The Nerd in Me, Looking out for Mr. John Venn

So my friend posts this picture via Instagram. There's a place in Portland, OR, called Blue Star Donuts, that sells glazed donuts and fried chicken.
 


True to my math teacher spirit, my immediate, instinctive reaction was: "I dig it, but this is a misuse of the Venn Diagram. YUMMY should really be the superset of glazed donut and fried chicken. Their intersection should be... well, RIDONCULOUS!"

...I've already come a long way from my nerd roots, I promise. I just didn't want Mr. Venn to be rolling over in his grave. Who has two thumbs and is always looking out for the dead mathematicians?

How I Do IB Test Prep

I don't really like teaching the IB, because I feel as though there are a lot of topics to get through and it's hard to build all the concepts from the ground up (given the time constraints), so you really have to rely on the kids having a pretty strong foundation in the fundamentals prior to arriving in your class.

HOWEVER, I am definitely getting better at it. Kids are pretty happy in my class because they understand the topics and even the convoluted IB problems are becoming more accessible to them over time.

I was thinking about what made the big difference between last year and this year, and one thing is that I am giving consistent practice quizzes at the start of class on a non-trivial skill, through the course of 4 or 5 consecutive class meetings. Following that as a sort of long Do Now, we pick up with "regular" instruction for the rest of the period, on a usually unrelated topic that is concurrent. The quizzes and the instruction run parallel, but they're not usually on the same topic. After 4 or so such practice quizzes, I give a graded quiz on those skills. (Which may or may not be around the time when I give a coherent unit test on the concurrent topic. The kids don't seem to mind that there are always two topics going on at once, because they understand that the quizzes are only to build up specific test-prep skills, and the unit-based learning we do is more coherent and also much more time-consuming.) The key here is that the practice quizzes have to be sufficiently complex, yet similar each time in format and content. The first time they see a practice quiz, they usually cannot do it and they require me to explain the problem to them thoroughly after they try it. But if they're attentive and they keep good notes and they actually try all the practice quizzes earnestly, they can get close to 100% by the real quiz even as I add slightly more complicated parts on the final quiz. This is very important, because 1. it builds their confidence with complex problems, 2. it gives me an opportunity to reinforce the integration of multiple concepts.

For example, recently we did a set of practice quizzes in Grade 11 that looked like this (while we are learning logarithms and log / exponential functions in our "regular" instruction):

"For the function f(x) = 3x^2 - 2kx + k + 1, find all value(s) of k that would...
a.) Cause f to have two x-intercepts.
b.) Cause f to have one x-intercept.
c.) Cause f to have no x-intercept."

As part of this problem, the kids would define the discriminant function in terms of k:
D=(-2k)^2 - 4(3)(k+1), graph it in the calculator, and then use the graph to analyze when the discriminant value is zero (k = -0.791, k = 3.79 when rounded to 3 sig figs). They will then conclude that when k is those two approximated values, f has one x-intercept. They would then test it, by substituting k back into the equation for f... f(x) = 3x^2 - 2(-0.791)x - 0.791 + 1 and f(x) = 3x^2 - 2(3.79)x + 3.79 + 1 both look like they have only 1 x-intercept.

They will then look back at the same discriminant function they had set up, D, and then visually conclude that the discriminant function has a positive value when k < -0.791 or k > 3.79 (the discriminant graph is above the horizontal axis, which means that it has a positive value). And in order to test their hypothesis that k < -0.791 or k > 3.79 would cause f to have two x-intercepts, they will then replace nice integer values k = -1 and k = 4 into the equation for f, to verify that f(x) = 3x^2 - 2(-1)x - 1 + 1 and f(x) = 3x^2 - 2(4)x + 4 + 1 both have two x-intercepts. (And they do.)

Eventually, they repeat this process for the hypothesis that -0.791 < k < 3.79 will cause the discriminant function to be negative (below the horizontal axis), which will cause f to have no x-intercept. They can test it with a nice k value such as 0, 1, 2, or 3.

Well, what is the point of all of this?

1. Notice that I base the problem-solving around graphical analysis, even though we know that we can solve quadratic equations and inequalities without a graph. Research shows that the more you emphasize graphical analysis on the calculator, the better the kids get at visualizing equations and inequalities as graphs. In the long run, when you take the calculator away, they will still be willing to manually simplify their discriminant function, to sketch the graph, and use it to aid their analysis of variable k and its relationship, ultimately, to function f. And that ability to visualize is a powerful tool to have.

2. Specific to this topic, the notion that k and x are related but not the same can be often confusing to students who are not used to doing such sophisticated algebra analysis. A lot of times last year, my (former) 11th-graders used to ask me, "But why can k have two values when they're asking for there to be only one x-intercept for the function f?" Breaking down the process as we did above addresses that specifically, because when the kids substitute the values of k back into the original equation, they can see that the graph they're shooting for works BECAUSE the parameter k has taken on appropriate values. It helps to separate the meaning of k from the meaning of x.

3. For your weaker students, repetition breeds familiarity, which then breeds confidence. They also get to hear me explain the same concept 4 or 5 times, over the course of 4 or 5 classes, which is very helpful for them. They also get to use their notes on all the practice quizzes following the first one, which gradually forces/allows them to be independent while building up to the real quiz. You cannot achieve this type of comfort level by practicing/drilling completely different-looking problems everyday, even if the problems are essentially on the same core topic. Less is more, I think. Once they build an in-depth understanding of a single problem type of sufficient "juice" and complexity, I believe that they can then more easily transfer that understanding to other problems.

I know this method works, because not only did my 11th-graders do quite well on the mid-year mock exam this year, but generally speaking, when I think back on the topics that we have learned this year, I feel that they don't have collective holes/gaps as a class. We've gone through and filled them all in, using this consistent quiz policy that runs alongside our regular instruction of new topics.

I also use the same practice-practice-practice-practice-then-quiz policy to help my 12th-graders do spiral review from last year. The difference is that I try to integrate even more topics for them. On one set of practice quizzes, for example, they needed to: write a sine function from a graph; describe the step-by-step transformations from y=sin(x) to that graph; graph another quadratic formula within the same grid, labeling all important info; shade the enclosed area between the two curves; write an integral that represents the area between the two curves; evaluate the integral by calculator; then show how they can get the same integrated result by hand. I gave them 5 or so practice quizzes leading up to the quiz on this one, because there were so many different skills involved that they needed to practice. This is how I pull it altogether for them, using a combination of mixed concepts and (still, much needed) repetition.

I feel that we (meaning, my Grade 12s) are in much better shape this year, going into the review period, as a result of this quiz policy that I adopted. We had started the spiraling quizzes on last year's topics, all the way back in August. By now, I have gone through and drilled most of the particular weaknesses that I felt that they needed to see again. This method is systematic, more focused, and much better than just giving them random mixed practice! I feel that, even though we're not yet done learning all the topics (we still have one more to go, which is Vectors), my kids are already pretty OK now with doing mixed IB practice on their own and needing only moderate support from me. If only they can keep up the stamina for doing extra practice on their own during the remaining weeks, then I know that they'll be in great shape by the end of April!

I thought I would share this, you know, because I really think it has made all the difference in my IB classes. In fact, I have been trying to do the same in my MS classes, by spreading out test practice over the course of a week or two, leading up to the test, instead of just giving a single practice test. It feels kind of like a huge waste of time, because the practice quizzes take about 20 minutes each class. But, in the end, you're actually saving time because you're increasing mastery by providing more regular feedback and multiple opportunities to self-assess. This year, I didn't have to spend any extra time reviewing equations in Grade 7 after I taught that unit, because the kids all had the algebra skills down pat. So, yea, do try it if you're not already doing this in your class, and I hope that it helps to address a variety of learning and mastery issues!

Visualizing Concepts

Here is an MS update. I feel pretty productive lately, as I always do during the second semester. I also feel quite productive with my Grade 11s, and as a result I'm taking on three new kids potentially, at least for a while. Grade 12's are doing OK, but the pressure is sure ramping up for their IB exams, so there's not a whole lot of "cool" instructional things that I can be doing with them...

Grade 8:

Referring to the system {5u + v = 21, 10u + 3v = 48}, one middle-of-the-road kid explained excitedly to her friend, "Oh! I got it! If 5u + v = 21, then 10u + 2v must equal 42. If you put that here into the second equation, then you will still have 1v on the left side. Which means that 42 + v = 48, and v must equal 6!"

Not bad for being day 1 of systems algebra. (Of course, they've been doing the same reasoning using shapes and visual substitution for several days now, so the transition to symbols was seamless. I didn't bother with even any examples on the board this year, and it worked out just fine without one.)

I went up to a boy who was one worksheet ahead, and pointed at the equation {y + 6x = 10,
y + 2x = 2} to ask him what immediate conclusion he could draw based on inspection. He immediately said, "X is 10 minus 2, divided by 4." I had to backtrack to ask him, "How do you know? Because 8 is equal to....?"

He finished my sentence, "4x." Great, now we're talking about the process.

I showed him the traditional notation for showing that work / reasoning via elimination, even though I know that in his head he's doing substitution from one (smaller) equation into the other.

  y + 6x = 10
-(y + 2x = 2)
        4x = 8

He was not impressed, nor surprised. When you introduce systems via pictures, the symbols become just part of the bigger concept. I explained to him that on the previous worksheet, I had let them just do much of it in their heads, but now we're going to work on the written communication piece, to carefully show our work on paper. I also gave him the hint that sometimes, instead of subtracting the smaller equation, he'll notice that he wants to add the equations instead. I said that he'll know when addition is necessary, because those equations will just "feel different." I left him on just that hint, and sure enough, 20 or so minutes later, when I came back, he had already identified the equations that needed to be added, and he was ruminating over the explicit mathematical reasons why that works. At that point, I felt that he was ready to discuss that if you have additive inverses, then you can just add them to cancel them, so we had a 1-minute discussion about that and I left him again.

I am very pleased with how this unit is going. I think I've gotten it down pretty pat; if I remember correctly, last year I didn't have to change any worksheets at all, and so far I haven't had to alter any worksheet either. This is one unit where I can just sit back and watch the kids' thinking unfold, more or less, and I can be very hands-off in their own building of the concepts of algebraic substitution and elimination. At some point, I looked around the room when I noticed the noise level rise, and saw that it was because literally every pair of kids is engaged in some sort of intense mathematical discussion with their neighbor. Awesome-o!

Grade 9:

In Grade 9, we've been working on learning / understanding / memorizing geometry area formulas via cutting and pasting. We each re-arranged, glued down a parallelogram to form a rectangle, to help them understand why A=bh for a parallelogram. Then, we each re-arranged a trapezoid into a thin parallelogram, to help them understand why A=(b₁+b₂)(h/2) for a trapezoid.  This year, I went a step further and highlighted the base 1 and base 2 sides using a green marker, so that the kids can see that they line up within the parallelogram when we do a "move and flip" of the top half of the trapezoid. This simple highlighting technique is superbly visual for my visual learners to see why the new parallelogram base MUST be (b₁+b₂).

Then, I proceeded to give them two practice trapezoid problems. In both cases, I had them draw out the parallelogram that it becomes, labeling the side lengths to emphasize the numerical connection between the trapezoid and its resulting parallelogram. It worked great! My concrete thinkers really latched on after the first example. One of the weakest kids in my class went up to the board and bravely (and correctly) drew out the trapezoid, its new parallelogram dimensions, and the calculated area, without any previous verification that he was on track. I was proud!!

I just love geometry lessons like this, because they involve both tactile and visual learners and help the concept "stick" in their heads. Slowly, I'm getting their geometry concepts up and running in order to do the 3-D project this year.

Addendum March 4, 2013: Here is a visual that Lara has made on Pinterest. Thanks, Lara!

Grade 7:

In Grade 7, we've also been doing Geometry... I'll have to report on that at another time, but I wanted to say that recently (prior to the Geometry unit), I discovered the best technique for teaching setting up of proportions. For ages it used to bother me that kids would not check that two ratios have matching units across the numerators (and across the denominators), and so half the time they would set up incorrect proportions. I figured out a trick this year. The first day we learned to set up proportions, I had them highlight the matching units across the equal sign. For that whole first day, they had to always highlight the units inside the proportions, and check that the matching colors lined up horizontally. After that first day, the highlighters went away and I never referred to them again. In the end, I didn't have a single kid mix up the positions of the numbers inside the proportions on the test. I think that mentally, the colors stuck with them and they're always visualizing that check, even when the highlighters aren't around anymore. 

Again, slowing the kids down in the beginning definitely pays off, I think. The highlighters are a trick that I'm trying to use more and more, to incorporate hidden visualization techniques that some of us "math people" tend to internalize in our minds but that teenagers who are used to rushing through things, could need more explicit instruction on. So far, so good.

Substitutes

I am such a perfectionist; I just hate being absent. 

I have to say that I am almost never absent. In the 7 years that I have been teaching, I can count the number of days that I have been absent. If I am absent, it's not because I was so sick and weak that I needed to crawl home. For those cases I would stay at school. If I am absent, it's because:

1. I'm going to a conference or job fair: I missed 1 day this year for a conference, and 1 day 4 years ago because I needed to attend a job fair.
2. I need to catch a flight for a wedding that is out of town: I miss 1 day every 2 or 3 years for this.
3. I'm getting married myself: I am taking 2 days off for my own wedding this year. Hopefully this is a once-in-a-lifetime type of thing.
4. My doctor absolutely isn't available in the entire month except during work hours, and the situation is urgent. I've missed about three or four half-days for this in all the years that I've been teaching.

However, sometimes we need to be absent and we need to have substitutes. And I HATE THAT. I cannot seem to substitute-proof my lessons. I try to give them all the details: all lesson material, with extra copies and neatly labeled answer keys and instructions on what to write on the board and how many answers to check. I try to make the lessons SO easy to run that a non-mathy person can still do it, and I even talk to the kids in advance and prep them for the activity. If there are support teachers in my class, I brief them in advance, send them all the lesson material, and make sure they know what should be happening in case the substitute teacher is unclear. In the end, I come back and the kids are like, "The sub didn't write anything on the board. They didn't do anything." Last time, I made spiffy pencasts and tested them on the computer under another teacher's login, and everything, and the substitute teacher didn't have access on their own login account, and the whole thing was a fail even though in my mind, all they had to do was to hit Play. That was not their fault... however, moral of the story: I cannot substitute-proof my lessons!!

ARGH. So, tomorrow I'm out for a half-day. I planned it so that I'd miss only one single-period PM class, thereby minimizing the damage. I've made all my answer keys and even color-coded them and everything. Keeping my fingers crossed that whoever takes my class will defeat all odds and make me proud.

What do you do to substitute-proof your lessons?

Visits #2, #3, #4

I've been insanely busy! OMG. This February break will go down in history as the most stressful break ever. At night, we were going out to meet up with old friends in Seattle, but because I was bussing to all these schools, I had to wake up at 6am every morning to get ready (and Geoff also woke up at 6am for moral support). But, I think it was well worth the effort, because I got to see a cross section of different schools and to talk to a bunch of math teachers in the area.

Besides the great school that I saw during visit #1 (which, by the way, now has an official MS opening posted on their website), I saw another great school during visit #2, in a location that is not within walking distance of downtown but is fairly central, within easy bussing or carpool distance. I took the bus there and it took me about 30 minutes. It's a small and pretty new school, maybe just about 300 kids from K to 12. They seem to have really wonderful administrators, and they even asked me to go back to do a demo lesson the next day. (This felt funny to me, of course, because I typically think that most of the hard work is in preparing for the lesson, not in delivering it. And it had been many years since I had been asked to do a demo lesson, so I was actually a bit nervous!) In the end, the demo went fine, I think, but it's too early in their interview process for me to know if they're really interested. (They actually hadn't posted the job yet, so I was the first candidate they were speaking to about the job, and that was only because I had emailed them out of the blue, requesting a visit.)

Visit #3 was to a large public school, and this one was pretty far, well north of the U District, if you are familiar with Seattle. The principal was very down to earth and kind, and she showed me around their math hallway (there was a lot of integrated technology -- smartboards, doc projectors, etc.) and told me that they have a few National Board Certified math teachers. I was pretty excited about the possibility of teaching at a good public school, but unfortunately they're not hiring at the moment. She said that if I am willing to, I could apply to the district and get an offer from the district (city) level, if I am willing to be dispatched anywhere in the district. She also alluded to the fact that they don't do cool, integrated teaching of mathematics like they do in all the other subjects, because math is high stakes and kids have to pass end-of-course state exams in order to graduate. That's a shame to hear, and it reminded me that I have to be very careful about public schools, in order to not end up somewhere that forces me to teach to the test.

Visit #4 was much more relaxed, because I visited a friend from PCMI on her home turf. She teaches at a coed Catholic school, in the boonies. It took me close to 2 hours to bus out there, and then on the way back, because I had missed the 1:10 bus, I had to wait 2 extra hours for the next bus to come. At least I got to check out the ponies at the street corner while I waited. (No joke.) But, her school was lovely! The campus was huge and beautiful, the technology was wired through the roof, and the kids seemed well-balanced with interesting personalities and "strong academics", as she said. It was a nice school to check out, because it helped me realize that no matter where I work, it's always going to be a tradeoff -- schools in the boonies will be equipped with great facilities, but the tradeoff is that I'll have to commute that far out from the city. Schools closer to the center will be less space-equipped, so when I evaluate them, I need to use a different set of criteria...

But, anyhow, this trip has been very productive! It has helped Geoff and I make the decision that we'll definitely go through with this cross-pond move, even if I may have to be flexible and to take whatever may come (including doing part-time gigs till I find a full-time job). So now, it's off to wedding planning and job searching all at once. Yikes.

Visit #1

I went to visit my first Seattle school today. It went well, I think. I loved the school. The kids were very nice; the teachers went out of their way to make me feel welcome; and I saw a lot of great math going on, from Grade 6 math to Multivariable Calculus. The location of the school is also fantastic -- within walking distance of my old hipster neighborhood. If this school is interested in hiring me, I would be thrilled!

But, so that I am not keeping all my eggs in one basket, I am visiting at least two more schools this week in hopes of just networking, seeing what is out there.

Job search is always tricky. I know that if they decide to hire me, I will adapt to whatever situation it is, but the question is always how to get that foot in the door, when you don't know what other candidates have to offer?

Anyway, I can only be myself and then hope for the best! Good thing there is that whole wedding planning thing to distract me.

They Like Logs!

I don't know if it's a coincidence, or if there are other forces at play. But today, during class, I noticed that all of my 11th-graders are solving all kinds of log and exponential equations fluidly without accepting any help from me. They were even completely comfortable finding inverse equations given a function like f(x) = a*b^(cx - d) all by themselves.

Amazeballs. This is the first time that I think my students as a whole really understand logs!

I still believe that the credit goes to this. Sometimes, it just pays to slooow theeem dooowwwn.