Assessment Group

In preparation for an “Assessment Group” meeting after work today, I’ve been reading about formative assessment.

“What we need is a shift from quality control in learning to quality assurance. Traditional approaches to instruction and assessment involve teaching some given material, and then, at the end of teaching, working out who has and hasn’t learned it–akin to a quality control approach in manufacturing. In contrast, assessment for learning involves adjusting teaching while the learning is still taking place–a quality assurance approach. Quality assurance also involves a shift of attention from teaching to learning. The emphasis is on what the students are getting out of the process rather than on what teachers are putting into it, reminiscent of the old joke that schools are places where children go to watch teachers work.” (from “Classroom Assessment: Minute by Minute, Day by Day” by Leahy, Lyon, Thompson, Wiliam)

I chuckle at the idea of students going to watch teachers work – I do sometimes feel like they put in so little effort and here I am stressing myself out, working at home, having nightmares about school…

I really do agree with the ideas of formative assessment. I guess I just keep finding myself conflicted in how to do it.

For example, I made a quiz over the graphs of exponential functions for my precalculus classes. My whole goal with the quiz was to make questions that could get at the shapes of graphs relative to other graphs but so that the kids couldn’t just plug it into their calculators. It’s like when I get into “test and quiz making” mode, I come up with these tricky nice questions, but I almost wish they weren’t being graded on them. I honestly call it test or quiz mostly so they will take it seriously!

I think they can learn from the questions, but how do I administer the quiz and have them learn from it? I would rather just not grade the quiz at all and use it for discussion, but I feel like they expect a grade and who am I to completely change in the middle of the year?


Never Passing Through the Student’s Mind

“U.S. Senator Jeff Bingaman recalled what his father, a chemistry professor, used to say about traditional lectures: the notes of the teacher go straight to the notes of the student without ever passing through the student’s mind.”

I stumbled on this in an article about a local charter school, and it relates directly to a situation I had yesterday. There’s a 10th grade girl in my geometry class who struggles in math. She told me this at the beginning of the year, and I tried to inspire her that this year could be different. She’s a really interesting girl who perks up at any mention of fractals or Fibonacci and turned in a really eloquent piece of writing for my first foray into making the students write in math class.

So she ends up pulling up to a B- last semester and was absolutely ecstatic. Then yesterday was the first test for this semester. A couple days ago I was having them work on a review of similar triangles while I went around and talked to each of them, and I started to become a little worried about her. Similar triangles usually involves dealing with fractions, and it became clear to me that her fraction skills were shaky at best.

So anyway, yesterday right before the test, she comes into my office and has this big confession to make. She says that she likes my class, but for some reason this whole unit on similar triangles has just not stuck with her. She says that she just feels like no matter how hard she tries to listen, she can’t focus on the lecture. When she can focus, she understands what I’m doing, but then she tunes out again and then when she’s looking at the homework problems, she has no idea what to do. She kept saying that she thought she knew the concepts but “couldn’t express it” the way that I showed, which basically meant she doesn’t solve proportions the way I do and was concerned about having to show her work. When I saw her working in class the other day, she actually knew what the answer was supposed to be but couldn’t explain why to her friend sitting next to her. She asked me “how to do it”, which I interpreted to mean how to set up the algebraic proportion and solve, and that’s where she was really lost.

I encouraged her to take the test and answer the questions “her way” to the best of her ability, not worrying if the “work” looked like what I showed in class. What I saw on her test was actually quite interesting. She was looking at the relationships between the sides in an addition sense – so that if one side increased by 10, she wanted the other side to increase by 10. She obviously missed the entire idea of increasing by a multiplicative factor, and I never managed to figure that out before the test.

So it makes me really wonder what to do now. I don’t feel right giving her a low test grade when I can finally just now see that she has a huge conceptual misunderstanding. Honestly, I’ve been questioning testing in general lately, but this really makes me wonder.

I know that when I give back tests, most students hardly look at it. The ones who scored poorly look embarrassed and are usually the ones that put it away the quickest. I kind of doubt most of them go home and pore over and read the comments and try to figure out how to improve.

I did a test corrections assignment earlier this year with mixed results. I’m still not sure it really shows them what to focus on; they just figure out how to fix that one problem in hopes of getting some points back. I mean after all, it’s hard for someone to look at their own paper and pick out their areas of weakness, but it’s fairly easy for me to do provided they show enough of their thought process in writing.

So I have this idea and I’m a little hesitant to do it because it would take longer.. but I was thinking of writing each student a short summary of their test that mentions the areas of deficiency, and giving them that before they ever see a red pen mark or a grade. Then, those that score below a certain threshold can meet with me to work on that deficiency and possibly earn back some credit.

I don’t know. It’s still playing in the “points game” which I am really starting to hate, but I’m not ready to go completely off the deep end with the anti-testing stuff yet. At least I feel like they’d have more of a chance of getting feedback before they shut off when they see a low grade.

Am I going to use calculus in real life?

Over winter break, I went a little crazy with watching TED talks relating to mathematics and education (and somewhat rarely, both). This is a really nice short one (in case you’re like me and don’t normally feel like watching a 20-minute video that some random blogger thinks is cool).

The thesis? That all of our high school math really leads to one summit: Calculus. But will most people really use calculus? Not really.

Of course, I have to insert a comment here about whether we should care if something is useful when teaching math. As a pure math person, I have to say that I don’t really care whether something I learn is useful, just interesting and beautiful.

Then again, I didn’t particularly like calculus. I know there are some beautiful ideas about the infinite and rates of change, but the way I learned it, it was anything but beautiful.

Anyway, Benjamin goes on to say that most people would benefit more from a statistics class than a calculus class. The general public’s lack of understanding of statistics is actually quite problematic; we are swayed by numbers that have little meaning coming from sources as ubiquitous as our banks, politicians, and news sources.

So would the world really be better off if the masses were forced to learn math that prepares them to better understand statistics rather than calculus? What do you think?

I like the idea of teaching more probability and statistics. I don’t really see why calculus is the holiest of holies. But if it were my way… we’d be teaching seniors advanced algebra and number theory. 😉

A Mathematician’s Lament

“Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics… Mathematics is the purest of the arts, as well as the most misunderstood.” – Paul Lockhart, A Mathematician’s Lament

I’ve been on a “humanist” math philosophy reading kick lately. I’ve gotten myself partway through two of Reuben Hersh’s books and many of his articles and found many interesting TED talks supporting the view that teaching, regardless of the subject, has changed in many ways and we need to adapt.

Lockhart’s article is one of the better pieces I’ve read on what is apparently a fairly subversive view of mathematics and mathematics education. I sat reading it, completely absorbed, getting more and more fired up with each page. The article addresses ideas which have been circulating in my mind a lot lately; one semester into my teaching career, and I’m already seriously throwing into question 80% of what I thought I knew about math education.

I am a brand new teacher – graduated with a bachelor’s in pure math less than two years ago. I never went to education school, which Lockhart refers to as “a complete crock”. He argues that “teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children.”

I can see this pretty clearly in my own experience this year. I teach geometry, which Lockhart calls the “instrument of the Devil” and precalculus, which he refers to as “a senseless bouillabaisse of disconnected topics” (is it bad if that quote makes me hungry?). Just halfway through my first year of teaching each of those courses, I already see frustration in the students with the traditional way of teaching. They shut off completely during geometric “proofs”, which I have now almost completely abandoned after realizing that not only did the students not understand them, but apparently lost the ability to explain simple concepts in their own words because they were so preoccupied with having to write a “geometric two column proof”. Precalculus has lots of review, and many of the students are completely shut off by having the same material they’ve seen before just lectured back to them in about the same depth.

It’s too easy for me to “be a passive conduit of some publisher’s “materials” and to follow the shampoo-bottle instruction “lecture, test, repeat” than to think deeply and thoughtfully about the meaning of one’s subject and how best to convey that meaning directly and honestly to one’s students,” as Lockhart describes it. So easy, and so boring, that in less than one semester, I am already itching to get further away from stand-and-deliver-teaching and the mind-numbing textbooks we use.

“Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them, that we think it’s too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon, why not about triangles? I think it’s simply that we as a culture don’t know what mathematics is. The impression we are given is of something very cold and highly technical, that no one could possibly understand— a self fulfilling prophesy if there ever was one.”

“All this fussing and primping about which “topics” should be taught in what order, or the use of this notation instead of that notation, or which make and model of calculator to use, for god’s sake— it’s like rearranging the deck chairs on the Titanic! Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion— not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.”

Many teachers are afraid to let real math happen. They want to stand and lecture and prevent the kids from trying anything new or ever making any mistakes. It seems they believe math is a set of absolute truths and they think they have been appointed by the gods to stand and deliver these truths to the congregation. Remember the lessons of medieval churches; congregations can’t read the holy word themselves and make sense of it, so we need the holy priests to deliver the word unto them. Looks like we need a real Reformation for math.

I could go on. But I think this is good enough for a first post. I’ll leave you with one last line from Lockhart (who really is a smart cookie, by the way – I laughed, got angry, got inspired, and felt very stimulated by the article).

“It’s perfectly simple. Students are not aliens. They respond to beauty and pattern, and are naturally curious like anyone else. Just talk to them! And more importantly, listen to them!”

If my ramblings have piqued your interest in the article (which is a little long at 25 pages, but truly excellent to read), you can find it online here.