Spinmeister, Dept.

San Francisco’s Unified School District decided to eliminate access to algebra for 8th graders even if a student is qualified to take such a course.  The latest article to justify the action is one written by Jo Boaler (whose self-styled approach to math education in my opinion and the opinion of many others in education who I respect has been ineffective and damaging) and Alan Schoenfeld, a math professor from UC Berkeley whose stance is consistent with math reformers. I.e., “understanding” takes precedence over procedure, among other things.

The article states:

“The Common Core State Standards raised the level and rigor of eighth-grade mathematics to include Algebra 1 content as well as geometry and statistical topics previously taught in high school.”

This is not true. A high school level course includes rational expressions (i.e., algebraic fractions), polynomial division, factoring, quadratic equations, and direct and inverse variation. The 8th grade standards do not include these. I teach an 8th grade math class as well as high school algebra for 8th graders. The latter is far more inclusive. Elimination of access to algebra in 8th grade is certainly not strengthening math ability for those students who are qualified to take such a course.

The article also states:

They  (i.e., San Francisco USD) found a unique balance that is now seen as a national model. They decided to challenge students earlier with depth and rigor in middle school. All students in the district take Common Core Math 6, 7 and 8, a robust foundation that allows them to be more successful in advanced math courses in high school.  The key is conceptually rich courses that benefit everybody, including those who go on to STEM majors in college. In-depth instruction helps all students and provides a more solid base for later math courses. All students get a solid foundation, and acceleration is offered in the 11th and 12th grades.

Translation:  For those students who wish to take calculus in 12th grade, they can double up math courses in 11th grade, so they can take Algebra 2 and Precalculus.  As far as what they mean by “conceptually rich courses that benefit everybody”, it’s anybody’s guess. I work with the textbooks that adhere to the CC standards for 6th, 7th and 8th grades.  I supplement freely with a pre-algebra book by Dolciani written in the 70’s and other materials.  The emphasis on ratio and proportion in 7th and 8th grades is rather drawn out and can be done more concisely, rather than harping on what a direct variation and proportional relationship is. Traditional Algebra 1 courses present direct variation in a much more understandable way, rather than the “beating around the bush” technique that defines such relationships as straight line functions that go through the origin, and whose slope equals the “constant of variation/proportionality”.

So much time is spent on trying to make the “connection” between slope, unit rate, rate of change and constant of variation, that students think they are all different things and are largely confused.  While Boaler and Schoenfeld may say that the confusion arises because teachers don’t know how to teach it, I assure you I know how to teach it. I use an algebraic approach in an algebra class, when students have the algebraic tools with which to grasp the concept more easily.

But the real goal of San Francisco’s elimination of algebra in 8th grade is to close the achievement gap as evidenced by the last paragraph in the article:

Groups that traditionally underachieve — for example, students of color, female students, students of low socioeconomic status, bilingual students and students with special needs — have all experienced increases in achievement. We congratulate the district for its wisdom in building course sequences that serve all students increasingly well.

For those students whose parents can afford it, they take algebra elsewhere in 8th grade and circumvent the system.  Those whose parents cannot afford outside help are stuck with what Boaler and Schoenfeld, and the SFUSD think is equity for all.


Scaffolding Dept.

The late Grant Wiggins was adamant about “authentic problems” and “authentic problem solving”. He felt that scaffolding problems was a cheat and that it short-circuited understanding. That is not the case.

In solving word problems, worked examples provide students a direct access to solving problems that are similar, and in the same category. By scaffolding such problems–that is, varying the problems slightly beyond the initial worked example–students are forced to stretch and to make connections.

Students do best with explicit instruction, starting with simple problems. They then begin to develop the knowledge and skills to solve increasingly more difficult problems with novel twists. Without explicit instruction in problem solving, many just give up and don’t try the problems. Students benefit by seeing how to think about the problem before actually working it. Imitation of procedure therefore becomes one of imitation of thinking.

While people may criticize this as mere imitation and rote learning it is not. As anyone knows who has learned a skill through initial imitation of specific techniques such as drawing, bowling, swimming, dancing and the like, watching someone doing something and doing it yourself are two extremely different things. What appears easy often is difficult–at first. So too with math.  Imitation of thinking is a level of understanding as one goes up the scale from novice to expert.

For example, students may be shown how to solve this type of problem: Two trains, 360 miles apart, head toward each other, one going at 100 mph and the other at 80 mph. How long will it take them to meet? The student can be shown that the sum of the two distances represented by 100t and 80t (where t is the time traveled by each train) makes up the initial 360 miles. A variation of this problem is: After the trains pass each other, how long will it take for them to be 90 miles apart? In this case, the same concept is at work: the sum of the two distances represented by 100t and 80t makes up the future distance of 90 miles.

In the words of Dylan Wiliam (Emeritus Professor of Educational Assessment at the University College of London Institute of Education): “For novices, worked examples are more helpful than problem-solving even if your goal is problem-solving”


Previously, about Common Core

I wrote a series of articles for Heartlander, that over the years have changed URL locations. My dedicated readers (as well as those who intensely dislike what I have to say) therefore cannot find what was once readily accessible.

For those of you who wish to revisit what I have said about Common Core, the articles are here:

And for those of you who missed it, my talk on math ed in the US is here, with my comments on Common Core starting at minute 19:24.