We were great fans of how EPGY (http://epgy.stanford.edu) did their K-6 curriculum.

It used multiple, parallel strands to try to teach similar principles. From a very early age it would combine set theory, logic and geometry with more traditional arithmetic.

Set theory made a great deal of sense IMO because a very young child could grasp "the union of two sets" of toys and use that to understand the more abstract but commonly taught x+y=z

Its been a number of years but I seem to remember them teaching a triangle as the union of 3 points on a plane and other interlinked ideas.

Being online, the program allowed students to progress more rapidly in one strand than another if one mode of learning was a better fit (although they were eventually encouraged to catch up across all the strands).

EPGY did assign grade levels but they were easily ignored and the progression from K through "sixth grade" was very fluid. Even when we encouraged older elementary school students to take the class we would suggest that start at the K level. In part this was because most traditionally taught second or third graders would be ignorant of logic and set terminology but also because the K-2 sequency could be completed in just a matter of weeks and treated as a game.

Before each lesson (which by default ends after about 20 minutes) there is the option to play a speed based game that roughly corresponds to the topics covered in the lessons. This was considered fun by my boys although it did reward speed a bit over knowledge.

I know the EPGY math sequence has its detractors but we found it to be a great experience for both of our boys.

Was it nonlinear, or was it just linear through many different topics? It's been a while since I looked at EPGY, and while it did hit a lot of topics, it did so in a linear fashion.

What I mean by nonlinear is that the student has many choices about how to progress through the material, and there's not a clear default "do this then do this then do this then do this". Basically, I'd love to have a system from which we as educators could learn about ideal sequencing for different types of learners. I don't see anyone doing that, but maybe there's a good reason no one is doing it. It may simply be very, very hard to implement well.

Maybe EPGY is best described as "parallel and linear." Each of the strands is linear, but the software does adapt to the student's invidual strengths and weakness.

A similar model seems to have been adopted by the ALEKS folks (http://www.aleks.com)...yes, we have written a lot of checks!

They use a pie chart interface for each grade level, which more than EPGY tries to explicitly align to school standards and grade definitions. It is still ultimately linear but giving the student some choice about whether to tackle geometry or algebra in a given session was a positive.

Perhaps one of the problems with a purely non-linear approach, I'm ashamed to say, is that we check writing parents are looking for something to signal whether or not our students are making "progress." Perhaps a system that finds a balance between linear progression and student autonomy is the best we can achieve.

Btw, the video at the end of the following blog post has some interesting things to say about how we are motivated. The basic premise is that the same techniques that make well designed games impossible to resist will be co-opted to motivate us as consumers, workers and students. I think you have already applied many of these lessons effectively with ALCUMUS and FTW.

http://gigaom.com/2010/03/19/why-everything-is-becoming-a-game/

That's basically the paradigm I have in mind -- to capture some of the aspects of games that make them sticky and use them in education.

When you say "parallel and linear", does that mean that it is easy for students to switch from strand to strand, or do you have a path laid out for you in which the students do X from one strand, Y from another strand, then Z from a third strand, then back to the first strand, and so on?

Yes, I think that is basically how these two programs work out. Ultimately the student is meant to finish all of the strands but has some discretion in which order they focus.

Where this might work is with a concept that could be understand through geometry, set theory and algebra. If a student could understand it best visually they might start with the geometry strand. With some understanding of the concept they would be better able to understand it in set theory or algebra assuming those are the more difficult subjects.

Its been a long time since my kids did either of these programs so my memory may be hazy.

This does remind me of a Richard Feynman anecdote. When he was asked for his secret of being such a great teacher he admitted he had none and believed that each student was so different that no technique would work for everyone. Therefore in any given lecture he would use a variety of techniques in the hopes of connecting with each student's preferred mode of learning.

I'm sure AOPS can go well beyond this parallel strand technique.

The best example of non-linear math teaching for younger kids that I can think of is the Montessori school (3-6 year olds) my older son attended his kindergarten year. The classroom contained a number of math materials each consisting of a self-contained activity that the teacher would introduce. (There were non-math materials as well.)

It would take a while to describe the materials but they were various sorts of activities or puzzles that taught something specific such as place value or skip counting or squaring and cubing and that had some sort of self-correction built in. If the child made a mistake things wouldn't come out right, or else the child was able to check his/her answers on the spot. Most were hands-on (appropriate for the age) but there were also pencil and paper exercises such as adding and subtracting four-digit numbers.

The children could use any material that the teacher had introduced and could observe other children and ask to learn additional materials. In my son's case, at first the teachers were concerned he wasn't doing enough math; then a few months later they were raiding the elementary classes for math stuff for him and a few other kids.

I'm very interested in what you're doing as I haven't been able to reproduce the experience for his homeschooled younger brother. The homeschool programs that purport to be "Montessori-inspired" are some of the worst for being overly-scripted and linear (even if they entail multiple topics).

The structure of Alcumus, with instructional videos of some sort and puzzle-like or game-like problems in areas of the child's choice, might be a good model for what you're describing in a computer-based format. A parent-accessible record of topics worked on by child/level of mastery might be reassuring, too.

(Are you familiar with Descartes' Cove? I've only heard about it -- but I think it's a set of computer games that teach math. CTY sells it.)

I'm only somewhat familiar with Descartes' Cove, having seen an early version of it several years ago. I don't think I had enough exposure of it to fully appreciate it, but I'm (very vaguely) imagining something much more flexible and something that uses technology as more than just a visually engaging medium. But I may not have a full appreciation of what Descartes' Cove is.

I mainly think that we have a lot more flexibility with the material we want younger kids to master -- the mathematics at that age is much more natural than the math we teach later. That is, it's much easier to place the concepts in puzzle contexts, or in natural contexts so that the students aren't confronted with tons of palpably synthetic problems.

We have Descartes' Cove. If I remember correctly it has 6 discs each representing a different island that focuses on a different mathematical subject. It is non-linear in the sense that I think you can skip from disc to disc but once there I think it was a pretty linear progression to complete all the puzzles. It wasn't a favorite of my boys.

What didn't they like about it?

I'll have to ask them, but I think it set the expectation of being more of a "game" but ended up consisting more of static screens and drill questions. My guess is that they were modelling after the old game Myst.

Tough to impress the kids once they have played WoW

Actually Jacob had some thoughts for an MMORPG type of platform for teaching math that would have different disciplines represented by different interlinked lands / zones. Just thought you might be interested in how a product of the 21st Century thinks about a "game environment."

That's one of the fears I have about trying to make an educational video game -- we could never touch the "real thing".

I suspect Jacob's ideas are pretty similar to some of the things I've been thinking about. Making the game *social* is a challenge. (A real pipe dream is that of a virtual economy, but that's a story for another day.)

When my son was little we would do math on walks. We had a game called "dinosaur olympics". He chose a dinosaur to play, he chose an operation and he chose a difficulty level (easy, medium or hard). He chose his least favorite dinosaurs to do the more challenging problems (just in case they got them wrong).

There is a very cool early math program out there called dreambox math (K-2). It's not exactly non linear but still pretty worth looking at. It does incorporate a lot of choice.

I have an idea.

Involve websites like this.


Academicearth.org

and mit open courseware

and, look at some material from youtube edu.

You can learn a lot by just bouncing around from math topic, to history, philosophy, comp sci. I think sites like these will be involved for smart students in the future.