Would begrudgingly recommend his book

I did not recognize the name at first, but after reading the first few paragraphs, I realized I actually own a copy of this guy’s book. He writes in this very recognizable way as to have me rolling my eyes while simultaneously agreeing with most of his thoughts. So Mr. Lockhart can be a little hyperbolic, I think. But again, he raises many good points, and I especially enjoy the bits of dialogue between Simplicio and Salvati, which I think does a better job of winning me over.

So I don’t know if it counts, but I largely disagree with what I see as exaggerations on his part. It’s his analogies in particular which rub me the wrong way. Right out of the gates, he describes a musician’s nightmare and then a painter’s nightmare, which honestly to me do not sound so far fetched. I think it's actually a very common experience for young musicians to have their love of music diminished by their experiences with music education. In general, I agree with the comparisons made between math and fine arts, but I think it more so says something about society’s take on education in general, rather than a problem which is unique to math. With these comparisons to music and art, he’s also maybe a little dismissive of the utilitarian aspect of math which has played such a critical role in scientific and technological developments. I don’t think it’s “wrong” to promote those aspects of mathematics, and I think there’s a lot about the modern curriculum which generates interest in the subject for a lot of people (including myself). Again though, I must acknowledge that he walks back a lot of his more radical ideas in those little play scenes between Simplicio and Salvati, where they at least come off as more sensible.

Despite my finding his tone annoying at times, it’s much easier for me to find things I agree with in Lockhart’s writing. For instance, I strongly agree with the idea that the creative, self-expression aspect of math is totally minimized in the current system. I personally always enjoyed math class, but I can’t help but wonder if I would have enjoyed it even more had it been more like Lockhart’s vision. And even as someone who has always enjoyed the subject, I often think the material doesn’t actually get very interesting until the university level. There’s such an emphasis on technique and method, rather than mathematical thinking. It’s as if “they” want us to become engineers (lame, jk). It’s all about productivity, after all. This is getting back to an earlier point which is that the state of math education is a symptom of not just government policy, but also sociocultural values. Something Lockhart hints at here, but I think states more explicitly elsewhere in his book, is that he believes math should be optional in school. I’m not sure how I feel about that, I sometimes agree and sometimes don’t. But I do think about that point quite a bit, so again I must respect Lockhart’s ability to open my mind to the possibilities.

Locker Problem

Super scratchy work on the problem shown below. I was basically thinking out loud on the page. The first question I asked myself was "how many times will a given locker have its state changed?" I realized that it would be determined by the number of factors of the locker's number, and went from there. You'll notice that I started with student 1 closing all of the lockers, and then basically worked on the problem starting with student number 2 and all lockers being closed. I get a little hung up on excluding 1 until near the end. I began starting to work out how to find the number of factors a given number would have, based on prime factorizations. I'm thinking it would involve counting the number of combinations of prime factors, but I realize it falls apart because it double counts factors.

At this point I've finally started to realize how to count the number of factors nicely based on the prime factorization, and I also keep on excluding 1 for some reason (I think it's just how the problem was pitched in class that got me stuck thinking like that).

I have a nice formula for giving me the number of distinct factors, and finally decide to include 1 because I realize the formula automatically does that. I swap around even and odd in my "final answer" to account for this change.

Ran through an example number just to try it out. Realized I would need to run every number through the formula to get a complete list and decided to call it quits. I should have known from the way the question was phrased that the answer would turn out to be an easily described set of numbers, but it honestly never occurred to me that only perfect squares have an odd number of factors - cool problem!

Favorite and Least Favorite Teachers

    I always find these sorts of questions difficult to answer. I think I was lucky enough to have many good math teachers and it doesn't feel fair to pick one out as my favorite. I do know that a lot of people have had negative experiences with their math teachers... And it seems way too common to hear stories about them being either particularly strict, unempathetic, or even incompetent. I often blame such teachers as a leading cause of the all-too-prevalent stigma against mathematics. Folks just seem to hate math. And if you ask them about it, I guarantee that 9 times out 10 they will start to tell you about one teacher who single-handedly ruined the subject for them. So anyway, I was lucky to not have any such teachers. And it's not even like I ever had a super mathy teacher that took my interest in the subject to new heights or anything dramatic like that. My teachers were simply good and decent people who made learning math a positive experience rather than a negative one. I believe I do have at least some natural inclination towards the subject, but in comparing my experiences with others, I can't help but think that an interest in math is the default factory-setting for people, which then gets crushed out of us by negative experiences at school and home.

    At the moment, I can only really think of a single time that I had a negative experience in math class... It was grade 9, and I had only completed about half of the homework. I think I had just left it until the last minute, and had basically only worked on it that morning - probably on the bus ride to school. So when I was called upon to give an answer to a specific question, I had nothing. My teacher asked me in front of the class if I had done my homework to which I replied, "kind of." In my memory, she yelled at me, asking "How do you only 'kind of' do your homework??" So anyway that was super embarrassing, and I've made a mental note to try and never embarrass students like that. Even then, I did like her as a teacher, and she was otherwise very nice and supportive. But I can definitely see how enough experiences like that can make math a bit of a sore subject for many people.

Response to Eisner

    I really enjoyed this article. There were several 'stops' for me personally. For instance, early on in the article, with regards to children answering questions from their teachers, he says that "children compete for a place in the sun." Now that's pretty funny if you ask me, which may have been the intention, but also I wonder if maybe he was thinking about young elementary school students. Even then, there are definitely younger students who don't necessarily feel much gratification from answering questions correctly in class. And then high school students, who are much more skeptical of the school system, and who are definitely aware of both the null and implicit curriculum, they may feel even less enthusiastic about the school's reward systems. Having read the rest of the article, and then revisiting this thought, I believe you could categorize a student's lack-of-faith in the school system as an unintended consequence of implicit curriculum. That's not at all to say I think implicit curriculum should be "hidden" from public awareness. It might actually be pretty neat to teach high school students to be mindful of implicit curriculum and its effects (actually, such a class is a part of the null curriculum).

    Overall, I very much respect Eisner’s balanced take on these concepts. One analogy I found particularly apt was his analogy comparing the school system to a road created by a well-trod path. I think this not only highlights the utilitarian focus of schooling, but also the fact that the formulation of our school system is the result of generations of trying to meet the needs of the day, while also building on top of a pre-existing foundation. This is a helpful reminder for me personally as I can tend towards an attitude of “everything is broken and we need to start over.” It’s certainly important to consider the practicality involved in making any changes to such well-established institutions. As Eisner also points out, there are undoubtedly “real-world” benefits to how we organize and structure our schools, even if some of it seems outdated to me. And I can’t remember if he mentions it specifically, but he certainly hints at the fact there are both positive and negative implications inherent in any formulation of a school system. The concept of the null curriculum is perfect for illustrating how it is in fact impossible to have a school which isn’t lacking in one respect or another. That being said, we can certainly always do better. 

    Looking through the posted BC Curriculum overview, I was first reminded of how incredibly vague and non-committal these types of government documents can be. It’s like they’re just latching onto a handful of keywords, repeating them over again to appeal to some sense of conventional wisdom. Two such key words are “inquiry” and “competency.” I do think that the idea of inquiry-based curriculum is pretty well in line with the thoughts of Eisner, especially his point on educating kids about how to pursue and investigate their own interests. So that’s potentially a move in the right direction, but I also think that any mandated teaching philosophy is bound to run the risk of being abused and becoming a bit of a crutch. Competency-based curriculum on the other hand, to me sounds much more in line with a very traditional approach to school, which again has its pros and cons. 

    Truthfully, I feel as though there is much more to say about this article. And I wish I had the time and capacity to respond to more of it here. I will definitely be on the lookout for his book and would love to read the rest of it one of these days. On that note, it amazes me that people have been thinking so deeply about these issues for so long, especially in considering the dates of some of the publications cited by Eisner. For instance, in 1938, Lewis Mumford writes that “pervasive instrumentalism places a handicap upon vital reactions which cannot be closely tied to the machine” - I mean, that's just good stuff (it sounds like something the Bene Gesserit would say, and I wonder if Frank Herbert read much Mumford...). Again, I think the age of these publications speak to how the roots of our education system are quite deep, and substantial change takes substantial time and also probably a steadfast rejection of complacency. I think the ideas of explicit and null curricula essentially go hand-in-hand and are certainly worth thinking about. To me however, the implicit curriculum deserves more consideration, and I find myself particularly interested in how we might re-structure and re-organize our schools to realign the implicit curriculum with our educational goals.

Group Discussion on Skemp

We had a lovely discussion about Skemp's article in the garden the other day. Going through the questions really quickly...

1) Are the two kinds of understanding distinct/separable?

We pretty much all agreed at the outset that the two types of understanding are both distinct and separable. That being said, near the end of the discussion, there was talk of both types of understanding as lying on opposite sides of a spectrum. While I think there is some truth to this with regards to how one might approach teaching a math topic (more relationally, or more instrumentally), I believe these two types of understanding exist independently of each other, and what's more accurate to reality is that we all have varying degrees of each type of understanding depending on the topic.

2) Is there a "best" order to teach them?

At first, we were discussing how teaching instrumental math is a great way to introduce a topic, and then after students learn the "how to's" it's best to develop more relational ideas. However, we concluded that this is not necessarily always the case, and different topics probably lend themselves more to beginning with one approach over the other. Near the end of the discussion we briefly toyed with the possibility that we might be limiting ourselves in thinking in terms of only two types of understanding - very exciting.

3) What kind of activities promote each one?

This was the point we spent the least amount of time discussing... But if I remember correctly, we briefly touched on how traditional-style worksheets generally reinforce instrumental understanding, while discussions can be more conducive to promoting relational understanding.

4) How to assess understanding?

Similar to the thoughts on the previous question, we discussed how traditional forms of assessment (tests and worksheets, say) can be easily employed to assess instrumental understanding. On the other hand, relational understanding might be better assessed through discussion, or through some well-designed questions. For example, you might design a question for which a learned algorithmic approach would lead to an error. Probably if a student has only ever solved equations of the form x + a = b (which had maybe been taught/learned as "one-step equations"), they would need some relational understanding to solve even something like 10 - x = 4.

Shoutout to Andy, Jasmine, Jacob, Krystal, and Raymond for making it a positive space for discussion.

Response to Skemp

    While reading Richard Skemp's article, "Relational Understanding and Instrumental Understanding", I was immediately struck by two things. First of all, that there is indeed an important distinction between relational and instrumental understanding, especially with regards to teaching/learning mathematics. And secondly, that it's perhaps a bit of a false-dichotomy-situation to believe "understanding" falls neatly into one of these two categories. In my mind, math can certainly be taught more "relationally" or more "instrumentally." It's tempting then to think that these two types of understanding exist on a spectrum. But even then, it's not difficult to imagine how one might simultaneously possess both types of understanding of a given topic. I think what's closer to the truth, is that there are degrees of both types of understanding.

    I imagine a human brain in a jar, hooked up to two gauges: one measuring instrumental thinking, and the other measuring relational thinking. As the brain gets fed different math problems, I think you would see the values of each gauge change, independently of each other. I believe we're all engaged with these two different "styles" of understanding to varying degrees, at different times, to suit our needs. I think part of the beauty of mathematics, is the sense that you can always push the boundary of relational understanding. Even something as simple as arithmetic can be generalized to sets and mapping rules, which can be further generalized to "basic" axioms. It's definitely not always desirable to engage in the most generalized level of relational thinking. But on the other hand, if you are seriously only interested in instrumental thinking, you might as well give up and let computers do all of your thinking for you (mostly joking).

    I believe an effective educator necessarily appreciates both "types" of mathematics. It's obvious that math encompasses both the relational underpinnings of it's topics, as well as all of its instrumental applications. I definitely tend to agree with Skemp and his view that there needs to be more of a focus on relational mathematics in schools. And I do really appreciate his justifications for teaching instrumental maths also. I've found that a lot students do tend towards the picture painted by Skemp, that of a student looking for quick and easy rules to pass their exams - very practical. The included passages from Sir Hermann Bondi especially resonated with me, and I think a lot of the satisfaction that you would expect a human brain to get out of studying maths is tragically blunted by an overemphasis on teaching the subject instrumentally. One of my personal goals as an educator is to pass along my passion for the mathematics. But to do that, you do need to make the subject matter appealing and accessible to your audience. I often see the instrumental approach to teaching as the "hook," which can then be expounded upon by all of the juicy relational ideas. Again, I think that's an oversimplification. There are no easy rules or 100% accurate "theoretical formulations," but I do believe that a truly excellent teacher is able to balance these two styles of teaching/understanding to suit the needs of their students.

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