If you are fighting a war then you need to understand which side to be on. This is hard, the maths war is complex, with wars within wars, false friends (often well intentioned), propaganda of varying quality etc.
A brief history of american K-12 mathematics education in the 20th Century by David Klein
This argues that progressivism or discovery learning or constructivism (fuzzy descriptors that are used interchangeably) has done enormous harm to maths education
- reducing academic content
- capitulating to utilitarianism
- restricting the teaching of algebra
- teaching the child and not the subject
- integrating subjects
- focusing on everyday living
- allowing children to learn what they want
- promoting freedom as profound and revolutionary ("Summerhill")
- reducing pencil and paper computation
- excessive use of calculators
- too much real world problem solving focus
- teamwork focus
- de-emphasising calculus and its prerequisites (algebra, geometry, trigonometry)
- offering a curriculum smorgasboard
- vague curriculum goals, eg. "to be a maths problem solver"
- too much learning by doing, hands on, inquiry based
- encouraging students to invent their own algorithms
- definitions and proofs gone missing
- books downgraded or missing
- promoting a false dichotomy between basic skills (bad) and conceptual understanding (good)
"Sifting through the claims and counterclaims, journalists of the 1990s tended to portray the math wars as an extended disagreement between those who wanted basic skills versus those who favored conceptual understanding of mathematics. The parents and mathematicians who criticized the NCTM aligned curricula were portrayed as proponents of basic skills, while educational administrators, professors of education, and other defenders of these programs, were portrayed as proponents of conceptual understanding, and sometimes even "higher order thinking." This dichotomy is implausible. The parents leading the opposition to the NCTM Standards, as discussed below, had considerable expertise in mathematics, generally exceeding that of the education professionals. This was even more the case of the large number of mathematicians who criticized these programs. Among them were some of the world's most distinguished mathematicians, in some cases with mathematical capabilities near the very limits of human ability. By contrast, many of the education professionals who spoke of "conceptual understanding" lacked even a rudimentary knowledge of mathematics.What I have learnt from this:
More fundamentally, the separation of conceptual understanding from basic skills in mathematics is misguided. It is not possible to teach conceptual understanding in mathematics without the supporting basic skills, and basic skills are weakened by a lack of understanding ..."
In war language is distorted as the advocates take a good idea and turn it into propaganda. Understanding will not come through repeating slogans like constructivism, either for or against. It will only come by delving more deeply into the actual curriculum and evaluating it from these points of view:
- does the author understand mathematics deeply?
- does the author understand child development, what is age appropriate?
I became confused about this myself. This article:
Basic Skills versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education by professor H. Wu.
The title is great, the author is a maths professor, his writing is open and transparent, he makes some valid criticisms of some aspects of discovery learning being taken too far, he makes a brave effort to make algorithms more understandable but falls down on the age appropriate child development criteria. I changed my initial favourable evaluation after being helped out by this comment on the squeakland list.
The evaluation process is not easy. To be good at both maths and child development takes some effort. The real danger is that you'll end on the wrong side of a fuzzy war.