Fantasy High School

@carloliwitter was tweeting recently about re-imagining High School mathematics and asking for ideas. The topic has caught my fancy even though its a bit off message for the blog.  High School is ripe territory for change while at the same time rife with difficulties for anyone attempting to do so. I’ll start with a model that I wish I could have had myself:  proof school.  The high schoolers there do the following:

2 math blocks every afternoon covering a single subject over 6 weeks including:

• Problem Solving + Combinatorics 
• Algebra + Number Systems
• Geometry + Topology 
• Analysis + Statistics
• Number Theory + Computer Science

The course descriptions sound interesting and they deep dive into number theory, abstract algebra, more advanced Geometry topics etc.

Note: Calculus is not covered.

Ok so that’s where my heart’s at but I don’t think that within a public school setting you could achieve the same thing and the more general challenge is providing a interesting and relevant path for everyone in the system.

So here’s some more ideas about changes with the caveat that these are only my impressions and desires and don’t reflect a systematic view of the state of the world.

• Reinvigorate Geometry. There’s enough material to split into 2 years on its own. De-emphasize all the taxonomy, volumes and surface areas. Instead elevate a proof and computer aided exploration approach. I’d particularly like to see more time to get to cyclic quadrilaterals, powers of a point, Ceva’s theorem etc.  That only seems possible if you stretch the course over more than 1 year.   Then its easy to branch out to circle inversions, tessellations etc.
• Explode Algebra II.  The narrative structure isn’t there and worse it often ends up being a significant  rehash of Algebra I.   What’s core here and what can we add in to make this connect more clearly to future topics? I don’t think touring absolute value, exponential functions,  polynomials etc. is working very well.  What’s worse is that pre-calc often bleeds into this material.
• Integrate programming more fully as first class component. Maybe the capstone of H.S. should be based around numerical computing.
• Stop doing stats as an add-on over several years. Make this a first class subject and perhaps a joint responsibility of the Science and Math departments. 
• Create more “labs” i.e. exploration opportunities.
• Have an applied pathway like Canada. Let’s figure out for real what we really need and no more. So Mathematics is not a barrier to those who aren’t going into STEM.
• More overall arc to the pathway. Students should have more of an idea of what they are heading towards and what the overall field looks like as well as some of the history of how it developed.

On the flip side what are the systemic issues making this hard to do:

• Choice models. Any model that envisions multiple pathways runs into several immediate concerns. First is that finding teachers to teach all these new subjects well is hard and anytime you have to schedule more classes it makes the logistical process more difficult. Its always going to be easier to schedule math where everyone does the same thing each year so all the teachers and period slots are more easily interchangeable.  Secondly, these models work best at large schools where there is a sufficient number of students to actually support them.  Small schools especially in low population areas may never be able to pull this off. Finally, status issues among classes easily subvert the aim of offering them. If one pathway is perceived to be more useful to getting into college then any such system will devolve into a high/low pathway with all the more ambitious students making the same choices. See: Calculus.
• Integration with College. Any such system has to not impede admission chances and that creates tremendous inertia as well as concern among students and parents.
• Tensions over tracking.   This feeds into my previous point about choice. More radical anti-tracking ideas mean you really cannot provide anything but a single pathway because one of them is likely to be come the de facto “high” pathway if left to develop naturally. 
• The realities of the wide difference in relative performance by high school. Kids come into ninth grade at vastly different places within the curriculum and their own personal journey. The system needs to move everyone forward but finding a way to do this is really hard. Most of the solutions so far involve slowing down significantly and basically hanging out in the pre-algebra and then algebra space for much longer than we currently do.  We really need a way to remove math as a gateway to everything post 12th grade. Because systemically we know that isn’t working.

[I deliberately didn’t really tackle Calculus’s place in the curriculum here. That deserves a followup.]