June 2001 letter about the Math Wars
I wrote this to my District Rep in preparation for our Professional Reconciliation hearing in June 2001. It was the most I’d thought about teaching math to that point; I was just completing my 4th year. And I don’t know that I love everything I wrote. But I’ll stand by it as in the main correct, and an honest, thoughtful attempt to find a place for teachers…
Those of you who have read posts here over the last 2½ years will recognize some similar thoughts, but in formation. You may even recognize early elements of my Outlook on Teaching Mathematics.
June 14, 2001
There are some things I didn’t say and we didn’t really cover in those math meetings. First, like it or not, battles similar to ours are being fought throughout the country. Second, when we got to what program(s) we want, we were fairly vague.
(Personal note here: I will be teaching Math Connections 3 next year. I will “fix” lots of stuff along the way. I have a feeling that that is what has happened in many places where it seems to be working. But this requires a certain skill/experience/mathematical ability level, requires hands-off or cooperative supervision, and wouldn’t it be better to start with a better curriculum in the first place??)
Let’s start with the “Math Wars.” It makes me damn nervous to be on the same side as what I would call right-wing kooks. It started as a California thing: “Back to basics” vs. “Constructivists” along roughly the same fault lines as the anti-Bilingual, and the anti-Affirmative Action fights there. There are big differences among these, but in each case the Education or Liberal or whatever establishment, in my opinion, took something that was decent, ran way way too far with it (to the point of abuse), and gave the right wing an easy target.
(much more beneath the fold)
I like to think about us as taking the reasonable center against the Ed nuts on one side, but then holding it against the back to basics cretins who will certainly be emboldened enough to start making real noise in the next few years. [Another teacher] has found the biggest back-to-basics site: http://www.mathematicallycorrect.com and their opponents are at http://mathematicallysane.com. They both make noise, they both point fingers at each other and say “you are extreme; we are reasonable” Anyhow, by being aware of the other extreme, we may pre-empt some of what the Supe might say.
If you look at what District 2 posted ([another teacher] e-mailed it to me) about the opponents of their curriculum, you may get a good clue to what could be argued against us. And, to be perfectly honest, unless we have some specifics, those sorts of charges could easily stick.
Next, here are some of what I believe would be elements of a strong mathematics curriculum.
#1 – Problem Solving.
We should be doing off topic mathematical problem solving with our kids, on a semi-regular basis. We should be teaching the skills, strategies, techniques, thought processes that all go along with being confronted with something that looks different from what we’ve seen before. Maybe take a class period once a month? Maybe every other week? But IMP does this work daily, and I doubt back-to-basics programs do more than routine mixture, train and work problems. We have the reasonable center.
#2 – Calculator Use
Calculators should assist in learning, not replace it. Book 1 of Math Connections (MC) opens with training kids to use the graphing calculator to find mean, median, and standard deviation, which they seem to learn to do accurately, but with no sense of what they are doing or what it means. On the other hand, I have seen graphing calculators well-used, after kids can sketch a sinusoidal curve, to demonstrate what happens as elements of f(q) = Asin(kq+c) are altered. Not anti-technology, but for proper use to maximize learning, while MC seems to use the calculator for its own sake.
There is a second area of concern with the calculator: What do you do with kids who reach high school and can’t subtract if there is a carry? Push him a calculator so he can solve an equation? Remediate the skill gap? Right now we do the former. I think we should do both (having a supplementary skills class, or a resource room kind of situation for the skills might help?)
#3 – Reading
At the right level (knock against IMP), with embedded literacy strategies (knock against MC), with literacy training for teachers. Especially in this borough, it behooves us to include work to improve reading everywhere we can, however, we are talking about mathematics, and the precision and accuracy of mathematical language should not be compromised in the process. The back to basics folks mock the idea of any writing or reading in math, while IMP and MC have far too much reading, and far too little emphasis on mathematically accurate language. Find the reasonable center.
A couple of weeks ago I was involved in a discussion. A girl posted a question on-line; she wanted to know what careers might use algebra (this was for a class assignment). The non-teachers who read this went berserk. They were utterly and unalterably opposed to this assignment for a variety of reasons, starting with opposition to any reading or writing in math class, and ending with opposition to doing anything they didn’t do when they took algebra (60’s and 70’s)
#4 – Discovery
A few times in each course I teach, I trot out rulers or scissors and ask the kids to cut, or measure, or estimate, tabulate their results, and see if they can figure anything out. The next day I review their findings, formalize them mathematically, and move on. Example A: measure the sides of a bunch of right triangles, examine their products, sums, squares, etc, and try to find where the pattern occurs. Example B: Pull out calculators, let them fill in table with first 20 square roots, and find what + what = which. I teach why 1.414… + 1.414… does not equal 2.000 the next day…. Example C: give them a bunch of 12” or 30 cm rulers, and ask them to construct triangles from them with varying sides. Let them convince themselves that the longest side can’t be longer than the sum of the shorter sides. And so on.
This Discovery Based Learning is great stuff, and I am glad that I have found some places to apply it. “Constructivism,” (MC and IMP both claim to be “constructivist”) take this nice idea and essentially say, this works well for these five lessons, let’s use these principles to teach the whole course. There is no, or greatly reduced, drill and practice. The “next day” where the teacher teaches is marginalized. (at the other extreme, back to basics folks mock any activity which doesn’t involve the teacher at the board)
#5 – Connections
The NCTM and probably most of us like linking parts of our curricula to real world situations, or connect up to ideas from other subjects, or even just link the graph to geometry, or the function to arithmetic. But IMP and MC have taken this idea of connecting math to such an extreme that every concept is placed in context, and some of those contexts are mathematically weak, or even misleading. It is good when our students understand how the concepts they are learning fit into the real world, but in order to do that we actually have to teach the math, and the real world is insufficiently abstract for many of our concepts.
Further, we introduce some connections with some classes, others with others. It is good if we know lots of ways that the notion of slope, for example, can be translated into a real world analog (I love doing position vs. time graphs, but not with every class). Some classes need more connections, some less. An ESL class might need extra to help with words, rather than concepts. But these books are one size fits all.
#5 – Drill and Practice
As you know from the “supplementing” stories, these new curricula come with vastly insufficient quantities of drill and practice. A back to basics approach might include drill to the exclusion of the more engaging parts of the curriculum. Again, drill is essential, but there is no need for it to dominate our subject, happy medium, etc. etc. On the other hand, in District 2 they say that their teachers supplement TERC and CMP and ARISE with drill and practice as necessary. I am not sure, without specifics, we should believe them. Further, we have the supplementing story from the Bronx (Lehman?)
#6 – Remediation
Students who move ahead grade levels while lacking mathematical (including arithmetic) skills, should have them remediated. However, neither pure “teaching for learning” nor “skills drills” but rather some sort of blend is most likely necessary. It is not acceptable to graduate students with big arithmetic gaps. There is no way to learn facts without drilling. But if a student demonstrates an incapability of learning through drill alone, variation in instruction, not pure repetition, is likely the best course.
#7 – Teaching for Understanding
This is probably the only thing these books really get right at all. It should be our obligation as educators to at least attempt to teach the students so they can understand what they are doing, not just rote formula and fact memorization (but see #5, still have to memorize those facts!) In theory, NY State style developmental mathematics lessons should be doing this, but we know that they often do not. Also note that at least Connections lets the graphing calculator interfere with understanding.
#8 – Good Process; Correct Answer
We want both. The Constructivists pay only lip service to the latter, the Luddites do the same to the former. We want a balanced approach that insists on correct answers, but that doesn’t simply begin and end with correct answers.
#9 – Mathematical Authority
In the classroom, ultimate mathematical authority rests with the teacher. We’d like the kids to figure lots of stuff out, but Connections training reduces the teacher to the level of a bright student.
Sorry about running on and on, but I hope at least some of this is interesting. And I’m more than willing to flush any of this out.
Our meeting is later today, but I have thought of a few additional details.
We’ve been concerned about finding data to back our contentions. I think the data simply does not exist. Both sides argue over a few particular cases that purport to show improvement or drops in achievement. Let them argue. In each case the improvement or drops are small, if they exist at all.
What a waste. There are ways to spend money to get results; Instead, why are we spending money on IMP and MC, which at best cause a small increase in achievement? (and there are studies that dispute that)
Reducing class size increases achievement. Significantly. With no arguments.
Slowing the courses down for lower-achieving students would raise scores.
Reducing teacher turnover would raise scores.