The @mathchat topic for Thursday October 28, sale 2010 is: “How can we facilitate vertical (cross-curricular) integration of mathematics teaching from early years to graduate-level?”

In this post I want to think about the vertical integration aspect of this question.

Below are some issues that I feel need to be addressed in a genuinely productive and transformative mathematics curriculum K-20.

1. Number-sense – a strong understanding of numerical operations – does not stop in elementary school. Number sense needs to be re-visited, there in increasingly deeper ways, help K-20.
2. Geometry – particularly the visual aspect of geometry, – needs to be integrated with algebra from the very beginnings of algebra. This is especially so for “Early Algebra”. Visual processing and syntactic processing seem to be different brain activities – integrating them would seem to lead to deeper mathematical understandings, and longer lasting memories.
3. Computation and computational devices need a coherent K-20 focus. Computation is not just about getting an answer but knowing how to reason that the answer is correct. Computation involves necessary approximations ( ); machines make approximation errors; errors propagate; analyzing errors is critical.
4. Trigonometric functions are circular functions, not triangular functions. Focusing on the unit circle from the beginning of trigonometry paves the way for deeper understanding of, and facility in, calculus.
5. Statistics runs through everything: analysis of data is basic to all mathematical activity. Going back to statistical basics – mean, mode, variance, skewness, is critical every time data – from records, experiments, or situations – is analyzed.
6. Algorithmic thinking is basic to mathematics. Applying an algorithm, understanding  it, analyzing it, finding its run time are progressively deeper engagements with basic algorithmic thinking.
7. Thinking in terms of functions, from the early years. The organizing idea of a function is foundational to modern mathematics: it needs to take precedence over formulas.
8. Inquiry driven mathematical investigations.
9. Extended project work at least as often as repetitive homework.
10. Learning lessons from, and at, all levels. Teachers of mathematics should see themselves as part of a community that includes K teachers as well as graduate advisers. Research mathematicians and university professors need to take issues of teaching and learning in the early grades seriously. Equally, elementary and secondary teachers need to learn about mathematical topics out of their comfort zone, and learn about research problems. We are all learners in one way or another. We should emphasize our common interest rather than divisions.
11. Above all, teachers of mathematics at all levels should see mathematics as a very cool, very empowering subject, that can raise young minds to new heights.

Math is a critical part of the education of all people. We can write math worksheets, offer math homework help, design math games – even cool math games, find math tutors, and set challenging math problems. But until we come together as a K-20 math community to discuss our understandings of mathematics, to voice our concerns about teaching, we will not begin to tackle the fragmented and piecemeal nature of mathematics teaching and learning.