Math with Manipulatives

Another hat tip to Sherman Dorn for his comments about the "Springboard" math (and language arts, aka English) curriculum in his county school system based on an article in the local paper about it.

[By the way, when did they invent "language arts" anyway? Do they read poetry in the original Russian or plays in the original Greek? Aren't all of the classes based on English translations when they deal with foreign authors? It's English, folks, and your students need to be able to read and write it so they can deal with my lab class and know enough about literature to survive college classes in composition and the humanities.]

Based solely on the article, which spent most of its column inches criticizing the content of the HS literature courses (when it wasn't shocking me with the huge difference between their curriculum and mine as noted in a footnote), I think I came away with a misleading view of what they are doing on the math side. I find the example Sherman ridiculed to be potentially very effective in the hands of a competent teacher because its purpose was not what the journalist thought it was. Details are below the fold.

The article introduced the math content as follows

In math classes, just four SpringBoard lessons are required each year. Among them is an Algebra I SpringBoard lesson that starts with students arranging baby marshmallows and spaghetti on blue paper to discover different ways to multiply binomials.

Seventh- and eighth-graders in Jennifer Apgar's Algebra I honors class at Ferrell Middle Magnet School in Tampa couldn't wait to dig into their bags of marshmallows in January and were relieved to find extras for eating.

The marshmallows represent office computers; spaghetti separates sections of the office.

"I'm kinesthetic – I like to use my hands," said Giana Moore, 12. She said she prefers such hands-on activities to doing math problems on the computer. The seven computers lining one wall in her class are rarely used, she said.

Well, at least I know where the student mentioned in my previous article picked up on that Learning Style terminology! In Middle School? (Little does this girl know that this is actually a visual learning tool. Your hands are only used to arrange them. Kinesthetic learning would be actually connecting computers into a network, getting your muscles to learn how to pull cable and plug in the connectors.)

This description from the newspaper article sounded a lot like the "discovery" (aka look-say or guessing) way of not teaching math that has been used in some elementary math curricula. The sort of curriculum that produces students who cannot divide or work with fractions when they get to college. What changed my mind was seeing a video showing what it really is, in action, albeit in the hands of an experienced and clearly skilled teacher. That video is part of an article from a Tampa Bay Online story from February 2009 that turned up high on a Google search for the term. The video probably resulted from the same dog-and-pony show, since it featured the same teacher at the same middle school in Tampa (clearly filmed on January 15). It just took another month to get the print version of the story out.

Several things were immediately clear.

The marshmallows were not being used to discover equivalent ways of multiplying binomials, they were being used to represent computers in a cube farm that was described in a rather long and detailed story problem (correction, they are now called Word Problems because story problems are even scarier than nuclear power) that you can see on the screen at the start of the video. [Side comment: I love how up-to-date the problem is, dealing with physically networking computers. The simplest answer to this problem is to go wireless!] That problem would be way over the heads of most college students I encounter, who act as if they had never had to analyze a word problem. My colleagues at Wannabe Flagship say the same thing, and my guess is this results from it not being on the local HS exit exam. From what I can see, the trend in "college" algebra has also been to avoid word problems as much as possible unless they fall in a very narrow class so only one 'type' shows up on an exam and can be solved out of habit rather than by analysis. (They spend the time gained from dropping this topic to teach students how to graph functions on their calculator.)

Anyway, back to the main point:
The video makes it immediately clear to me that the marshmallows are not being used for what the reporter says, but rather to help visualize the geometrical situation that has been described in words in the problem. Even better, the developers of this module recognize that there might be a vocabulary issue with the word problem and make that part of the analysis explicit. As the teacher puts it, "A lot of good practices are in Springboard". Looks that way to me. The 'manipulative' (which has the advantage of being edible) is there to illustrate why the equations work out the way they do. I really like the emphasis on collaborative learning, which is a great way for students to work on problem solving by articulating their strategies.

I see this as the math equivalent of theory and experiment in physics. We set up a problem, we solve the problem, and then we see what happens when we measure the unknown in an experiment. Incorrect approaches are exposed by the harsh light of reality. You see an example of that in the comment of one student in the video story. I can see why the math faculty have had little trouble adopting this system. In addition to changing only a fraction of their curriculum because they added selected modules to supplement their existing approach (no doubt tailored to their HS exit exam), it does what experienced teachers already do. The upside is that it helps inexperienced teachers by giving them an entire exercise with printed worksheets to guide students through this process.

The unanswered question is whether they can transfer this skill to wholly new problems. If it is just one exercise with no reinforcement, it will be lost quickly. Experienced problem solving requires more time and additional coaching. (Do they still do "coin problems"? Those can be tested with play money.) Can they use everyday objects like marshmallows to model some other problem? To identify the marshmallow with a variable like "x"? That would be great. But unlikely except in the best classes.


* Long Footnote:
I really was shocked to see the huge difference between what they described as the "national" curriculum and the college prep curriculum I experienced decades ago. Bro, if you read this, please comment on the sequence your kids followed in HS. We had a 7-9 and 10-12 system that only changed to the middle-school scheme when my 2000 capacity HS went from overcrowded with 3 classes to capacious with 4, so I don't have a clear memory of what we did in 9th grade. I can still remember the trailer where I studied 10th grade American Lit, which was followed by a year called "world" lit that started with Greek plays and ended with British literature, so we got a good mix of plays and poetry that year. Senior year was devoted to a capstone composition class that featured (in my case) a wide range of fairly contemporary short novels and short stories with an existentialist bent, with major focus on learning how to research and write a 10+ page term paper. We read Conrad, Hesse, "Clockwork Orange", etc. No movies, no "iTunes", the modern books-on-tape, but (unlike one class mentioned in the article) we could all read at the 9th grade level or above. Even one kid notable for his lack of intellectual curiosity got connected in this class.

The curriculum described in this article lacks a senior composition class and puts world lit before American lit so they have limited exposure to advanced American English before taking their "high stakes" exit test. If this is a national curriculum, it is no wonder so few students can even write a decent lab report. I think they also struggle with reading problems because they appear to have never heard of the critical reading exercise of writing a pre'cis (a short summary of a text that contains only facts, no opinions) that was common in our curriculum.

The comments on this article were also interesting. The usual suspects show up attacking "government" schools, but I have seen little evidence in our area of a major difference between non-selective private schools and the public schools. They also place into 5th grade math. The public doesn't know how they measure up because they don't have to take the state exit exam to graduate from HS, so everyone assumes they are doing a good job.