“You are a slow learner, Winston."
"How can I help it? How can I help but see what is in front of my eyes? Two and two are four."
"Sometimes, Winston. Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once. You must try harder. It is not easy to become sane.”
-- George Orwell, 1984
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Recently schools in Oregon
are sent toolkits about teaching maths in "proper" way and to "eliminate white supremacy". I usually don't care about lefty nonsenses but this time they went too far and touched my beloved mathematics. It went so far that I can't stop myself writing a defend against it.
Macroscopically this is just another batch of money wasted onto the "political correctness" industry, where you will only find in the States. Their way of philosophy and action are quite similar to that those from the cultural revolution in China in the sixties. Even back then when rocket scientist Qian Xuesen had to lie about agricultural production, their revised mathematics curriculum is still mathematical in context, only with a political twist in it. You get questions like "in accord to instructions from Chairman Mao we seized X acres of land yesterday and Y square meters of land today. How much land did we recover from the damned landlord in total?", and yet they are still doing proper mathematics.
We need to stop the horrible wave of decontextualization, in all possible fields. It's not just only for mathematics or science, but also in history, literature, social science and so on. Today as an math educator I feel like I should break the myth of the so called equity maths.
First of all, what is mathematics? Well this is a very complicated question with diversed answers. According to Wikipedia mathematics can be treated as a symbolic logic or its dual pair as operation of a formal language (the logical or mechanical one, not the linguistic one). Or if you like constructivism then there is also one definition for intuitionists. However if we lower the bar and ask the same question for primary/secondary school level mathematics, then it will for sure be the operation based on Peano axioms on integers, which is then extended to the rational (or real) number system. From such aspect the logic induced is consistent and unique. For every question you ask within the logic they is one and only one answer. If a quadratic equation has two roots then you don't call another root be "an alternative" to your first root. The answer is to include both roots. Giving only one of them is mathematically wrong.
For the rest of the article I want to focus on the part of solving questions within the mathematical logic. That means questions that can be completely described in propositional logic. A famous example is in Laudau's book "A Course of Pure Mathematics", where he wrote in the preface that he expects the deduction of "2+2=4" to be:
2+2 = (1+1)+(1+1) = ((1+1)+1)+1 = (2+1)+1 = 3+1 = 4
which uses the definition that 1,2,3,4 are right next to each other on the sequence of positive integers and the axiom of associativity. Similarly arithmetic calculations are within mathematical logic. High school algebra are also within mathematical logic using statements like "for all x if x^2-1 = 0 then x = 1 or -1". Geometry is a bit tricky, but we can also form a first-order logic using Euclid's axioms (with parallel postulate) and so on. I stress that this is just to give a more concise definition of the part to be taught in the subject of mathematics to be discussed in the following, but not the axiomatic approach that should be taught. For kids 2+2=4 just because "two steps after two" is four and that's it, but the problem itself can be described as in mathematical logic, that it is within scope of our discussion.
The truth of a proof for a question in mathematical logic, is purely objective. This is due to the uniqueness and consistency in the logic. In a more practical level when we assess student's work, logical errors are one of the two types of mistakes (or imperfections) from students' work. The other type of mistakes would be stylish problems (presentation, key steps being skipped etc.) which are subjective. That does not change the fact that the logic or rather the core knowledge of the subject is objective. Relativism does not apply in mathematics or other sciences.
With precise definitions, a grading scheme can also be objective by exhausting possibilities. While grading remains objective under proper settings, this is by no means an attempt to promote quantity over quality -- it is just a statistic showing the estimated understanding of the students. Grades can be used solely for the purpose of improving teaching quality. This is just one of the many standardization for the sake of efficiency, which we shall cover again later.
The goal of solving a question within the mathematical logic is of course to complete the deduction. Focusing on intermediate steps does not mean the final answer is not important. In fact, the answers obtained in previous questions are often step stones for a consequent question. In such sense the answer is of no less importance than the middle step. Getting the right answer, is of course a major goal in a maths class. (And right here means correct(ortho-), and does not imply that political or social rightness is of the same correctness. Such discussion is a pure waste of everyone's time and resources.)
One very important consideration when you teach someone mathematics is called mathematical maturity. Roughly speaking, it is your exposure to different levels and fields in mathematics so that you can understand a new topic quickly without spending 10 minutes on every definition. Due to the correctness and uniqueness of mathematics it is always possible to introduce someone a highly sophisticated tool and he can do mechanics out of it without sufficient maturity as long as he can follow the logic. However his lack in maturity means that the process will take exponentially longer, and that is a waste of time. When I was year 1 in university I joined a open courseware site and took a course in functional analysis. Yes I can understand every word about the Arzela-Ascoli theorem, but I had truly zero idea what it was doing. This is because I simply did not have enough exposure in function spaces as well as point-set topology that I couldn't visualize what is going on (unlike now I can casually use that a few times in a single article).
For primary and secondary school level mathematical maturity applies as well. At a lower level this is almost equivalent to arithmetic fluency which is gained by repeated practices on calculation, hence the teaching sequence is almost linear. At high school level there are various topics that are indeed taught in sequence. While they may or may not be taught independently, these are designed in the way that previous topics would boost the exposure and maturity of the students to equip better against the next topic.
Topic sequencing, alongside with many other existing standards and practices, are based on years of professional experience (bottom-up) and research (top-down). They do not come for whatever supremacy theories, but rather efficiency in teaching (and learning). We focus on formalism and mathematical rigorousness because these are the core of mathematical maturity and the key to master those topics. We grade the assignment as to reflect the portion of question that students are able to tackle. We standardize tests because that gives us an objective, scientific and reliable way to understand the progress of the students. And yes we need objectivity, just because this is how the world works.
Talking about efficiency, there are always idealistic imaginations on how education could be at its best: teachers tailoring materials for every students, personalizing feedbacks and updating goals constantly; school preparing the best peer environment and passive resources (e.g. library) for them. Is this possible? Yes and it exists -- in the few top high schools over the world, particularly in the States and Scandinavia (note that top secondary schools in western Europe and Asia took completely approaches). They mimicked what Euclid or Socrates did in the ancient times with the best possible teachers and resources around. That comes with a cost though: the tuition fee would be up to hundreds of thousand dollars if not millions per year, and this is open to the only half-hundred geniuses every year.
Taking a step back, there are new AI technologies which automates some of the assessing steps freeing more time for the teachers to work on more inspirational stuffs. In China they took a step further by monitoring students' reaction during class as part of the assessment -- I am against such measure though, as it hinders the students' privacy.
But now...we are talking about reforms of public schools, and most of them won't have the resources to take any of these measures. Sadly this is the fact as supported by the highly valued private education sector both at the institutional level and also the individual level. Over the past 30 years, teachers have spent more and more time on out-of-the-classroom matters. It is almost impossible to squeeze out more time to boost the quality of teaching. Unless significantly more resources are being put into the sector, what we observe nowadays is basically the equilibrium point of public school teaching.
I believe most math teachers are in the classroom to teach mathematics, and I expect most students are there to learn mathematics (at least, not politics). We separated the curriculum into various subjects that teaching and learning are specialized. What we do in the classroom can be evaluated, but only in the context of mathematics. We do not need the interference from dumb politicians and their fanatics.
I tried to write the article without real criticism on the reform. But since I mentioned the cultural revolution at the beginning, I shall
quote from Andrea Widburg something that I also observed from that horrifying cultural revolution: 'the only way to achieve the left’s beloved “equity” is to lower the standards or abolish them entirely'. Think about a world where hospitals were run by youngsters without medical knowledge, and bridges build by "engineers" who had no idea about civil engineering. As scary as it sound that really happened in China, just 50 years ago. Would you like to see the same in the western world?
