A couple of days ago I gave my Algebra II students a summative exam. My exams consist of about 10 standard multiple choice questions and 2 or 3 more involved problems. One of the “thinking” problems went like this.
[Pretty simple: ad – bc]
[Not too bad: (a-b)d – (c-d)b = ad – bd – bc + bd = ad – bc]
(c) Compare the two results
[Way easy – they are identical]
(BTW, I got this problem from Weeks and Adkins, “A Second Course in Algebra”, Ginn and Co.,1962 )
What I got was most kids multiplying 783 by 783 by hand and subtracting from this 782*784. Some kids succeeded in doing this without error and got the correct answer, 1.
In going over the problem today, I worked it out using the simplification and also got 1 – the easy way! Imagine the students’ surprise and indignation when I told them they would get only partial credit for getting the correct answer the laborious way. One girl, who usually has a bit of an attitude, called out “So you don’t care if we got the correct answer!?!”
I told the class – “…no, I do not care. I care about how you think, not about whether you can multiply three digit numbers, we have calculators for that. This is not third grade, this is Algebra II – I expect more out of you; I expect you to think!”
I don’t know what effect this had – they saw that I felt deeply about the issue, that I raised my voice and I was “in their faces” and they were quiet. Will they try to think more? I doubt it, but I think this exchange goes to the heart of what is wrong with math education today.
I work in a district with one high school and two middle schools. These middle schools are good – they score above 800 on the California API. I know the teachers who teach Algebra I at the middle schools, the ones who teach the better kids, those kids who go into Algebra II as freshmen at the high school. These are good teachers – solid, enthusiastic and experienced. When I get kids from these teachers I can count on the students not having trouble with solving equations, doing most arithmetic correctly and doing their homework regularly.
What I can not count on is these students having been challenged to think. My impression is that in middle school the time is spent on the mechanics – e.g. doing 50 problems on graphing a line until you can calculate a slope and a y-intercept and you can graph a line correctly. But heaven forbid that you try to teach the kids to graph a line using x-intercept and y-intercept. “We haven’t been taught to do it this way – I am going to stick with the old method.”
I do not object in teaching mechanics per se or only one method of doing things. What I object to is that in the process of doing all those 50 repetitious problems, the time and opportunity to challenge the students is gone. We do not emphasize or “grow” critical thinking in middle school math, so when they are asked to do this in high school (if they are!) – the kids fall apart.
I understand perfectly well that cognitively kids must acquire some basic knowledge before they can be asked to think critically. I also understand that there is tremendous pressure to reach and surpass that 800 score and therefore we are going to teach to the test – the kids are going to be able to do the problems on the state exam – we are going to drill them until they do!
But I also understand that we live in the world of Wolfram Alpha and Siri. We no longer ask cashiers to calculate change (because they can not and we have come up with machines that can). Free software such as Wolfram Alpha can answer some pretty sophisticated mathematical questions – certainly the high school level. It’s no longer that difficult to get the correct answer – what is still difficult is to think critically. This is a quality that future employers will seek, this is a quality that makes a nation competitive – and we are not doing it.
So what is the solution (assuming I am correct in my diagnosis)? One thing is more articulation between the middle schools and the high schools. Beyond the face to face talking this means sharing exit and entrance exams for correct placement, being familiar with each others tests and most of all instilling a culture that glories critical thinking at as an early age as possible. We need to take away some chunks of time and to move some emphasis to critical thinking – we need to do that from 6th grade on and we need to do it pretty quickly.