As I read these, I have been put face-to-face with the concept of courage. That quality of courage, that truly rare and admirable quality, is one I wish I possessed in greater measure. Because in my little world of mathematics education, there are so many things to learn, and so many things that we could be doing better, and the rapid interactions between the development of new technology and the progression of culture, that the courage to face up to these new realities and to

*truly*try to change things is something that is badly needed by all involved.

So instead, I sit at tables and in meetings and at professional development workshops and discuss foolish and distracting things:

- Should all students use calculators in their classes?
- Do boys and girls learn differently? (See this article at the American Mathematical Society for a takedown of the idea that boys and girls have different abilities in math. I think the authors should get a Nobel, if they could.)
- How will our students pass the future Common Core Standards requirements (which are currently algebra 1 only, but will eventually include geometry and algebra 2) before graduating from high school?
- Related question: how can we get more of our students to understand algebra 1 at an earlier age?
- Is Khan Academy the school of the future?
- What's the best way to teach (insert favorite/least favorite mathematical topic here)?
- Why can't students understand fractions? (or decimals, or percents, or scientific notation, or....)
- Should more of our students be taking Advanced Placement courses?
- How do we teach the dyscalculic (actually, I am not 100% sure that that's a word, but oh well) student?

Because, truthfully, we know a LOT more about how people learn mathematics, how they mentally process mathematics, and why certain mathematical topics are more difficult than others than we ever have. The brain research that has been done (yes, you must immediately purchase Dr. Stanislas Dehaene's

__The Number Sense__; it explains most of it for you) over the past 15 to 20 years puts the lie to many ideas that are fashionable in education, such as Piaget's "stages" concept. Which is to say that we now know enough to rethink how we teach and at what ages we should teach particular topics - at least from the "neuro" direction.

But the neurobiology of mathematical cognition isn't the only thing a real, modern, quality, math-educational experience should consider. Another facet is that of "curriculum". That is, what should be taught? Our solution in this country, and in most others, has been for there to be a

*de facto*(if not

*de jure*) standard curriculum: algebra 1, followed by geometry, followed by algebra 2, followed by precalculus, etc. But perhaps, just perhaps, there is another way...

What if, instead of thinking of curriculum as a linear sequence, it were thought of more as the Web - with links that were logical and coherent, and centered around one "big idea", but one that a student could follow via whatever path was appropriate to their knowledge and interest? A simple example of this might look like:

Obviously, there are nodes within nodes, and lots of other possible connections I've ignored, but I hope this figure conveys the idea. The "big idea" at the center is fractions, and the number of things that can be built off of it is enormous. Students can explore all of the different connections from the center, and jump around until they find something that connects to their own interests, perspectives, or points of view.

Possible candidates for "big central ideas" are:

- Fractions
- Addition
- Multiplication
- Functions
- Congruence
- Algorithms

Finally, the remaining pieces to consider if we are going to develop a 21st century model of mathematics education, in my view, are:

- How technology will play a role (lots of people involved in this; I like reading Ihor Charischak's and Phil Wagner's brokenairplane blogs on this topic)
- How teaching will have to evolve to manage these new approaches (this is a "third-rail" kind of issue, as it will require some major rethinking of math education for teachers of all age groups, especially including teachers of pre-adolescents)

With courage, and some luck, and the organizing power that inheres within the Web, perhaps such change can occur.

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