In my previous post, "If Massachusetts was a Country," I discussed the reasons Massachusetts students scored well on the Trends in International Mathematics and Science Study (TIMSS) exam, intensive hands-on instruction. This approach to teaching science challenges calls for school "reform" based on scripted learning and high-stakes testing. In addition, according to TIMSS Massachusetts, Minnesota, North Carolina, and Indiana were top scorers on the eighth grade math exam and the United States as a whole scored above the international average. In this post, I examine what good math teaching looks like in the People's Republic of Massachusetts and also in the Country of California.
There is a scene in the movie Stand and Deliver where math teacher Jaime Escalante pleads with one of his Mexican-American students at Garfield High School in East Los Angeles, California to "fill the hole," to balance the equation, to recognize that the positive and the negative of a number together must always equal zero. It is a very powerful scene.
I had a similar "ah-ha" moment when I was helping my son study for a chemistry test. He was trying to memorize chemical formulas without success and I was not much help because I had the same problem with chemistry when I was in high school. Suddenly we both realized that we were looking at algebraic equations and the key to every chemical formula was balancing both sides of the equation. The only difference was that energy was a new unknown; it had to either be added to one side to make the equation work or released on the other side after the chemicals bonded into a new compound. If you add energy in the form of heat to a block of ice, you get water. If you mix carbon (C) from wood and oxygen (O2) from air in a fire (heat energy), you get the gas carbon dioxide (CO2).
I found an ally in my campaign to morph content, skills and understanding in online videos by Dan Meyers, a former high school math teacher based in the San Francisco Bay area. Meyers argues in "Math class needs a makeover" that math classes need to focus on conceptualizing and solving problems rather than memorizing what are for students meaningless formulas
I believe Meyers' ideas fit in nicely with The Algebra Project, an approached developed by Cambridge, Massachusetts-based Bob Moses, a Civil Rights activist from the 1960s, who promotes the idea that all children can master math. Because of its Civil Rights connection, Algebra Project teachers and staff are now involved in a number of schools and school districts including the Recovery School District of Louisiana in New Orleans, Halifax County, North Carolina, Jackson, Mississippi, Petersburg, Virginia, Clarendon County, South Carolina, and Edison High School in Miami Florida.
In the Algebra Project, Bob Moses lays out a five-step approach to teaching and understanding algebra. If it were up to me, I would write Moses' approach to pedagogy directly into the Common Core and mandate it, not only for mathematics instruction, but in every subject area. The five steps are:
1. Students participate in a physical experience, like a trip, where they see examples of what they are studying (e.g., arches, geometric shapes, suspension bridges).
2. Following the trip, students draw pictorial representations or construct models of what they have observed.
3. Next, they discuss and write about the event in their everyday dialect or intuitive language. Moses calls this stage "People Talk."
4. Their oral and written reports are then translated into the standard dialect or structured language as part of "Feature Talk."
5. In the last step, students develop symbolic or Algebraic representations that describe what they have learned. They present these representations in class and explore how they can be used to describe other phenomena.
I use a similar approach in middle school social studies classes. I take students on a walking tour that includes the Brooklyn Bridge that connects the New York City boroughs of Manhattan and Brooklyn. From an overlook on the Brooklyn side of the bridge they "see" how the completion of the bridge in the late 19th century made possible one integrated city. They sketch the bridge, take photographs, and walk across it. When we return to class they construct and present model suspension bridges using their images. Expert groups prepare reports on the history of the bridge, the geography of New York City, the people who constructed it, and the technology involved in creating a suspension bridge. They work in teams to assist each other in formalizing their reports and translating them into "Feature Talk." In the last step, students transform their reports into interactive electronic presentations.
The Algebra Project and the Meyers video reminded me of "math" experiences I had at a summer work camp for neighborhood teenagers from Brooklyn in the 1970s and 1980s. It illustrates the importance of real world problem-solving as part of mathematics. I was working with teams of six teenagers and our goal was to re-roof a cabin in the woods about 50 feet from the nearest road. We had to load 90 pound rolls of 3 foot by 50 foot tar paper on a flat bed truck, drive as close to the cabin as possible, carry the 90 pound rolls of tar paper to the cabin, and then carry 90 pound rolls of tar paper up the ladder to the roof. I think you get my point. These were heavy rolls of tar paper.
I asked my work team how many rolls of tar paper they thought we needed for the roofing job and suggested we could calculate the amount we needed using geometry. They did not want to be bothered. It would be easier to carry the tar paper from the road to the cabin and up the ladder to the roof than it would be to solve the problem, or so they thought.
By the time we got the rolls of tar paper to the work site they were rethinking their decision. It was hot and the tar paper was heavy. The first teen struggled getting up a few rungs of the ladder but finally made it. He called down "this is too hard. We better learn the math." We measured the roof and then calculated exactly how many three-foot wide fifty-foot long rolls of tar paper we needed for the job. It was still hard getting them up the ladder, but we only took as many as we actually needed. Real world math had triumphed over student resistance.
In a recent op-ed piece in The New York Times Sunday Review, a professor of social sciences at Northwestern University argued that the "great stagnation of American education" and poor quality of United States schools places the nation's future at risk because it "hurts our economy's capacity to grow." I think the author places the cart before the horse. Poor school performance is the result of economic stagnation and poverty, not the cause.
A close look at the TIMSS report on 8th grade mathematics reveals some interesting data that would be useful to American policy makers if they were actually interested in scientifically-based conclusions instead of just finding ways to support preconceived beliefs.
In the People's Republic of Massachusetts every ethnic group scored above the international average of 500 on the 8th grade mathematics test. The United States student average on the test was 509 and every group but one topped this score, and the other group almost matched it. Asian students in Massachusetts were the highest scores (599) followed by Whites (572), Multiracial (567), Blacks (516), and Hispanics (507). While there clearly is a racial/ethnic divide, at least students in every group are learning math.
But what was even more interesting was class divisions measured by the percentage of public school students in a school eligible for free or reduced-price lunch that is an indicator of poverty. In schools were fewer than 10% of the students were from low-income families, the average score was 584, but in schools were over 75% of the students came from low-income families the average score was only 491.
While student scores tended to be lower across the board in the Country of California, they followed a similar pattern. In schools were fewer than 10% of the students were from low-income families, the average score was 524, but in schools were over 75% of the students came from low-income families the average score was only 455.
We need to have more teachers like Jaime Escalante teaching children how to balance equations by "filling the hole," but the best way to improve math performance is to address poverty in the United States.