So school has begun and despite my palpitations about the boredom of only two familiar preps, I’m pleasantly busy. Last year was a hell of a lot of work, and given the nosedive that my writing time took, I should maybe not be so eager for a less…familiar schedule. So instead of demanding new classes, I accepted the first semester, threw a minor temper tantrum when no one listened about second semester and all is well. Algebra 2 in particular is proving a delightful challenge, given my new emphasis on functions.
In no small part because of this planning breathing room (is anyone noticing I’m saying my panic was a total overreaction?), the senior Water Park Day registered in my awareness ahead of time. In prior years, I didn’t heed the warnings that half my class would disappear, and so would be forced to dump my lesson plan on the Day itself, when the smaller classes would just have a day to practice. But thanks to this old, familiar schedule that gives me more time, I anticipated the impact.
So for the first time, I was able to give serious thought to having a day to pursue math without regard to subject matter or schedule. I could have a “math day”! Then I remembered Grant Wiggins’ challenge to math teachers everywhere in the form of a conceptual knowledge quiz.
Grant proposed this as an actual test: I will make a friendly wager: I predict that no student will get all the questions correct. Prove me wrong and I’ll give the teacher and student(s) a big shout-out.
What math teachers think their kids would know the answers? I certainly didn’t. In some cases, they probably were taught, but in others, I doubt an elementary school teacher would ever think to bring them up. But even if all the concepts were taught by fifth grade, how many kids of that age could really appreciate the questions?
Most of the questions tease at the paradox….wrong word? tension? between the functional day-to-day applications of arithmetic, and the amazing truths that underlie them. John Derbyshire wrote, in Prime Obsession, that “arithmetic has the peculiar characteristic that it easy to state problems in it that are ferociously difficult to solve.” (I was rereading Prime Obsession last night; there’s tons of useful thought material for math teachers. I need to go get his book on algebra.)
Arithmetic looks easy. (And certainly in the last twenty years, the rush to shove everyone into calculus has led to a certain contempt for “basic arithmetic” classes.) But even if elementary school age children are capable of understanding its ideas fully (and most of them aren’t), they haven’t experienced several years’ utility of arithmetic. They haven’t had time to get bored of the routine rules that they are expected to remember (mind you, many don’t, but leave that for another day.) Yeah, yeah, invert and multiply. Yeah, yeah, you can’t divide by zero. Wait, what the hell do you mean multiplication isn’t repeated addition?
To really enjoy this test, to be fascinated by the underlying truths–or misconceptions–behind certain everyday math tools, requires familiarity with “the rules”. Time spent in the trenches of doing math just because.
That’s when a teacher can spend an enjoyable hour taking the kids back through a re-examination of the basics and what they really know. I’d much rather discuss these concepts with adolescents who have survived two or three years of high school math than try to force sixth graders to “demonstrate conceptual understanding” of dividing by zero.
I had no real expectations—no, that’s wrong. I had hopes. My sense was the students would be interested in the exploration, if I didn’t take on too much or dive in to the wrong end of the pool. But which end was the wrong end?
So for each of my four classes–two Algebra 2, two Trigonometry–I gave them the test and 20 plus minutes to write down their thoughts. I was alert to the possibility that kids would use five minutes to doodle and fifteen to giggle, but in each class the bulk of students asked for and got an additional five minutes to finish up. I collected their answers and will share some of them in later posts; they were often detailed and thoughtful.
After the writing time, the students had a few minutes to “share out” in their groups, so they could learn what questions puzzled their classmates—and also as reassurance that they weren’t alone in their befuddlement. Again, this seems different from Grant’s intent; he considered it a real test that the students would either answer correctly or leave blank in confusion. I listened in on many conversations; they were rich with exchange as the students realized they weren’t alone in their uncertainty.
But certain questions also sparked genuine debate and interest. More than a few students offered up multiplying negatives as an example of multiplication being something other than repeated addition. In every case I witnessed, their group members, who had written something to the effect of “isn’t it always repeated addition?” instantly recognized the roadblock that negative numbers posed to their definition. I came across more than one group arguing whether multiplying by zero counted as repeated addition (“yes, it does. If I have zero groups of five, I have zero!”). Interestingly, no one came up with the roadblock I was interested in, and I’d never once considered negative numbers until my students brought it up.
Their discussion time was about ten minutes. My goal wasn’t to have them determine the answers; rather, I wanted them all to have a shared experience before we discussed them as a class, and I gave them the “answers” (to the extent I knew them). That way, there’d be more of a sense of “we”–yeah, we thought of zero, too! yeah, we all have 3F=Y–that’s not the answer? yeah, we think dividing by zero gives you zero–it doesn’t?
So then we went through the answers as a group.
I had taken a subset of Grant’s list, ignoring the last three items. Doing it again, I would have swapped out question 2 for question 11 “appropriately precise”), because while question #2 is good, it really requires its own day. The rest of them are easily covered and discussed in at most 15-20 minutes each.
The questions I really wanted to spend time on, to explain in at least introductory depth, were 1, 3, and 5. From a practical standpoint, I wanted to be sure everyone understood why they got questions 4, 6, and 8 wrong, assuming most missed at least one of them. I was genuinely interested to see what they had to say about 7 and 9 but was going to take most of my lead from them. Question 10, I wanted to know if the trig students knew it; obviously, my algebra 2 students learn about imaginary numbers for the first time.
My trig classes are quite different in nature. Both are small, just 25 in each. Both are doing quite well; I have no kids who simply shouldn’t be there, as I did last year. My first block class is stronger, on average, but has more surly kids who mouth off. It’s very irritating, frankly, since the five or six kids giving me quite nasty sass are seniors who are doing relatively well (Bs and Cs), and who openly acknowledge that they think I’m a hell of a teacher. Two of the surlies had me last year for algebra 2, when they were much less trouble, and had been switched into my class because they were failing with another teacher. But these other teachers, who they didn’t like (and often failed, forcing them to retake a fake summer school course if they couldn’t switch to my class), didn’t get nearly the lip. I’m a tad flummoxed. My second block class has more kids who are amiable and interested but not taking the class as seriously as they should, so several more low scores on the first test. First block has a stupendous top tier, but it’s just three or four kids. Second block has a top tier of close to eight, but they aren’t quite as strong.
Anyway, I was expecting more interesting conversation from second block, and I had it backwards. First block was on point, even the cranky ones. They loved the test, wrote detailed responses, discussed it thoroughly in group, and were wildly participatory in the open discussion. Easily 90% of them came up with the correct response to imaginary numbers (and the ones from my algebra 2 class identified multiplying by i as 90 degree rotations in the complex plane, which was quite gratifying, thanks so much). Second block, the amiable, mildly uninterested ones pulled things down slightly, goofing around and making jokes while the stronger kids would have preferred more time to explore things. The conversation was still great, the students learned a lot and enjoyed the discussion, but I had the enthusiasm levels backwards.
My algebra 2 classes, I nailed in terms of expectations. Block three is a fairly typical profile, except I have a lot more sophomores than usual (which is due to our school successfully pushing more kids through geometry as freshmen). But still a good number of seniors who barely understood algebra I, a lot of whom are just hoping to mark time til graduation without ending up in summer school. (One of my specialty demographics.) And in between, juniors and seniors who are often thrilled to find themselves actually understanding math and succeeding beyond anything they’d ever hoped (another specialty of mine). Typically, many of the seniors were in class, as they lacked the the behavior or grade profile (and sadly, in some cases, the money) to go to the water park. So I expected conversation here to be a bit lower level, with less interest. Happily, everyone engaged to the best of their ability and many told me later how much they loved just “talking about math”. I spent much more time on questions 4, 6, and 8, and could see them all really registering why they’d made the mistakes they did. But they still were enthralled by questions 1, 3, and 5, which is great because it’s going to give them some memories when we review percentages in preparation for exponential functions.
Last up was block 4 algebra 2, a ridiculously strong class; only five students are of the usual caliber I expect. The seniors are all well above average ability level. Two of the kids are so skilled that I’ve already introduced three dimensional planes and the matrix, while still forcing them and the other really strong kids to deal with complex linear word problems (mixture questions! I usually skip them, so it’s a trip). They stomped all over the test, writing at great length, discussing it with their teams and then shouting out to other groups to see what they’d answered for multiplication. The class discussion took so long that I actually allowed it to continue for 20 minutes into the next day, when I invited one of my mentees to watch. He came away determined to try the test in his honors geometry class.
Look, the whole day was teacher crack. Take a day. Try the test. I’ll be discussing individual questions and my explanations in future posts, but this introduction is offered up as invitation. High school teachers working in algebra 2 or higher would be a good starting point. Honors classes in algebra and geometry would also benefit. Every math teacher can find links from this test to their math class—but then, that’s not the point.
As for me, I started out the day with hope, but also a determination to see it through as part of a way to honor Grant Wiggins, who felt very strongly that students needed to do more than just march through curriculum. I promised myself I wouldn’t abandon the effort even if it went wrong. It didn’t go wrong. Quite the contrary, the test sparked delighted interest and intellectual curiosity among students who are often hard to push into exploring mathematics in depth. So hey, Grant, thanks for the idea–and the inspiration.
September 27th, 2015 at 12:39 am
[…] Source: Education Realist […]
September 27th, 2015 at 4:03 pm
I think that when people imagine “the brighter students will help the not-as-smart ones achieve understanding”, this is the picture they have in their head. Unfortunately, none of what you did here came from a curriculum statement or a teaching class; you invented it partly in your own and partly with an idea from some guy on the internet (which you tailored to meet your needs).
September 27th, 2015 at 5:01 pm
Well, I’m not sure about the first. What they would have in mind is something like this:
1. Each group gets a question.
2. The group puts together a poster explaining their answer to the question.
3. They present the poster, and the teacher will ask them all questions.
4. If they all don’t know their answers, then they all get Fs.
September 28th, 2015 at 10:26 am
“Both are small, just 25 in each.”
??? I think, growing up (35 years ago), that 25 in one of my classes would have been the biggest class I can recall. I’d say a typical class would be around 20 (these would be in large, suburban high schools). How many kids do you have in your classes? How many kids in a class are typical-either in your school, or in public schools in general? I have heard from several sources, that the local high school has math classes of 40-50, and I was utterly appalled, but I had assumed it was an anomaly of this particular place. Is it not?
Incidentally, as an adult with kids (and as an adult with college and graduate school behind me), I now think that even 20-25 per classroom is absurd: classes should be around 8-12. I realize it is financially impossible, but the ideal is the ideal. When I see college classrooms of several hundred in an auditorium, I see ‘waste of time and money.’
September 28th, 2015 at 12:25 pm
I don’t think anywhere has classes of 40-50. Most high school contracts have limits of 35 or so. I commonly have between 30 and 35, and consider anything under 30 to be a “small” class. Anything under 20 I consider a waste of a teacher.
September 28th, 2015 at 8:23 pm
Public schools in Utah can (or least used to) have enormous class sizes in the 40-50 (or even 50+) range. The birth rate in Utah is the highest in the country, so they can tax themselves at a pretty high rate and still have less money to spend on each student than other states. I worked for years in an intervention program that provided supplementary services to struggling students in various Utah high schools. My actual work site was at West High School in Salt Lake City, Utah. The last year I was there (2005-2006), the students were on their third schedule of the year by October. Guidance kept redoing the schedules over and over in a desperate attempt to keep all the English classes under 50 kids per class.
(Your math lesson sounds fabulous by the way.)
September 29th, 2015 at 12:52 am
Thanks! God, I can’t imagine 50 kids in an English class. English teachers in our area are the ones most likely to fuss about class size, and who can blame them? That’s a lot of essays to read and correct.
September 29th, 2015 at 7:32 am
I will be vague in order to protect my anonymity, but I am new to a town (and I have young children, so I don’t have direct knowledge), and two people have told me that the local high school has math classes of 40+.
The oddity is this: those math classes are intentionally set at 40+. Apparently, there were dramatically differently skilled math teachers at the school, and the school, in order to guarantee an ‘equal’ math opportunity, increased the math class sizes to give all the students to the better teacher(s).
My kids are in private school.
September 28th, 2015 at 6:17 pm
That sounds wonderful. I hope my kid’s teacher reads your blog & takes up this challenge. (but…sigh, it’s unlikely)
September 29th, 2015 at 5:35 pm
[…] definitely one thing I’m facing as I work to support teachers at school. Seeing @ED_realist recent post gives me a clearer picture of my level of conceptual understanding. This post showed me at […]
September 29th, 2015 at 10:34 pm
How long is your prep period? My prep is 50 minutes. I’m teaching for 300 minutes except for one day a week when I work 220 minutes because there is a hour and twenty minute common planning time that day. Prep is not long enough for my liking and I come in over an hour early everyday to prep. The problem mostly lies in the fact that no curriculum is provided by the district it is all teacher generated and we follow the awful balanced literacy approach in ELA. Common planning times are almost always admin directed and provide for no actual lesson planning. It is mostly just wasted time talking about state testing, district testing or the admins personal lives.
September 30th, 2015 at 1:33 am
I’m actually writing a piece on preps right now. I teach a 4 x 4 block schedule, so I have a 90 minute prep when I have one. But right now, I don’t. So I’m teaching from 8-3 with just 30 minutes for lunch.
September 30th, 2015 at 3:03 am
[…] was initially horrified at my schedule when I first saw it last June. Having since conceded the possibility–just the possibility, mind you–that I might have overreacted, I thought […]
September 30th, 2015 at 11:13 pm
[…] notes. I recently put up a post here about the blogger Education Realist. Well, ER has a math quiz for you. To be precise, simpatico educator Grant Wiggins, who died May 26th, had one and Ed Realist is […]
October 1st, 2015 at 7:56 pm
I really doubt that your students answered question 10 correctly. Most people think that imaginary numbers were invented to deal with solutions to quadratic equations. But historically, people had no problem with unsolvable quadratics. It was the general solution of the cubic that required people to develop and accept imaginary numbers:
Click to access ShortHistoryComplexNumbers2006.pdf
If I remember correctly, Mr. Derbyshire does cover this in his book on Algebra which I recommend.
October 1st, 2015 at 11:01 pm
Yeah, I’m looking for something like “solving problems that don’t have real solutions”.
May 2nd, 2018 at 4:25 pm
Though it might be interesting to mention to them that they were motivated as being useful to *finding* the solutions to problems that *do* have real solutions (the cubics mentioned).
October 6th, 2015 at 1:06 am
I;m finding it hard to believe that your kids got the imaginary numbers invention correct. Did you really teach them all about cubic equations?
October 6th, 2015 at 5:53 am
I already answered that. We usually cover it in pre-calc, so all I was looking for was “to solve problems that previously couldn’t be solved”, with the example being the one they knew about.
November 25th, 2015 at 1:33 pm
[…] who could disagree? Arithmetic has, as John Derbyshire notes, “the peculiar characteristic that it easy to state problems in it that are ferociously […]
December 20th, 2015 at 5:36 pm
[…] then staying afterwards to chat. This last week he showed up to my first block trig class, with the surly kids who mouth off. We were in the process of proving the cosine addition […]