Abstract
This chapter describes and comments on the large qualitative differences between curriculum intentions and outcomes, within and across countries. It is not a meta-analysis of research on international comparisons; rather the focus is the relationship between what a government intends to happen in its society’s mathematics classrooms and what actually does. Is there a mismatch? In most countries there is. Why? This leads us into the dynamics of school systems, in a steady state and when change is intended—and, finally, to what might be done to bring classroom outcomes closer to policy intentions. Two areas are discussed in more detail: problem solving and modeling, and the roles of computer technology in mathematics classrooms.
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Notes
- 1.
My thanks to Kaye Stacey, Michel Doorman, Berinderjeet Kaur, Akihiko Takahashi and, particularly, Gabriele Kaiser, the editor of ZDM.
- 2.
This, like every other statement in this chapter, is a trend; it is not true for everyone in each group.
- 3.
Modeling, the now-standard term for the use of mathematics in tackling problems from the world outside mathematics, uses the same practices as mathematical problem solving—plus a few more.
- 4.
These examples were developed by the Shell Centre/Berkeley Mathematics Assessment Project, see http://map.mathshell.org.uk/materials/index.php. The “expert tasks” under the “Tasks” tab epitomize problem solving. Boomerangs, Figs. 2 and 3, is from a MAP formative assessment lessonlesson on problem solving.
- 5.
None of the solutions in Fig. 3 is fully correct and complete—a design choice that makes them a better stimulus for classroom discussion, because the students are put into a critiquing “teacher role”, which is more proactive than merely understanding someone else’s solution. The more sophisticated solutions are beyond most students’ problem solving at this level, but are there to show the potential of more powerful mathematics.
- 6.
Equally, until the 1950s the Geometry examinations for the highest achieving 20 % of 16 year old students included proofs of standard Euclidean theorems, each followed by a non-routine application of the theorem—an example of solving problems with a well-controled “transfer distance”.
- 7.
The authors add “However, there is little empirical data available to confirm the promise of ‘teaching with variation’ ”.
- 8.
The earlier examples, though they are related to practical problems, have been taken to this “well-posed” stage—the reason to call them problem solving not modeling.
- 9.
This contrasts with the attitude of teachers of English, who welcome the opportunity to link the technical and stylistic aspects of language with the student’s world.
- 10.
T∼tap on/off, P∼plug in/out, M∼man in/out, and S∼sings/stops singing—because it is important to recognize that there are some variables that do not affect the quantity of interest, in this case water level.
- 11.
For example, in the 1980s the National Council of Teachers of Mathematics developed “standards”, setting out curriculum goals, the National Science Foundation funded the development of curricula and assessment in the 1990s, while substantial impact in classrooms began from 2000 onwards.
- 12.
The first 4-function calculator I used cost $450 ∼ several thousand dollars in current money.
- 13.
Against the advice of the mathematics education experts, the British Government insisted on retaining fluent “long division” as an essential skill in the National Curriculum.
- 14.
As the Mathematics Working Group finished its design of the original 1989 National Curriculum in England, I asked a senior civil servant why we should expect it to happen; she replied “But it’s the law of the land”!
- 15.
Paul Black (2008) describes the process of consensus building across communities behind a successful curriculum innovation, Nuffield A-level Physics.
- 16.
The Blue and Red Boxes are still widely regarded as classics. In 2008, one of the first “Eddies”, the $10,000 prizes for excellence in educational design of the International Society for Design and Development in Education, was awarded to Malcolm Swan, its lead designer, for The Red Box. (The other went to an Editor of this book.)
- 17.
Japan, where a substantial part of the teacher’s week is spent in lesson planning and lesson study, has larger classes. In the US and UK teachers and their unions are profoundly skeptical that the trade-off would be sustained. “They’ll cut the PD again after a year without reducing the class sizes”. This exemplifies a whole set of other system issues.
- 18.
I estimate that a thorough formative evaluation of some NSF-funded curricula and some traditional comparators would require funding comparable to the original development program, roughly $100 million.
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Burkhardt, H. (2014). Curriculum Design and Systemic Change. In: Li, Y., Lappan, G. (eds) Mathematics Curriculum in School Education. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7560-2_2
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