The standard middle school / high school course sequence in much of the US is Pre-Algebra, Algebra 1, Geometry, Algebra 2, Precalculus. This is pretty much a US-only concept: elsewhere, algebra and geometry are taught every year in middle school and high school. This American tradition leads to many endemic problems:
- The traditional Algebra 1 includes an enormous amount of highly technical material, with no regard for age-appropriateness or pedagogical insight. This is true whether it is taught to ninth graders or eight graders. As a result, that course has a high failure rate. (See my many articles on this subject.)
- A year of Geometry between Algebra 1 and Algebra 2 allows students to forget much of what they learned, forcing Algebra 2 to include too much material that is essentially a repeat of Algebra 1. One way to deal with that is to include some algebra review in the Geometry course, but that comes with a cost: the reduction of geometry content. (Alas, this is a long term trend in US education. See my Defense of Geometry.)
- The traditional Algebra 2, again, is highly technical and supremely boring, making it a favorite target of those who would reduce the amount of actual math taught in high school. (See my Defense of Algebra 2.)
In acknowledgment of some of these issues, and perhaps others, the Common Core State Standards proposed an integrated path through high school math. However, altogether abolishing the traditional Algebra-Geometry-Algebra “sandwich” was never on the table, given how difficult it is to make any changes to high school math in this country.
As department chair, well before Common Core, I tried to address these concerns. Given that I never went to an American high school as a student, I was not particularly attached to the traditional sequence. And I was lucky: my colleagues and administrators were open to new ideas. Here are some of the lessons from our gradual transition to a somewhat integrated program.
- We creatively called the courses Math 1, Math 2, and Math 3.
- We did not go for extreme integration. A program can be so integrated that you cannot unravel the threads, which would be disastrous for faculty buy-in and clarity of organization. Instead, we just moved specific units up or down the grades, based on experience.
- The change was based on actually observing what happened with actual students, as outlined above: some topics were too hard for 9th grade, some topics were made easier by the advent of technology, the Algebra 2 books were too hard for some kids and too easy for others, and so on… Those issues are well-known to high school math teachers everywhere and not special to our school, but I can assure you that integrating the sequence really helped.
- Specifically, we moved the end-of-book Algebra 1 topics to later courses. Solving systems to Math 2, and most quadratic stuff to Math 3. This turned our Math 1 into a very accessible bridge to high school math, which was skipped by the stronger students, but was almost life-changing to the others. The course focused on how to work in a group, how to use tech tools, and all the foundational algebra topics (distributing, factoring, solving, graphing). Having time allowed us to go for deep understanding rather than superficial proficiency. Math 2 and beyond were untracked, and the fact that some of the less prepared kids had had our Math 1 was key to making that work.
- Having some algebra units in Math 2 allowed us to focus the geometry on geometry, not shallow algebra problems about supplementary angles, etc.
- Teaching all of trig in one semester to 11th graders had been a disaster. Kids could do it, but they forgot everything as soon as the course was over. In the integrated system, we did a little trig every year, which worked really well, because each time we got to review the previous year’s content in homework.
- Integrating also allowed us to make good use of our schedule, which involved long periods. (70 minutes or longer.) On a typical day we could do some new geometric stuff for part of the time, and review some algebra stuff for the rest of the time. Or vice versa. Or insert trig instead of one of those. This is a huge answer to the fear that kids can’t do math in a long period, that they burn out after 45 minutes. It’s just not true if you have two units going at any one time.
I hope these ideas are useful to any of you who are considering moving to an integrated sequence. See also my more general article on Big-Picture Planning, as its content complements this blog post.