Monthly Archives: October 2011

What do we teach and who should we teach it to? (I)

Two days ago I had a parent/teacher/counselor/student meeting for one of my Algebra II students. The father wanted to know how come his daughter, a straight A student in Algebra I Honors in middle school, was doing so poorly in her freshman Algebra II course.

We batted it back and forth, but what it came down to was that the student understood all the concepts, was applying them correctly, but in the process of solving the problems she was making mistakes in arithmetic and elementary algebra that were killing her final answers. Things like not distributing correctly a minus sign, or making an error in a simple division (35/7 = 6) – these small mistakes were propagating through her problems and yielding wrong and sometimes nonsensical results.

By the way – this student was not the only one. During class work, I circulate among the students and I look at how they do a problem, where they are stuck, why they don’t get the right answers. Often the problem is the same – small algebra missteps, and poor work with fractions.

These are not low level kids – they are not in Honors Algebra II, but they have had a good Algebra I course which they passed with an A or a B. Their middle school teachers were experienced and demanding. The schools they came from are highly rated by the state.

So where is the disconnect? Why do these good students make elementary or even pre-algebra mistakes? Why do they have so much trouble with fractions?

Talking with some my fellow high school teachers elicited three hypotheses:

(a) Elementary school teachers hated math when they were in school so they are not well prepared in math both analytically and emotionally e.g. they are not well prepared to teach fractions

(b) Middle school teachers teach to the test and because of the number of concepts they have to teach, there is no time for review and cumulative practice and testing.

(c) Middle school teachers think of their students as children and treat them as such, whereas in high school the expectations are significantly higher – suddenly!

Of course these hypotheses are based on anecdotal evidence and they can not be generalized to all the math teachers – especially from what I have seen in the blogosphere.

I do have two additional hypotheses however – hypotheses based not only on what I observed in my years of teaching in the US, but also in what I have seen in Europe and especially Eastern Europe/Russia.

The first hypothesis states that in trying to cover so much ground, so many concepts, we do not allow sufficient time for the kids to internalize and practice the fundamentals and we do not expose them to problems that involve critical thinking.

The second hypothesis is that we have gone way too far in the concept of “all kids must go to college”.  Students who do not care about math, who do not see around them that math can be useful, who are not comfortable in an academic environment are shoehorned into a three year math program. We do not offer these students another way of finishing school, such as an apprenticeship and we are disappointed when they do not perform well.

So it seems to me that we have a two part problem here – figuring out why our better students are not doing as well as they ought to and figuring out what to do with the student population that should not be forced to take logs, roots of polynomials, inverse matrices and so on. In the end the question boils down to “what do we teach and who should we teach it to?”




“I don’t care if you get the correct answer”

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.

(a)    What is the value of  ?

[Pretty simple: ad – bc]

(b)    What is the value of  (Simplify your result)

[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]

(d)    Use the previous results to solve

(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.

What can SBG do for us?

It’s been a month since I posted something new in this blog. I have had enough time (I think) to be able to reflect on the use of SBG grading in Algebra 2 and AP Stats. First, I would like to address the use of SBG in Algebra 2 (mostly freshmen).

I have divided the course in units, approximately one chapter in length. For each unit I have put together 5 to 6 Learning Objectives (LOs). Every week, I give my Algebra 2 classes a quiz that covers 3 to 4 LOs. The quizzes are cumulative, and they consist of simple problems for each LO covered. The grading is 0 to 5 for each problem – sometimes with partial credit – and the grades are entered into a spreadsheet. The grades are automatically color coded in the spreadsheet – red for 0 to 3, yellow for 3 to 4 and blue for 4 to 5. Turnaround is usually 48 hours or less. Once a week, I eliminate the names from the spreadsheet, sort it by ID numbers and send this “public” spreadsheet to students and parents.

In general I am not overly enthused about what SBG accomplishes. First, the grading of the weekly quizzes and the entering of the data is very time consuming  despite the help of two TAs.

More importantly, I have found that for my students, SBG does not seem to lead to improved learning. The big advantage of SBG for me is that by looking at a spreadsheet, I can see right away (color helps a lot here) which LOs the kids have bombed and which they have done decent or better.

While this helps me as a teacher, it doesn’t seem to help the kids. Even after reteaching and retesting, many kids still don’t get the concept and indeed repeat the same mistakes. If I reteach again, I still may get poor performance in some of the LOs.  What I find most distressing is that we end doing the same kinds of problems so that an increase in performance is to the detriment of higher order thinking.

Here’s an example. I wanted to get across the idea that linear equations (can) arise from patterns where there are equal changes in two variables. I started with problems such as this:

Pattern #                                 20                     21                      22

The idea was to find a relationship between the number of triangles (y) and the pattern number (x). It took a few weeks of giving this type of pattern problems on quizzes, homework, in class work, before most kids were able to get the correct equation. Even then, asking a follow up question, such as how many tiles were in Pattern #5, gives rise to nonsensical answers or just plain wrong ones (sometimes due to simply faulty arithmetic).

Then, if I switch the question to one like: “Given the table below

















find the value denoted by the question mark”, it seems like a brand new problem, with no connection whatsoever to the pattern problems.  We end up doing now table problems, until a majority of the kids get it – but it still looks to them as a separate type of problem. There is no synthesis, despite my calling attention to the similarities of the two types of problems.

To conclude, while SBG is helpful in alerting ME to weaknesses, it does not seem to be able to move the kids to higher order thinking. I am beginning to think that I could do just as well with summative tests and then reteach those areas that most kids get wrong in the summative tests.

There are a number of SBG-related issues that still are unclear to me. First, how much do SBG results depend on the population of students one deals with? My students – generally – are very poorly prepared for being in Algebra 2. True, most of them are freshmen with all the developmental issues this implies, but above and beyond the question of maturity (what do you mean you haven’t reviewed the results of the previous quizzes?) there is a stark issue of math preparation – or lack of it. For example, my students can not do fractions. Two problems in a recent quiz had to do with solving a system of two linear equations – standard fare for Algebra 2. One problem had the usual type of equations, in the other one the equations appeared as fractions. Result? 75% correct answers in the first problem and 5% in the second one. Would a population of better prepared students benefit more from SBG?

Second, how much of a “burden” should one put on SBG? Is it fair for me to expect that SBG will lead/help in getting higher order thinking from the kids?  Perhaps SBG delivered all that it was meant to deliver – alert me to my students weaknesses.  I ask this question, because in AP Statistics I am happy with detecting the weaknesses and reteaching – but then again, that is different level of course with a different population of students.

To be continued….