Mathematics
Introduction written circa 21:13:52
I'm going to say something. I'm going to say many things. This is a workshop. This is a workspace. I'm going to make statements. They may not be final. This is a workshop. I'm going to say something and I'm going to tinker, I'm going to fiddle, I'm going to refine. But I need to start. I'm going to say some things and you may not agree. But run with me for a bit while I tinker, fiddle, and refine. I'm in workshop. Run with me while I figure it out. You may not agree? I may not even agree in 5 minutes, but we'll refine. Run with me while I workshop, while I think thoughts, and figure it out. [21:20:55] I'm going to speak and I'm going to move. Move with me. [21:22:55]
18:32:46 4 Nov 2024
Mathematics is arguably the most important class in education, if not one of, obviously. While I say obviously, I also note how common the sentiment "I didn't need any of that after high school" is. Mathematics is not important in itself alone, it is important because of what it forces you to do. Few other classes or streams of discipline in U.S. curriculums require the degree of fluid reasoning seen in mathematical classes. Fluid reasoning is strengthened almost nowhere else in education, and this kind of thinking, this kind of critical thinking is critical.
Even if you never do another calculation again, if you actually engage with mathematical thinking, that engagement may be one of the most important things you ever do in education across your lifetime. It is not about math. It is about thought. It is about how you think and teaching you to think. It is about fluid reasoning.
The idea "you'll never need this again" reflects a hyperfocus on crystallised intelligence and an undermining of the value of fluid reasoning. It reflects a lack of awareness about where fluid reasoning touches your life and how it is applied. It may also reflect difficulty with applying reasoning abilities across domains, meaning, even if you did noticeably make use of and strengthen your fluid reasoning in mathematics, you do not regularly apply this strengthened reasoning ability to non-mathematical domains and subjects, and thus do not see its full use in your life.
I have previously said Educators who give exams on topics, questions, or skills not taught or previously prepared for in class, but which they intend to have their students reason through based on fundamental skills, are testing fluid reasoning ability and testing for fluid intelligence. I have said They are likely inadvertently administering partial IQ tests in some sense. I said This practice should be re-evaluated if IQ testing is not the intended activity for a classroom's setting. All of this still stands.
I value thought and reasoning in education. I believe it can be taught and strengthened. I believe it should be taught. I believe in testing, the objective is generally to evaluate a student's growing understanding of the curriculum, of what they're taught. Consequently, it may be a bit strange to evaluate fluid reasoning in a room where it's not taught.
Fluid reasoning can actively be worked into instruction, but it's not often that it is. What sometimes occurs is a series of information transfers, casual tests of crystallised knowledge, discussion of crystallised knowledge, hyperfocus on crystallised knowledge, emphasis on crystallised knowledge, review of what was taught by the educator as it was taught (again crystallised knowledge), and then, a formal examination that requires fluid reasoning and references the student's crystallised knowledge minimally or in a manner that is disproportionate to how it was reviewed.
I note that although I mentioned mathematical classes are prime opportunities for strengthening fluid reasoning, this practice or behaviour I've described [of prioritising crystal and examining fluid] often occurs in mathematical courses, more than others. I believe it's because, pretty clearly, independent understanding of mathematics requires fluidity to some degree, so educators do not feel they can adequately examine your true understanding of math without testing fluid reasoning. I'm sure many feel it's too easy if they do not.
I don't know, it seems a lot of people are threatened by ease. There is a kind of fixation on struggle, harm, isn't there. How do we get through this? How do we tell people it's okay to allow someone ease.
It's a projection of a lack of self-confidence, in some sense, as many things are. The idea that all must suffer beyond reason, with "beyond reason" referring to events such as heartbreak and grief. Mandating unnecessary difficulty and struggle, even creating it when it doesn't exist, suggests an inability to fully envision a world where people are okay and they're happy. Happiness is almost threatening, I guess. It suggests a personal experience with suffering, in fact, it suggests more than one, and it suggests an inability or refusal to see other options. It suggests an inability to see a way forward without the suffering of others or even oneself. It suggests a lack of imagination. A lack of creativity.
The amusing thing about this is, as I've recently realised, this idea of suffering and harm, difficulty in educating is a reflection of limited fluid reasoning, in some sense. Believing it is not education if you do not struggle. It is not a sufficiently rigorous education if you do not suffer awkward systems or educational practices founded in disjointed sense or limited reasoning. Believing students are meant to feel bad, suffer and then overcome. Limited fluid reasoning. Students may be penalised for not demonstrating a skill their educator is themself failing to demonstrate in another domain.
In any case, on this practice of prioritising crystal and testing fluid in education. I believe it's quite common in mathematics, particularly. In fact, I'm fairly confident it contributes to a lot of the stigma around mathematics, academic or intellectual anxiety around maths, severe dips in self-confidence, self-image around maths, abandonment of proactive maths as soon as possible, etc. This sort of behaviour, practice, of under-preparing and over-critiquing shifts responsibility from educator to student (remember, you can never truly shift responsibility, but you can attempt to) and distorts a student's sense of self. It is deceptive. Not intentionally, usually, but it is deceptive.
Students may be penalised for not demonstrating a skill their educator is themself failing to demonstrate in another domain.
Again, there is a lot of distortion, deception, identity shift, responsibility shift in these sorts of behaviours and thoughts in education.
On abandonment of maths, you abandon it because you associate it [consciously or otherwise] with diminished self-image, diminished hope, diminished belief in your future. A sense of you can't do anything right, you can't succeed, you're not good, etc. hardly ever stays in its own domain in your mind. You may think you only have these thoughts about yourself with mathematics, for example, and nothing else. You may think you're fine. These sorts of thoughts tend to spread to other domains without your knowledge. They're very smooth, very slick. They threaten your ability to go on, slowly, over time. They threaten your sense of safety in the world, again your sense of hope. It is only natural you begin to abandon any areas, and domains, and disciplines, where these thoughts are strengthened. Where they are fortified.
There's something about third grade, ages 7-8. I've noticed this in education. My experience is primarily with ages 5-14, and I have noticed quite clearly there is something about age 8. Third grade. Many people experience difficulty with mathematics from their first experiences with it, but I've noticed around age 8, third grade, begins a divide. This seems to be the moment when students are regarded "good" and "bad" and it's taken for who they are. I've also noticed this seems to be when students begin actively disliking math. This is noted both from students of a young age and adults reflecting on their past. I loved math as a kid, but... Before this, it challenges them, but not too much more than anything else they're learning at the time. The critiques and corrections they receive on mathematical performance do not too much differ from what they receive on writing, English language arts. And yet, this is when any difficulties seem to begin being treated as fundamental traits. Not good begins to become a fundamental trait. I've noticed, if you don't somehow pickup and recover specifically around age 8, third grade, difficulty is likely to continue until you graduate or leave secondary education. I'm not sure why this age but I have a hypothesis this is when we begin testing fluid reasoning. Not teaching, testing.
This is actually the age I believe we should be teaching fluid reasoning. Unrelated, but somehow the numbers line up. I would like to teach Discrete Mathematics in grade 5 (age 10), begin preparing for it with basic set theory and the concept of possibility, in grade 4 (age 9), which leaves some kind of introduction to how to think on what you do not know around age 8 or earlier. Truthfully, even earlier might be easier. Young children are relatively tapped into imaginative abilities anyway and have tendencies to do without being told what or how to do. The thinking exists, they just need to be shown how to transfer the same thinking to another domain.
Anyway, this is around when we seem to begin testing fluid reasoning. I believe a lot of standardised testing also begins here. Amusingly, this is also when we seem to begin restricting fluid reasoning in other domains. Around 7-8. Maybe not restricting, but challenging. The question is can you challenge without restriction. What I mean is, as I noted before, certain thoughts are slick, they're a bit viscous, slow but they move, and they seep into other domains, regions of your mind, with time. Depending on how you challenge a child's fluid reasoning when it is presented to you, they may learn to begin restricting fluid reasoning in all, across all domains. They don't know, they may not articulate it, but they know. They don't know, but they know. You don't know, but they know. You are not going to say Do not use fluid reasoning in this moment, you will not know, but if what you punish or challenge is an expression of this reasoning, they will know. They will know, Whatever thing my mind just did, done. We don't do that here. They won't even know that they know, but they'll know.
Anyway, fluid reasoning. In some sense, many educators are 'right', meaning I agree with them, I find their reasoning sound. I understand believing a student's mathematical reasoning and understanding are not truly being strengthened without some strengthening of fluid reasoning. It makes sense to want to continually evaluate fluid reasoning, but again, it then needs to be taught. It is a distinct skill to be strengthened and learned. Not everyone will necessarily be able to strengthen or access this skill to the same degree, but if you want it tested, it would make sense first to attempt to teach this skill to the degree it will be tested. Otherwise, you are evaluating your student not on their skill level based on your teaching and probably not even on their skill level based on prior teachings, but on some more fundamental skill that has nothing to do with your education. Not fundamental as in unchanging, but fundamental as in rudimentary structure, what exists without you, what they would be able to do whether or not you were there, whether or not they had ever participated in your "education". Again IQ testing, in some sense, specifically because the IQ is not rigid, it is not absolute, it is not ultimately defining of what a person does, but it is an attempt to isolate cognitive operations and evaluate what your mind does without relying on academic knowledge, taught knowlegde, crystallised knowledge.
It's a fun little paradox. Because of the nature of fluid intelligence, fluid reasoning, to teach fluid reasoning and then test it is not a test of crystallised knowledge. You are testing exactly what you taught, but it is not crystal. Fluid reasoning is expansive, it compounds on itself, it grows. Because of the nature of fluid reasoning, even if you teach it, even if you teach how to apply it, you can then test it and still be testing fluid reasoning. This is for educators who fear they may make it too easy. Who believe they need to test fluid because math but do not teach fluid because that would be giving it away that would be easy that would be giving you the answers. This, again, reflects an educator's own misunderstanding or misapplication of fluid reasoning.
And on easy. Learning is easy. It is time we moved on from this*. Learning is of our most innate functions we could not stop it if we tried. There is perpetual learning, attention, focus, regional difficulties, and there are blocks. I say regional to suggest parts of the mind, regions of the mind, not regions of the world, the planet. Learning is easy. A pursuit of making learning not easy is frivolous. You are not fundamentally changing this thing, this entity, this idea we call Learning. You are shaping an individual's experiences and perceptions of it. You are not making Learning not easy, you are making their experience not easy. You are not changing the shape, the qualities, the structure, the characteristics of Learning. You are changing the shape, the qualities, the structure, the characteristics of their experience, their life, and of them. Your student, them.
Anyway, I think this is now, all I have for now.
I'm sure there will be more, in time.
*I say this in jest. I respect and honour that we collectively learn and we take time. We will move on as and when we will, I do not decide the time. Learning is easy. And the fact learning is easy is something else to be learned, with time.
21:21:18
Note, the very specific statement I thought I myself could find many ways to argue against was "Mathematics is arguably the most important class in education, if not one of, obviously." Most everything else, specifically after the first three paragraphs reflects thought and reasoning I'm more secure in. Of course, still a workshop, as it always is.
21:25:17
21:34:40
I did not say this before, but also these thoughts are fairly disjointed. This is notetaking. This is documentation. This is reference for future work and thought. The thoughts in themselves are quite firm, I say this is notetaking not because I am unsure of them, but they may otherwise be organised and arranged a bit differently. This is workshop. This is thought.
21:37:22