Eisner’s Curriculum

I was reading Eisner’s article, and right at the beginning where he was explaining that curriculum is what we teach and that we teach compliance, I realized that this would be a point that would have made me stop and think, but that idea of teaching compliance had already been expressed many times in just two weeks of BEd that I didn’t even skip a beat. But then the article got to the competitiveness, and mentioned physical activities and games, and yeah, that’s some competitiveness, and I think I understand some of the negative impacts, but then Eisner went deeper into the classroom dynamics, and normal distribution of grades, and university admissions, and honours, and all that, and that made me stop. Coming from a country with a different school culture, that was almost shocking. The concept wasn’t unheard for me, but I honestly thought most of it was just Hollywood exaggerations to add to the plot and highlight the good guys vs the bad guys stereotype in high school drama movies and shows. Possibly coming from a country with many standardized tests, including National Standardized Admission Exams for universities, there was none of that extreme competitiveness Eisner mentioned. That was quite something.

Another contrast, and it’s a huge section, is the entire segment on the Null Curriculum. The things you don’t teach. I’m honestly not familiar with education during Eisner’s time, but my experience growing up, your entire school schedule was set, and even university was pretty much laid out. During my time as a student, art class was just for kids. I think my last art class was grade 5 or something. There was also no philosophy, or virtually any thing Eisner mention is missing. My experience volunteering in a high school in BC in 2026 was completely different. I saw that students have many options to choose from, together with a core mandatory course list. That seems way more in line with what Eisner thought was missing.

I think the BC curriculum approaches most of those issues with an open mind, and trying to solve the issues mentioned by Eisner. Maybe the different streams of mathematics could be taken as some sort of honours classes, but I guess that is a way to make some part of math mandatory and some more complex part of math optional. From what I noticed (but I’m not exactly sure how the BC curriculum states that) students opt into those streams. The teacher may suggest to a struggling student that they could take the workplace math stream, but it is ultimately decided by the student. I guess that is possibly healthier.

The Good, the Bad and the Ugly

During my studies I’ve come across many math teachers. Many different styles, personalities, views, approaches. Some were more in sync with me, some were less.

The Good was a high school teacher that I had. Ironically it was the teacher that gave me my worst math grades before university. I’ve always liked math, it was like a game to me. A puzzle. A challenge. So I’ve always been interested in math and was good at it. Regardless of the difficulty I may have had in other classes and subjects, I’d always get 95% or more in math. That teacher was the one to break my streak. They would challenge me more. Not only more than the previous teachers did, they would push me beyond what they would expect for the rest of the class. I asked if they could make the tests harder for me, and they were happy to. That means the tests were way harder and that reflected in my grades. But instead of getting a boring and expected 100%, I’d get a hard fought sweaty 80%. And the challenge was worth way more than the grades for me. It made math fun again. No regrets!

The Bad, on the other hand, was a thermodynamics teacher in university. Not exactly a math teacher, but it was a math heavy subject. That was a teacher that expected the students to memorize all the formulas. And as I mentioned earlier on a previous post, my memory is not that great, so that didn’t really work well for me. To be completely honest, it is possible that they didn’t really expect the students to memorize all the formulas, as they would seemingly “accept” that the students would cheat. They’d just turn a blind eye, leaving the classroom unattended for most of the time during exams. Another problem that I have is that I can’t cheat. My mind just blocks me. I tried once. I had a piece of paper inside my shoe. Just knowing it was there made me unable to focus on the test because I’d just be sweating, shaking, and drooling. I know that was just an incompatibility between me and the teacher’s style, but other than just personal capacity, I think that style silently passes the message of “this subject is so hard that regardless of how much you study it won’t be enough and you’ll have to resort to cheating”. I’ve had other physics teachers with equally hard topics that would provide the students with the formulas. That was the physics department’s policy, but was not the same for the engineering department.

That leaves the Ugly to me, I suppose. Ugly throwing shade at my future colleagues…

The Locker Problem

When I first read the blog post presenting the problem, I immediately concluded that each student changes the state of the lockers that have a number that is a multiple of the student’s number. That didn’t help much. I had to think of it from the lockers’ point of view.

Yeah, that’s not that hard. The locker will be changed by the students whose numbers are a factor of the locker’s number. Okay. If the lockers start open, lockers with an even number of factors would stay open and lockers with an odd number of factors would be closed. Easy. Final answer. Done.

Then I started thinking if there was a better answer for that. But how do I know which numbers have an even or odd number of factors? And I thought I’d have to brute force my way through it. No, it can’t be! That’s not how we do things. So I started thinking about factors, and as it usually goes I thought that prime numbers have only 1 and themselves as factors! That’s it. Prime numbers have an even number of factors, everything else must have an odd number, right? Then just to check I thought of 6. And it turns out that 6 has 1, 2, 3, and 6 as factors. Four factors. Four is even… Wait. Of course! Factors come in pairs. Always! The factors of a number n always come as a×b = n. But then does that mean every single number has an even number of factors and therefore every door is open? Can’t be!

I had to try and brute force the start. Starting with 1 I found that it had only itself as a factor. Wait a second there. 1 is a weird number, so maybe only 1 is closed and everything else is open? Let’s keep checking before we conclude that… Continue brute forcing: 2 has two factors, 3 has two factors, 4 has... Three? Wait! How? What about the factor pairs? 4 is 1×4 and 2×2. OOOH! One of the pairs is a number times itself! So that is a pair, but that number only shows up once in the list of factors. That is the odd factor! The factor that doesn't have a pair other than itself! The square root of the number! It has to be it! Perfect square numbers have an odd number of factors. Also, 1 is a special number in many ways, but also it is a perfect square. Of itself! Brute forced my way to 10 just to check and, as expected, locker 9 was closed!

That’s a beautiful problem. That is a problem that would make me enthusiastically tell the students “that’s why math is the most beautiful thing in the universe!” To which I know most students would just sigh, but such is life…

I built a small interactive visualisation here to help see the first 50 lockers and the first 50 students (we don't need more students because no student after the 50th would touch the first 50 lockers).

Interactive Visualisation

Instrumentality, relationality, and terrible memory.

I've always had a problem with pure instrumental understanding. I imagine part of that is because I have terrible memory, but also because I always wanted to know why. I was curious and loved the feeling of finding things out. I understand some people just want to know the easiest way to solve something, a "just give me the formula" kind of approach. Shortcuts, mnemonics, and memorizing formulas can all help with efficiency, but for me that never really stuck. Having a really bad memory could make me remember the formulas wrong. Memorizing the whole formula and mistaking a plus with a minus sign could sound like a small mistake, but that could also be as big as mistaking the gravity going down with it going up. I've seen a lot of sign mistakes on memorized formulas or on calculations but never found someone who thought gravity was pushing things up. That just shows how fragile the instrumental understanding can be, with small mistakes being huge disconnects with reality. Not that focusing on relational understanding makes calculations fail proof, but at least, if you know what the big picture is, you can look at the result and say "this doesn't seem right", and double-check your work. If you just follow a procedure with no understanding of the relations between that and the world around you, the result is just a number, and a wrong number and a right number are both just numbers.

I agree with most of what was said, including the part where the author mentions the possible reasons for instrumental understanding. But I think when the author mentioned the difficulty of assessing relational understanding, the biggest mistake was trying to do that with traditional math questions. "Solve this" and "calculate this" questions will only check the execution, not necessarily the thinking. Maybe assessments should just be different. Make some math assessments be more like humanities studies, asking students things like "what would be the next step?", or "why would we do this?", or "when would we use this?" instead of just asking for final answers. Math has number problems and word problems, but those are usually just answered with numbers. Maybe math also deserves word problems with word answers.

Hello World!

 Hello everyone.

Let’s try to have some fun and hopefully not divide by zero.