In early December, I attended the California Math Council Northern Section conference in Asilomar, as I’ve done almost every year since the mid-1980’s. In my last post, I discussed my session on fractions. For various reasons, I only attended two other sessions.
The first was by Eric Muller, who works at the Exploratorium, San Francisco’s amazing science museum. He presented an extraordinarily entertaining middle school activity about air pressure. You can read all about it here and here.
The second was Gail Burrill’s excellent talk on “What belongs in Algebra 2”. This is a crucially important topic in an age where this essential course is under constant attack in the mass media, and even from some in our profession. I have discussed this several times on this blog:
Those who oppose the course have three main arguments. One centers on the claim that the course inherently undermines equity and pushes marginalized populations out of math altogether. Another is that data science should replace the Algebra 2 / Precalculus / Calculus sequence. I answered those points in the above-linked posts, but I encourage you to also read this interview with Adrian Mims. He is the founder and leader of the Calculus Project, a very effective program to support Black, Hispanic, and low-income students in advanced math high school courses.
The third argument against Algebra 2, unlike the first two, has some validity: it is aimed at the traditional version of the course, one which ignores the existence of technology and prioritizes mindless manipulation of symbols at the exclusion of meaning. Burrill’s presentation offered a clear alternative to that — an approach based on sound pedagogy, solid mathematics, and close attention to classroom and societal realities. I will summarize some of her main points, and link to relevant material on my blog and website.
Instead of a huge list of disparate topics, Burrill emphasizes the importance of a conceptual understanding of functions, which is best achieved by experiencing functions in multiple contexts and in multiple representations:
Playing “what’s my rule”
Geometric transformations
Describing a function using English words
Mapping one set into another using arrow diagrams
Ordered pairs
Cartesian graphs
(I would add function diagrams to that list.)
This is solidly in the professional consensus: Burrill referenced NCTM’s Catalyzing Change in High School Mathematics and the new AP Precalculus “Math Practices” as providing useful guidelines. Unfortunately, the professional consensus has not reached the textbook publishers — and thus many, many classrooms still deliver the stone age version of the course.
I was thrilled that Burrill omitted rational functions and logs from her list of what belongs in Algebra 2. This was largely what I did when I was a department chair. We removed rational expressions from Algebra 1, and rational functions from Algebra 2. For most students, those are inaccessible and a turn-off. Removing them did not seem to undermine our students’ preparation for standardized tests — quite the opposite, because we were able to spend more time on necessary foundational topics. Rational functions are much more effectively taught in Precalculus, where they are actually needed in preparation for Calculus. As for logs, we limited ourselves to base 10 logs in Algebra 2, using a very accessible approach. As recommended by Burrill, we saved a full-on introduction to logarithms for Precalculus.
Burrill also acknowledged that in the current technological environment it is absurd to spend a lot of instructional and practice time on factoring trinomials. Those can be factored at the press of a key. What remains important is factoring, as a concept. How is it related to distributing? How is it connected to the graph of the function? Can a given binomial be a factor of a given higher degree polynomial? (I would add the geometric representation of factoring using manipulatives. I wrote about this here.)
She also suggested challenging ChatGPT to some non-straightforward questions, and assessing the validity of its response. The example she gave was “which of these functions could yield a parabola opening upwards with positive x-intercepts?” As it turns out the AI was only right some of the time.
For symbol manipulation, Burrill correctly emphasized the basics:
For rules of exponents: “when in doubt, write it out” — e.g. x^2·x^3 = (x·x)·(x·x·x)
For the square of a sum vs. the sum of the squares: graph each and discuss (again, I would suggest including the area model in that discussion)
For rate of change: draw slope triangles, analyze successive differences
Discuss notational subtleties — e.g. f vs. f(x)
Do not waste time and energy on “proving” or memorizing arcane trig identities — put the focus on the unit circle and basic identities
She also prioritizes recognizing functions from their graphs and tables, and interpreting real world graphs and data, a basic ingredient of mathematical modeling. I completely support that for Algebra 2 and in fact for Algebra 1 and Precalculus. Beyond that, she presented a huge amount of basic statistical and quantitative literacy content to complement the work on functions. To her credit, she made clear this was in addition, not instead of the ideas she presented in the first half of her talk. I very much agreed with her points about this, but I don’t think they can all fit in Algebra 2.
In my view, doing all the needed work on functions well cannot be rushed, and there are topics such as sequences and series and complex numbers which I believe can and should still be part of Algebra 2. As department chair, to make sure our students had a chance to really understand crucial Algebra 2 concepts, I created a whole separate course on mathematical modeling, quantitative literacy, and basic probability and statistics. It was by far the most popular elective we offered. (Also, to be honest, I think a lot more of the quantitative literacy / basic data science content should be part of the science and social studies curriculum. Alas, that may not be happening any time soon.)
In any case, if you get a chance to attend a talk by Gail Burrill, you are sure to learn something and have plenty to think about!
— Henri
