To be fair the current driving trend in mathematics education is largely an extension of an existing trend in education generally. The idea is that we need to cater more to the students to better engage them in the material. There is a focus on making things fun, on discovery, on group work, and on making things "relevant to the student". These are often noble goals, and it is something that, in the past, education schemes have often lacked. There is definitely such a thing as "too much of a good thing" with regard to these aims, and as far as I can tell that point was passed some time ago in the case of mathematics.
A couple of prime examples, in terms of textbooks and material for instructors, are brought up and suitably lampooned in a YouTube video by a Washington state weather presenter who encountered, and was appalled by, these particular teaching programs. The material in question is the TERC Investigations "Investigations in Number, Data, and Space", and the University of Chicago School Mathematics Project "Everyday Mathematics". The focus of the YouTube video is on these math programs complete aversion to teaching students the classic methods for performing multi-digit multiplication and division. Indeed, these programs not only fail to teach such a method, they go so far as to actively discourage the method ever being taught, preferring that students didn't learn it outside class either. What sort of methods do they teach? Well, for example, to solve the problem 26×31, a student might use the following approach: we can write 26×31 as 20×31 + 5×31 + 1×31 since 20+5+1=26; Now we know that 10×31=310, and 20×31 should be twice that (620) and 5×31 should be half that (155); so the solution is 620+155+31=806. Note that the student could break the problem up differently, and thus there is no single approach that consistently works on all problems; each new multiplication is an entirely new problem. To be fair the methods they do teach, such as the above, are interesting, and I myself tend to use them (or variations thereon) for quick mental calculation. My complaint is not so much to the methods taught, but to the failure to first provide a solid grounding in traditional systematic algorithms for performing multiplication and division. Indeed, in my view, the real problems run much deeper than this particular symptom.
At this point I should perhaps provide a little background as to who I am to complain. I am a mathematician, currently completing my Ph.D. in mathematics. My interest in math is mostly pure math and philosophy of math, but extends to math education and popular mathematics. I've been a TA for many years and have plenty of experience dealing with students. And I am not alone in my concerns with the current direction of math syllabuses, plenty of other professional mathematicians who actually look into the syllabus are taking issue too.
So what do mathematicians see as the problem? I would say that it is, in essence, that the individuals writing these new math programs have lost sight of the core skills that early math education should be instilling. In the drive to make the material "relevant to the student", what is being taught has become too applied. In the new programs there is a a focus, almost to the point of exclusivity, on teaching mathematics via real world stories using pictures, blocks, etc. Indeed arithmetic is done using blocks, and fractions and fraction arithmetic using "fraction strips". While such props and aids are useful in motivating the mathematics, it should be just a beginning. A key skill in mathematics, if not the key skill, is abstraction: the ability to abstract away from real world objects, and manipulate these abstractions to draw deep results, is vital. Abstraction is fundamental to mathematics; it is what gives mathematics both its power and its scope; it is the mechanism by which higher mathematics is built upon elementary mathematics. Abstraction and abstract thinking is one of the core skills that mathematics education should be imparting - and yet it is completely ignored by these math syllabuses.
Equally, in an effort to nurture students and foster creativity there is an effort to eliminate rote learning, and emphasize that there may be many ways to arrive at a solution, and letting the students invent their own procedures. Often these invented procedures are very problem specific - they may work for the particular problem at hand, but fail to generalize to other cases. Ultimately this, combined with the very visual (as opposed to symbolic) approach results in the students having limited exposure to consistent, systematic, algorithmic approaches. Again, a core skill that mathematics education should imbue, logical structured thought and a systematic approach to dealing with abstract objects, is being ignored. This is particularly poor in light of the ever increasing importance of skills in algorithms and computation brought about by the needs of modern computers.
The real tragedy is that, because mathematics is a heavily layered subject, each new topic building upon the previous ones, once students fall behind catching up can be a nightmare. Indeed, students often meet a rude awakening in late high school or at college when their limited mathematical repertoire fails to provide the necessary tools to fully grasp the next topic. Even worse, by failing to impart the core skills of abstraction, and logical systematic approaches to dealing with abstract objects, we are denying students the very skills necessary to even begin to expand their mathematical toolkit. At its heart mathematics is about abstract and logical thought, and without these core skills no student can hope to succeed in mathematics.
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