Algebra 2 is often a terrible course. It is consistently at the heart of anti-math diatribes, and a painful memory for many students. I will not defend such a course. But it doesn’t have to be that way. I wrote a defense of the course and a summary of how to improve it here:
https://blog.mathed.page/2016/05/in-defense-of-algebra-2.html

I taught this better Algebra 2 for years. After taking that course, maybe half of our students were able to take Calculus with a reasonable rate of success. But I’m guessing only maybe 10% of them could solve the equations you list from the entrance exam you quote.

The emphasis on highly technical and supremely boring manipulations is terribly counterproductive, for all the reasons you list. But a humane Algebra 2, which is interesting, and keeps students’ STEM options open, is entirely possible. I know, because I taught such a course.

As for statistics as a replacement… I have an MA in math, and I’ll be honest: I don’t understand the underlying math of, say, standard deviation, or confidence interval formulas. Those things are usually taught as black-box techniques, in a way that is 100% opposite to the sort of teaching for understanding we all strive for when teaching math.

What it boils down to is that we need better teaching of high school math, whether algebra or statistics or any of the other possibilities. NCTM’s recent document (_Catalyzing Change_) launches an in-depth conversation on how to do that. It will take a little more work to sort it all out than merely dropping the Algebra 2 requirement.

You won’t fine me carrying much water for spending much time on quartics or quintics until Calculus. In my Algebra II class we use “Algebra for the Practical Man” by Thompson as our text, who walks the kids through the derivation of the quadratic and cubic formulas, and then essentially ignores quartics and quintics beyond a warning that there be dragons.

> How about with logarithms?

I will schlep a lot of water for logarithms. It provides exactly the sort of 1% busting epiphanic moment you want if you teach them well. Here’s my progression for them:

Start with the concrete. Give the kids a worksheet that asks them to computer \log_{10} of a bunch of carefully chosen numbers using a calculator. Don’t explain anything about what \log_{10} means. (I use 1, 2, 3, 4, 6, 9, 10, 12, 16, 20, 30, 40, 100, 1000, 4000, and 10,000.

Pattern time. Ask them to look for structure and patters. Usually about ten talking in small groups, then brainstorming as a class. With 100% success, the class notices that there’s some sort of additive structure to logarithms. I say nothing.

Prediction time. What’s \log_{10} of 24? Of 60? Of 200? Of 240? Eventually the students realize that they can predict the values of logarithms using the additive structure.

Generalization time. Ask the kids about the logarithm of a times b. Again, the class always works out the theorem. A decent fraction of kids have an epiphanic moment here, the sort of thing that you want to erode the 1% business.

Why is that so valuable that everyone should do it? A couple of reasons:

1. The heuristic of moving from concrete to patterns to prediction to generalization is an enormously valuable one to internalize for everyone. It’s truly general education.

2. Logarithms in particular illustrate a nice move: identifying a truth preservation that operates differently after some sort of transformation. I don’t know that you get that in the lower algebraic forms. This pays off crazy when I teach stoichiometry and can explain the mole/gram equality relationships as analogously similar to logarithms.

3. Logarithms offer a lovely opportunity to talk about the relationships between truth-preserving rule sets. At first they seem arbitrary, but they flow necessarily from the properties of exponents, which themselves flow from the properties of multiplication, which themselves flow from the properties of addition. Exposing students to a concrete example of how complex rulesets can be related to one another and have important implications for one another is general education.

4. Even if logarithms don’t necessarily introduce new ideas about truth-preserving transformations (I think they do), practicing the internalization and application of those transformations is also general education, especially as they become more complex. How much of our adult life is soaking up some new rule set, figuring out how to navigate with it, and then undertaking projects that rely on it? Almost all of it.

There’s a guy whose work I’d love for you to read some time – Ravi Jain. There are scattered fragments available online, but when his book is out, I’d love to hear what you think of it. He presents well what I present haltingly and inarticulately.

William, I like the critical approach to change you describe in the first paragraph.

However:

I’d propose that what unifies intermediate algebra is the idea of a truth-preserving transformation.

IA offers students lots of practice applying transformations to lots of different kinds of statements of equality. But those ideas are introduced in beginning algebra and further developed with planar transformations in geometry.

In fact, I’m struggling to think of a new idea the course introduces about truth-preserving transformation. It just asks, “Hey, how about you try those ideas out on polynomials with degree five? How about with logarithms?” Those are valuable experiences, but only for students who elect into them IMO.

I’d propose that what unifies intermediate algebra is the idea of a truth-preserving transformation. That we can take statements and manipulate them in certain ways that are guaranteed not to change whether the statements are true or false. Intermediate algebra introduces kids to the idea that such transformations are possible, that abstract rules for the can (and must!) be learned and applied with precision. If offers them an opportunity to learn to communicate that transformation with concision and clarity as well. All of these are immensely valuable parts of general education, doubly so if we think of education as the process of elevating the soul to contemplate truth and not just get a job.

Would replacing that instruction with a survey of statistics have the same implicit effects? I suspect not. If it would have the same implicit effects, we’d expect that students who have been unable to master the transformational rules of intermediate algebra would be unable to master the transformational rules of statistics. We don’t see that, so by modus tollens, perhaps what’s going on in statistics doesn’t exercise the same abstract reasoning muscles.

If we wanted to replace intermediate algebra, I suspect we’d be best off replacing it with a deep dive into formal logic. A semester of the categorical syllogism backed up by a semester of propositional logic (and for the precocious kids, digital logic). That gets you a complex, abstract, rigorous set of truth-preserving transformations that students can master. Unlike intermediate algebra, it doesn’t rely on successful mastery of the rules of arithmetic, so some kids don’t start behind the eight ball.

Students learn Algebra from their teacher; outside influences don’t influence what a student knows. Were we to replace Algebra with some other logical system which is considered more immediately applicable, then knowledge of other areas or media or friends and family would give some students advantages over others.

I imagine that the Calculus requirement came from a place of expecting students to demonstrate conquering a course sequence decided at the time when a course sequence in Statistics wasn’t invented yet; what would that sequence look like, and would it end up with the same complaints?

BTW- I do enjoy teaching this course and the majority of my on-level students thank me at the end of the year.