of review could come as needed as the curriculum develops. And our book reviews all these crazy topics that won’t ever have much bearing on the future curriculum. Symmetry? Modeling** (the mathematical kind)? Animal Husbandry?

I got frustrated halfway through the week and decided that instead of just hammering out more material I would do a problem solving activity with my AP class that would remind my students of many of the things they needed to know while engaging them in deep problem solving at the same time. The sad part about teaching an AP class is that it totally felt like I was “losing” a day (my yearly schedule is nagging me), but it was totally worth it. This is something I am going to struggle with all year, as I have taught a very application based Calculus for a year and this is my first shot at the AP. Here is a mini-unit I organized about piecewise functions.

MOTIVATING PIECEWISE FUNCTIONS: I tried to get them to see why piecewise functions are necessary by giving them data of tourism arrivals and departures in Jordan and the US over the past 15 years and asking them to tell me the story that the numbers are telling them (thanks John for giving me the idea with your post about Telling the Story of a Number). Each group came up with 4-5 bullet points and wrote them on the board for the others to see. Quite unsurprisingly, there are major drops in both US arrivals in departures around 2001 and 2008, which most groups mentioned in their story, but others came up with some great explanations that I didn’t expect. For example, one group mentioned that Petra was named one of the new Seven Wonders of the World somewhere around 2008, so that might explain an uptick in arrivals (cool!). Another explained the rise in US tourism in the mid-1990’s to a Deep Purple tour… We talked about how it would be hard to fit ONE function to the data because it’s kind of all over the place, but we could fit a bunch of chopped up functions. The cool part about framing it like this is that the points where the function changes corresponds to major world events, which is because those events changed the relationship between the variables. I saw a bunch of light bulbs go off on that one.

ACTIVATING THEIR PREVIOUS KNOWLEDGE: Then, I played for them DJ Earworm’s 2009 United State of Pop mashup (I blogged about this about this on Sam Shah’s blog this summer). We made the metaphor between a mashup and a piecewise function and used that to give ourselves a quick reminder of how the notation works. This led into a few examples as a reminder, but none of the drill and kill – I just wanted them to remember that they knew how this stuff worked.

FINALLY, THE PROBLEM SOLVING ACTIVITY: With more-than-inspiration from Mimi’s Income Tax Unit, I presented them with how income tax works here in Jordan. They were really surprised – most thought it was some sort of flat tax. They were also confused. Why is it so complicated? So I presented them with the goal of the task, which was to make the Income Tax more easily understandable for the average person. I found the income tax for five different countries, and each group was tasked with graphing Tax Owed vs. Money Earned and then writing a piecewise-defined function that will give you your tax owed if you plug in your income. That way, an average person could either just find their income on the graph, or plug their income into the function. Trying to be “Less Helpful” à la Dan Meyer, I tried to provide scaffolding only where needed. This was so hard for me! I just wanted to give them little hints. I gave in to these urges every once in a while, but this was the most I have ever let my students really struggle. Most tried to start directly with the equation and had a lot of trouble abstracting the situation, but over the course of a period and half (everything takes roughly 24 times longer than I think it well), pretty much every group had a graph drawn and pretty much finished the equations.

(that’s Spain’s Income Tax)

REASONS THIS WAS NOT “LOSING A CLASS”

  • They realized that the keys to piecewise functions are the points on the boundaries of of the intervals. This will really help when we talk about continuity and differentiability.
  • After some experimenting, most groups realized that the slope of each segment was the same as the percent of taxed money on that segment. Any sort concrete exploration of the idea of slope is alright by me.
  • Many students became much more comfortable with point-slope form (or, more importantly, realized that this form of the line is much easier some of the time than slope-intercept), which will help when we talk tangent lines.
  • One group made connections between all the ways they could have solved the problem. They actually determined their equations analytically – CATEGORY BASE TAX + (TOTAL INCOME – CATEGORY BASE INCOME) * TAX PERCENTAGE – which was impressive to me, as I always do things other ways first (graphically, numerically etc). But they came to the realization that this form is pretty much exactly point-slope form if you rearrange it a bit and then got the added conceptual understanding that arises from point-slope form here. I thought this was great.
  • This was more genuine, engaging and thought-provoking than the rest of the week combined (and it’s not even a particularly rich problem).
I was feeling really frustrated and down from limping through review, so this class was exactly what I needed. My goal this year for my math classes is pretty simple: I want to really motivate everything we study. This was a great start and I hope to come up with more rich (but not necessarily “real world”) tasks like this. NEXT YEAR: NO MORE DRILL AND KILL REVIEW.

(PS One of the sweet things about working here is that when I leave I will get most of the tax that I paid the Jordanian government right back. And I don’t pay US taxes because you have to make a boatload abroad to have to pay. Buhahahah.)

**I actually love love love modeling, but doing textbook problems about modeling is like that first chapter in science textbook that “teaches” the scientific method.

My Dog Could Do Math (or perhaps just perform algorithms)

I totally understand the place of the algorithm in mathematics. But the argument about the use of algorithms reminds me of a deeper issue, how math education is currently trying to figure out how to adapt to major technological advances in computing that allow us to have computers perform these algorithms for us. I’ve seen a lot of arguments that the math curriculum needs to change drastically to take advantage of new tools like graphing calculators and Wolfram|Alpha (which, of course, both rock). Basically, that any sort of direct computation needs to be phased out of a 21st century curriculum – that we should teach students how to problem solve using these technological tools. Others argue that deep understanding is enhanced by knowing how these processes work. To which others counter that people drive cars all the time to get from place to place without ever having built a car or really knowing how it fully works.

Somehow I was reminded of this when I saw the family dog, Whiskey (who is absolutely hilarious), performing some of his tricks when I was visiting home this summer. Our family’s favorite is the “Bang, you’re dead” trick.

First, my mom puts her gun out, to which Whiskey responds by sticking his paws in the air innocently. Then, my mom yells “bang!” and Whiskey awkwardly flops to the floor, flips over and plays dead. As you can probably tell, it’s a pretty amusing trick, and pretty complicated for such a puny little brain. But… here’s the whole video from which I got these screen shots (no idea why it ended up so stretched out):

Notice that he tries just about everything before he gets it right. He has sort of a general idea of what he is doing, but he has no idea why he is doing each of the steps. When he accidentally does one too quickly, or jumps up instead of putting his paws up, he can’t diagnose his misconception thoughtfully and fix his mistake. He just blunders through trial and error until he figures out something.

The sad part was that this totally reminded me of a few students whom I taught last year (seniors taking Calculus) who had no idea how to do basic algebra because they had no deeper understanding of what was going on and somehow had no idea how to check their answers to see if they was on the right track. It felt like their previous math teachers had taught them how to do tricks, and perhaps I wasn’t doing any better. If they managed to plow through the right steps and stumble on the right order, I would reward them with the treat of a good grade, and that reward exists in both traditional grading methods AND in Standards Based Grading. But the problem for me wasn’t in the algorithms. It was that they had no deeper understanding of mathematics to accompany those algorithms. Teaching just computation, and not teaching it well.

So for me, when math computation technology proponents argue that you can drive plenty of places without every knowing how a car works, I always think about the one time when the car breaks down. What do you do then? You are stuck. You have to call someone for help. You can’t thoughtfully work out the problem on your own. I agree with the hopefully general obvious opinion that learning how to do computation is not the goal of a mathematics education. And I agree about the danger of accidentally teaching meaningless algorithms, which can easily happen unless you conscientiously dig deeper in checking for understanding. But the idea of throwing out the deep conceptual understanding of mathematical structure that goes along with learning some of these processes in favor of using computers for computing just doesn’t really sit well with me. The important, deep conceptual understanding of mathematics certainly doesn’t come from just learning algorithms, but it also not helped by never learning why algorithms work. Technology, though a great tool, is not a replacement for the human mind.

Why I am thinking about things like this and not the hectic as-of-yet-unplanned first week of school is beyond me…

UPDATE: If you haven’t checked out Matthew Brenner’s “The Four Pillars Upon Which the Failure of Math Education Rests” go read it – reading the whole thing is on my to-do list but everything I have read from it so far is wonderful. It was pointed out to me after I wrote this that he wrote something very similar (though about ten times more eloquent) on page 55. Agree to agree I guess! Now I must read the whole thing.