Response to "Battleground Schools"

As always with these readings, the first page always makes me stop and speculate. In this case, the reading mentions the "[oscillation] between two poles" in terms of mathematics education. For the purposes of this blog post, I will be using "conservative" and "progressive" much like the book has. When seeing this sentence, right off the bat I got déjà vu and thought:

"Wait a second, these are awfully similar to the Skemp and Lockhart articles! Though in Lockhart's case, it was more so a passionate lament. But both of these touched on a very similar dichotomy that I pointed out in my Lockhart's Lament response. While the side that Lockhart lamented over was instrumental mathematics that saved time, felt mass-produced, was sterile and completely devoid of art, the other pole that he longed for was one of exploration, inquiry, understanding, and everything else that the 'Progressive' column is represented by!"

This reading introduces yet another dimension to consider: the political dimension. In addition to my prior classifications, I can say that this too, is political. While right-wing politics favours the conservative school of mathematics education due to its emphasis on promoting workforce habits (ex. obedience, precision, correctness), I draw an association between left-wing politics and the progressive way. It got me thinking, now that this is the case: there has been a pendulum swinging of over a century between the two poles of mathematics education, with the same arguments being repeated. One would think that eventually, the soldiers would've settled on an agreement, so why does it keep oscillating? The realization here is that most likely, education will only be as stagnant as the political theatre surrounding it (i.e. not at all), and policies are all at the mercy of the political whims. This was further confirmed to me in the later parts of this reading, with political factors being at the heart of the reforms.

For my second set of stops, it was around the acknowledgement of math phobias and how they are formed due to various presumptions. When reading the presumptions listed, I could remember these being in play when I was in school (a time of more conservative-sided mathematics education). Yes, many people did see math as a stiff, soulless subject. Yes, there was an association between math and less savoury groups of people (ex. nerds). Et cetera. I've seen the stigma around mathematics affect people like my mother and sister who have resigned themselves to being "not math people" (despite them having a good grasp of everyday math like money splitting). I thought about the idea of math phobia here, and maybe the oscillations happen towards the progressive side because the math phobia buildup reaches a critical mass. From the reforms described later in the reading, symptoms of math phobia were addressed by reformists who wanted to reduce the stiffness of math and make it a subject of thoughtful experimentation. Math phobia definitely feels like the greatest argument against conservative mathematics education, since it's often the root motivator behind the reforms oscillating to the other side.

Finally, I had a stop at the start of the section on "Progressivist Reform". My first thought was that it's neat that these ideas of progressive-leaning mathematics education go back a long time! The things we are learning in teacher education today are actually not recent concepts. As a testament to this, I saw the name Dewey and vaguely remembered seeing it on a citation in one of my course's readings. These are the same readings that get discussed and considered by teacher candidates like myself in the year 2024, which reinforces the claim that similar arguments get repeated regardless of the time period. Considering that we are entering a swing to the progressive side in BC, the parallels are genuinely intriguing to see, and I feel seen by the people of the past.

Reading further into the history of these reforms, my question of how society accepts oscillations back to the conservative side was addressed. It seems that an urge to have a state's society be advanced for whatever reason (in the case of New Math, to not fall behind the USSR by raising students into potential rocket scientists) would have us inspecting the curriculum and declaring it too narrow. It was quite something to see that calculus in high schools was likely a result of this conservative math education reform. Unfortunately, the way New Math was set up made it so it was doomed to oscillate away -- no one could understand what was being taught! This was the funniest part to me, since the New Math reform felt so short-sighted in hindsight. It also assumed that most kids could and would become rocket scientists that attended university, which also brings up a good point: conservative math education has benefits in higher education, so there is a top-down dynamic happening here. If the politics deem it best for more people to attend university (like with the time of New Math), then there would be an oscillation towards conservative math. This was certainly not the case by the last reform in the reading, and the NCTM seemed to relax the pendulum swings while straying from the political landscape of their time. I liked that they tried to strike a middle ground by maintaining some demand for competence in common areas, but also promoted different types of assessment, formative assessment, meaningful activities, and more. It has many parallels with the BC mathematics curriculum.

Assignment 1 Write-Up

Click here to access the presentation slides.

We finished the first assignment of this course, and I have to say: it was more fun and less stressful than I thought it would be! I would like to give a special thank-you to Leon and TsáKtalay’pa for providing something unique to the project. To describe what each of us did, Leon was instrumental (pun not intended) in making the tensegrity table and coordinating the interactive activity where we gave groups of 3-4 the chance to make a da Vinci popsicle stick bridge with prizes, and TsáKtalay’pa provided his expertise on the history of bridges, vector forces and mathematical interpretations, running the activity timer, and answering questions. I also have to credit myself for providing the inspiration and diagrams for the mathematical and artistic extensions segment of the project, as well as conducting the extension experiment that can be seen on the last slide. The slide deck was made in an afternoon with contributions from all three of us. All in all, everyone contributed greatly and our project wouldn't have felt as good as it did if even one person was missing.

Pictured: The original tensegrity table math art by Owen Rohm, and Leon's duct tape cardboard tensegrity table which took 2.5 hours to make (according to him)

Pictured: The step-by-step visual guide on how to assemble the popsicle stick bridges for our interactive activity

Pictured: The theoretical diagram of our proposed extension of the tensegrity table - the Extended Tensegrity Arch

As for my experiences with working on and presenting the project, I have to say it went a little differently from what I expected. After working on the slide deck about four days before the actual presentation, the Extended Tensegrity Arch diagram (as shown above) bothered me a lot. Initially, I wanted to take that afternoon to assemble the ETA, but after Leon mentioned the sheer length of time it took to assemble even one tensegrity table unit, I decided to cut my losses. However, I had time on the weekend, so I took it upon myself to make a toy model of 5 smaller tensegrity table units.

Even making one unit was very challenging. That, or I lacked the dexterity.

After scrambling to get one unit assembled, I instantly flipped it on its side in accordance to the ETA diagram that I drew in order to learn more... and voila! It was staying still despite being sideways, which was in line with my hypothesis. I was inspired by spaghetti bridges made for my high school physics classes. The spaghetti bridges would have weights incrementally added underneath, and oftentimes the first snapping point was actually at the top where the compressive forces made the bridges buckle. As such, I thought: since the tensegrity table's main draw is its ability to resist compression while maintaining its form, why not use that same principle in a bridge? That was the hypothesis, and it felt great when it was proven true. Even better, I got to understand the need for the non-central threads to hold the tensegrity table together by snipping it with a scissor (as seen in the last slide's video). 

On presentation day, I think my biggest surprise was how hard it was for someone to make a popsicle bridge if they had no prior experience with it. None of the groups could form the 9 popsicle (3 middle segments) stick bridge within 5 minutes, and the longest bridge category was won by a bridge with only 2 middle segments. Other than that, the positive reception felt really good, and the payoff for my inquiry into my diagram's hypothesis was equally great. We were able to make an impromptu discussion on what should be done to the top of the middle unit and whether the arch was plausible or not, and in the end I was able to shed a bit of light on my findings! I still think it is unfortunate that we weren't able to make the full ETA, but I am still proud nonetheless for taking it as far as I did.

I also want to apologize to every group that came after us, considering how distracting the popsicle sticks were. If I ever hold any activities related to popsicle sticks, I will be sure to collect them back before starting something new.

As for the potential of this project in my own teaching, I feel that due to the foundation of physics that holds up the art piece, there is an inherent parallel with physics that would manifest. I found that the ETA diagram was a major inspiration for integrating the project into teaching, and one would notice that it resembles a complex free-body diagram which is a key element of understanding physics. Not only that, but due to the ambiguous yet plausible nature of the diagram, it raises a whole slew of questions that we can get students discussing! Even during the presentation, we've had people questioning whether the arch can really hold up against gravity and itself, questioning the placements of the strings, and theorizing about what would be needed in order for the top unit to stay in equilibrium -- all things that we can get students to think about in a case study fashion in order to build their spatial reasoning skills (I want to credit Leon for bringing up this skill to our group, which I also think is great for general relational mathematics). It would also be cool to get students to draw their own free-body diagrams as part of a debating process in order to defend their own tension-based construct designs, which would serve to exercise their critical thinking, visual and verbal communication skills. If we want to get wild, perhaps encouraging students to experiment with toy models to verify their diagrams like I did, which could increase the amount of "play" happening! There's also a tie-in to the historical aspects of the physics and the contextual premise as well; bridges come as part of humanity's long history, as well as being a long-used subject of applied mathematics (think of how much bridges are brought up in the field of engineering, and not just the Tacoma Narrows Bridge collapse). It would also be useful to promote elements of the history of mathematics through this extension. I could also see myself asking students on how they could further extend the ETA, using the same principles that allowed the ETA to be plausible. Could we make it 3-dimensional and create a super-extended tensegrity DOME? Maybe we could create a leaning tower of tensegrity tables -- the possibilities are endless.

In terms of what couldn't be used: I'm going to go with having students constructing their own tensegrity tables, at least within a short timeframe. It is a far more challenging task than making popsicle stick da Vinci bridges, which is saying something considering how difficult it was for many people to make da Vinci bridges (though da Vinci bridges may be more in the realm of possibility). It may also be challenging to find a curricular unit to naturally segway from and to this art piece depending on the course, but it could be possible to bring up tension-based art and structures during miscellaneous instructional time as a fun brain teaser (for example).

Response to "Lockhart's Lament"

As a general review, I found this to be the most entertainment reading experience so far. Lockhart's sheer passion for criticizing the state of mathematics teaching in the early 2010s was sky-high. I highly recommend reading this aloud in an equally passionate tone while listening to some intense, imposing music like O Fortuna by Carl Orff, or in my case, the Final Fantasy 7 Advent Children version of Sephiroth's theme.

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Reading the article, I came to the generalization that Lockhart was very unhappy with the state of mathematics education, and how it not only didn't reflect the artfulness of mathematics, but actively snuffed it out in favour of the practical, algorithmic aspects of the subject. My agreements and disagreements are really just where I lie on the spectrum relative to this article. Regarding the spectrum, it has many tie-ins to the spectrum of instrumental and relational mathematics that Skemp illustrated. While instrumental mathematics was concerned with things like rote memorization and algorithmic repetition in order to create workforce-ready people, relational mathematics concerned itself with the deeper into the "why" of mathematics, which is a word that Lockhart also used to describe what should be taught instead. However, Lockhart presented the additional component of art to the spectrum, as well as the playfulness and whimsical nature of mathematics as a discipline. The artistic aspect was completely tied to the traits of relational mathematics while the sterility of the status quo was tied to instrumental mathematics traits.

I would like to introduce the third component to this spectrum: time and effort. Although not completely related, even Lockhart admits that promoting the artistic, free-flowing aspect of mathematics and allowing children to explore mathematics like an art form would be impractical. On the other hand, the "path of least resistance" was found in the status quo through lectures, tests and the like. Coupled with the idea that instrumental mathematics does find its uses in the real world and specific fields, I found that my main disagreements with the article's messages and suggestions is that it is impractical, and also diminishing the promotion of instrumental mathematics. Lockhart would occasionally say things like "I don't think notation is wrong, just excessive notation" and other disclaimers about not swinging the pendulum to the other extreme, but I could get the feeling that he was tempted to lean that way anyway. My question is this: if we reserve the mathematics empty husk for promoting mathematics in Lockhart's sense of the word, then where does that leave time for the so-called sterile concepts? Simplicio has agreeable points that we'd still need to know the same basics -- some degree of standardization -- and even for later schooling, the polynomials and functions that Lockhart deemed unused in the real world should still be conveyed, as there will be people in the classes who do plan to enter STEM careers and need that foundation to thrive in the higher education courses. Even these sterile tools come from a history of people trying to simplify and abstract steps so that we don't need to deal with the long cumbersome process of discovery when sometimes we just need the tools (rather, instruments) to get the numbers we want. To get poetic here, it's not a bad thing to have sterile tools when we just want to get into the meat of things without contaminating it with extra mental clutter.

However, I do agree and sympathize with Lockhart's plight, especially considering that he wrote this in the early 2010s when mathematics teaching was far less progressive. It is true that even when I was in high school during the mid-late 2010s, we were still going through the curriculum that Lockhart described: lecture, test, et cetera. It is true that many of the things I argued in favour of the status quo are only beneficial because it's within the context of the status quo. I refer to the artistic analogies that Lockhart chose in the beginning of the article; while it is true that the painting student has the freedom to explore their creativity and present it, this is because their wider system allows it in all stages of the medium. On the other hand, Lockhart's problem is one that requires a top-down solution starting from stages after high school. Why is it the case that we need to prepare students for the next stage? For classes that come after? Even the progression from Foundations and Pre-Calculus 10 to Pre-Calculus 11 depend on foundations being built (excuse the pun). As such, I agree with introducing thought into the "why" of things, and that the status quo is responsible for many cases of boredom or self-labeling as "not being a math person". I agree that with every formula, we should give students the chance to explore problems without access to the formula first, and maybe they'll derive something differently, much like how the ancient civilizations were able to achieve the same conclusions for similar problems due to a lack of a unifying "curriculum" that they had to abide by. On the note of ancient civilizations, it would be good to shine a light on them and many other historical figures when the opportunities arise as a means to enrichen understanding of mathematics and its countless endeavours. Going back to before though, it should be done in moderation, with a respect for instrumental mathematics still existing.

The Locker Problem

Right at the moment of hearing about this problem, I thought, "This sounds easy! I already know what this question is asking about, and I already have ideas brewing." Little did I know, I would be spending a bit more time on this problem than I thought.

The first thing I did when I sat down was to establish the problem. As such, I wrote down the premise and drew a picture of some numbered lockers.

After that, I thought back to my initial brainstorming when the question was introduced: what if I used the fact that each locker is only opened when a student's number is a divisor of the locker's number? This property holds every time; for example, locker 1000 would be opened by the 2nd, 4th, 25th, 500th students, or another one along these lines. Thus I began to explore, as I now had a lead.

The premise here was to create a flow chart for each locker door. I quickly shelved the red text in the image since I wanted a more catch-all flow chart. I noted another truth: if a locker's state changes an even number of times, it doesn't actually change at all, whereas an odd number of times would change the state in the end. For example, closing and opening (2 changes) versus simply closing (1 change). Thus, whether we had an odd or even number of divisors was the key to deciding the locker's state.

So now the goal (note that there's a lot of narrowing in on goals here) is to figure out an easy way to get the number of divisors for a number. 

Fortunately, I remembered a very nice function definition from the number theory course I took last year (okay B.Ed breadth requirements, you win): the tau function to get the number of divisors of a positive integer! All that is needed is to get the n = p1^a1 * ... * pr^ar prime factorization of the number, with 'pi' being a unique prime, and 'ai' being a power.

Upon further inspection of the tau function definition, the conclusion I reached was this: if at least one 'an' were odd, then the number of divisors would be odd, which means the locker would be closed. The first 12 lockers were explored in the picture above, along with 180 since it was a number with two even-powered prime factors and one odd-powered prime factor.

Eventually, I came up with this flow chart to tell if a locker was open or closed:

Of course, we can simplify further based on the red text at the bottom-right corresponding to the first composite case. If every power in the prime factorization is even, then you could rewrite the factorization into a simple m^2 term. For example, 144 = 2^4 * 3^2 = (2^2)^2 * (3^2) = (2^2 * 3)^2, using exponent laws. Alternatively, if n's prime factorization only has even powers, then:

n = p1^2b1 * ... * pr^2br, where 'bi' is a positive integer

n = (p1^b1)^2 * ... * (pr^br)^2

n = (p1^b1 * ... * pr^br)^2

These steps are reversible, so the biconditional holds. The locker door will be closed if and only if it is a perfect square, and there are 31 perfect squares between 1 - 1000, inclusive.

Throughout these steps, I found that a picture wasn't too helpful in the end since I took a more number theory-like approach. The first step was to simply find a lead based on pattern recognition which gave way to custom-made rules. After that, it was a matter of narrowing down goals, which helped me know what to look for (in this case, the tau function). Using the tau function, I was able to make inferences with it in order to simplify it into a basic rule. So basic in fact, that we can just write "open" for every door except for the perfect squares.

Favourite and least favourite math teachers

Math has always been a big part of my life, with math classes being present in almost 20 of my 25 years on this planet. Throughout this time, I've had a variety of math teachers: my father who taught me the times tables after dinners, various YouTube videos and playlists, and high school math teachers. For this response, I want to highlight the first winter term of last school year, when I took two 300-level math courses at UBC to get into this program. During this time, I experienced two professors: one that I've hailed as the best, and the other being one I view far less favourably.

My all-time greatest UBC math course professor was phenomenal during my time in Calculus IV. While a basic point, I have to note his professionalism. He had a clear syllabus outlining everything we needed to know during the class, whenever he couldn't make it we would get ample warning and a recording or move lectures to Zoom (if he was sick, for example), and all notes and other reference materials were clearly organized. In addition, marking was rapid, which was great for getting feedback and knowing how we were doing. Exams were also very fair, with no surprises. Everything so far may feel like a bare minimum, but when contrasted with the other professor, the completeness of his professionalism shined greatly and I want to be a teacher who can do the same - no surprises, communicating, and well organized.

Aside from that, he was also personable, passionate, and had a sense of humour. (He also made sure the class knew he was a Swiftie, which added to the funniness.) Because of this, he was very approachable, and we all knew that if we had a concern or question, it would be addressed with utmost care and consideration. The course also had a Piazza forum set up, which he participated in very actively with great attention to detail. Before each exam, he would set aside time for everyone to practice with a set of NEW practice questions and walk around to observe and check in with students. As such, I and many others in the class could feel his genuine care for our learning, making us all want to do our very best for him and ourselves. 

If I were to use attributes to describe this professor, some things that come to mind are caring, competent, confident, and charismatic. A takeaway I got from this is that if you go above and beyond as a teacher, the students will return the favour. The high class averages for assignments, exams and the final grade are some pieces of evidence of this, and I want to note: scaling was very unlikely to have happened. Anecdotal conversations with my classmates also reflected these sentiments, as well as the large Piazza "helpful" or "thank you" counts when the professor replied to grateful thank you threads, and the perfect 5/5 RateMyProfessor reviews.

On the other hand, the other professor (for an intro course on complex variables) felt like he was lacking in many aspects. Sure, he got the bare minimum done: assign homework related to the lectures, do lectures, and administer exams. He was also a nice person, so at least he was approachable after class. This is where the similarities end. 

The biggest issue was his attention to the class, or lack thereof. One time, when my friend raised his hand up, I turned on a stopwatch to see how long it would take for the professor to notice. The professor had plenty of opportunities to pause and ask for questions (which he never did), but it took almost 4 minutes before even looking up to address my friend! While he was receptive to a small verbal nudge, this complete lack of attention towards the large audience felt inexcusable. Additionally, his speaking voice was quiet and his notes were particularly hard to decipher, which made basic note-taking and listening difficult. Exam and assignment feedback also took a long time, which made gauging ourselves a frustrating process. He also struggled to respond to Piazza posts at all, which made us feel disconnected. While the previous professor made us all want to do well, this one created a confusing and unnecessarily difficult experience.

This term was the only term in my entire 7 years taking UBC courses where I filled out professor evaluation forms (for both of them), partly due to the stark contrast of teaching quality in front of me. Both professors highlighted many things that would serve as good example of what and what not to do, from bare minimum tasks, to how I fit myself into the class' social ecosystem. The pains of the latter professor and the sheer joy and passion of the former have certainly inspired me to strive towards being a teacher more like the former, and I feel that I can look back at this articles for ways to conduct myself and my classes as a means to achieve it.

EDIT:

In my response above, I went into the strategies and methods that each of the professors exhibited. Here, I hope to elaborate a bit further on how they directly impacted my learning and the outcomes of said strategies and methods. 

With my favourite professor, I mentioned the many ways that he exhibited professionalism, confidence, and approachability. I feel that because of his professionalism, I felt confidence in him. In EPSE 308, we discussed the 5 parts of motivation, but I would like to draw attention to credibility, which concerns the capacity of a student to believe in their teacher. Credibility could be split further into care, competence, and passion, which my MATH 317 professor exhibited wonderfully as described above. Just because I was able to believe in my professor wholeheartedly, it made myself and my classmates intrinsically put in more work to improve. We were very active in helping each other, which gave us opportunities to teach each other (and teaching is one of the best ways to learn). As a result of me teaching and answering a lot of people on Piazza and on the course group chat before the final exam, I went in with a strongly rooted understanding. With minimal reference to my cheat sheet, I confidently finished the entire exam and got a final course grade that I felt reflected my learning at the time. Even now, with most of the content being forgotten, I have a strong feeling that I can relearn the material very quickly if need be, which I cannot say for MATH 300's content. To finish this part off, I also want to say that because the professor was so active on giving feedback on Piazza and adding helpful comments on returned exams, we gained the benefits of constant formative feedback. Knowing if we were off-track early meant that unnecessary stumbling time was saved, but at the same time we were given room to struggle through questions as a way to learn. I believe that this feedback component is particularly important, and once again, the "bad" professor lacked in this department as well.

The points I raised regarding the MATH 300 professor included inattentiveness, lack of confidence, and an absence of feedback within a reasonable timeframe. While the care, competence, and passion of the "favourite" professor above led to great results, the opposite led to poor learning, poor motivation, and overall a poor educational experience. For context, MATH 300 is an introductory course to complex variables, which is inherently a more challenging but very interesting topic especially once you explore zeroes, multifunctions, integrals, and all the intricate workings of the complex number system. Because of the professor's unpreparedness, I can say that the care and competence components were lacking. He was a passionate person about the topic, but it wasn't enough to save his credibility as my motivation dropped with my fellow front-row seated friends. There were frequent stretches of pausing to reread notes, which made me wonder if he was even qualified to teach! In conjunction with the traditional mind-numbing lecture-assignment-exam format that has shown to be weak for the learning process (compared to engagement and activity, which the professor wasn't able to meet the most basic criteria of eye contact engagement), the content was not delivered well with messy handwriting, soft speaking, and all these other factors that made myself tune out and doodle on my notes instead, which made me less prepared. For the final exam, my preparations were done by watching a YouTube playlist covering the same contents, and I feel that it was responsible for me surviving the course. Unlike MATH 317, I don't think I could relearn the content as quickly because my mentality was driven to a survival instinct where I tried to game the assignments and exams such that I could get a high score. The professor's pedagogy demotivated my learning for an inherently interesting subject, and instead amplified the difficult aspect of it, which in turn created a feedback loop of killing my motivation. My exams were disastrous attempts to "salvage as many points I can get" and I got bogged down by the UBC math exam specialty of time crunching, and when everyone got scaled up, the average was still low and I felt like I didn't earn my above-average grade at all considering how little was done by the professor to ensure retention and instilling of ideas. To sum up why I didn't like my 300 professor, it's because he was just a script-reader reading the script for the first time, which is in stark contrast to the confident, prepared style of my favourite professor.

Response to "Three curricula all schools teach"

The first thing that made me stop was reading that critics of schooling pointed out that schools fostered compliant behaviour over initiative. This was surprising to see because just last night, I was doing my readings for EDST 401 that focused on the purpose of schools. These readings had a central theme of the purposes of schooling since the 18th-19th centuries, being that of conformity and generating citizens who were ready for an industrialized world. Interestingly, this article seems more optimistic about it, as it views these practices less so as a result of direct intent, and more so unawareness from the people planning curricula. Regardless, the same ideas were being reinforced; in this article, examples were given such as reward systems implicitly teaching compliancy, or planning a curriculum out of tradition which perpetuated predictability and stability - all common themes with what I saw in my recent EDST 401 readings.

Another stop-and-think moment I wanted to highlight was when the reading started to question heavily on "why" things were, specifically in the literal curricula and its null counterpart. An inquiry, reflective thinking was taking place - I could see these processes that were discussed in my EDUC 450 readings happening in real time, so there are more connections being drawn here. In the EDUC course, the reading was on being a reflective thinker and its necessity in order to adapt in an ever-changing world, and I could see these ideas being put to practice here. If we are in a world of markets, why is the required curriculum not including economics? Similarly, if we live by the laws of our societies, what is it simply relegated to being an optional elective that students could choose to take? It makes me think of when I went through the provincial curriculum in the mid-2010's; in my Law 12 class which was designed to be a survey course, we learned of the many types of laws, how laws and justice were carried out in our society, and more - all very practical things that would benefit everyone in our society! The funny thing is, I chose to take it because "the teacher was better", not because of some utilitarian curricular ideal. While I came for that, I stayed for the valuable, relevant lessons. While there is further room for inquiry in whether it should be a mandatory course, there are good points made in the article about injecting the ideas into existing mandatory courses in order to enrichen the curriculum.

As a final note, I have to applaud the article for making the three types of curricula clearly defined - we have the explicit one that comes to mind first, an implicit one, and a null curriculum which could be subdivided into explicit and implicit categories, much like what the article has done. It made me consider that indeed, the curricula can have ripple effects. Perhaps one policy or approved method of teaching has changed in the explicit curriculum - then the implicit things being taught would be changed too. With these two curricula changed, so too does the list of things falling under the null curriculum. Before, I only considered the explicit as "curriculum" while the others were entities undescribed by the same term. Thinking about those entities as curricula as well would serve us well in knowing the full picture of what is being taught and mandated, and lead to informed inquiries and decisions.

September 9, 2024 - Skemp Discussion Reflection

Today in class, I discussed with some of my classmates some questions presented to us regarding the Skemp reading.

Are the two kinds of understanding distinct/separable?

If I were to define relational and instrumental understanding, relational understanding entails depth and explores the 'why' something works, whereas instrumental understanding is algorithmic and/or procedural mastery. We came to the conclusion that they are certainly two distinct terms on a semantic level, but they are also closely related under the umbrella term of "understanding". They are not mutually exclusive, but complement each other. Instrumental understanding can be improved by relational understanding, and there are also cases of the reverse being true, especially in more technical disciplines. For example, you could learn all the theory behind the usage of a woodshop tool, but it only proves beneficial once you gain an instrumental understanding of how to use it.

Is there a "best" order to teach them?

I believe that this is very dependent on the content that we are trying to get students to understand. Let us take the case of learning the formula for a circle's area, which involves the number pi. Of course, the relational aspect is to understand why every term is where it is, whereas the instrumental aspect seeks to train students to apply the formula masterfully. One could argue that instrumental understanding is better to introduce first, as it builds familiarity with the concept that would make relational understanding easier to acquire. There is also an argument of whether the students have the base knowledge to understand what "pi" is before being taught the 'why it works' for the formula... though a counterargument would be that pi is a simple enough concept to teach together with the relational understanding of the circle's area, As an example of relational understanding being better to teach first, take exponent laws. Showing how one law is derived can be akin to teaching students how to fish, as they can derive the rest of the laws by applying the same methods.

What kinds of activities promote each one?

For instrumental understanding, conventional exercises come to mind first. The repetition of using the tools and formulas given is designed to build mastery in using them. Such repetitions may be administered through worksheets, homework and the like. Simple tool demonstrations (example questions) are another activity that may be used.

For relational understanding, I was able to come up with a few ideas. First, I noted the sparsity of proofs in high school mathematics, especially compared to university. One activity could be to get small groups to try and derive a formula on their own (such as the exponent laws, which are an easier exercise due to how the laws are derived similarly). Another set of activities deal with the tangible; matchstick exercises to visualize concepts, or graphing calculator playground labs (such as Desmos, which has functionality with sliders and variable setting). In high school, I fondly remember a Desmos lab exercise where we played with function transformations to see how a graph changes as certain values are changed. This gave me a relational understanding of transformations, whereas simply knowing that "c = horizontal movement" and "a = vertical stretch" would've been helpful, but shallow.

How to assess understanding?

For instrumental understanding, I turn to the conventional once again. Much of conventional testing is what I like to call "applied regurgitation"; while the student doesn't simply repeat answers with exact values, they are repeating the steps needed to use their instruments properly. In a vacuum focused on instrumental understanding, this is a fine way to assess.

However, relational understanding has been considered more challenging to assess. My discussion group seemed to agree that talking and questioning the student was the best medium to assess relational understanding. Regarding the implementation, I had an idea that I liked: if mathematics is a language, then why not take a page from how modern language courses are taught? I recalled my old French 12 class where the teacher included a "casual conversation in French" section in the final exam process, and was scalable with a class of thirty. As such, the idea was this: once in a while, why not allow students to show their deeper understanding in conversation? We could set a rubric for what sort of things we look for in our conversation, and assess accordingly not just on their specific relational understanding of whatever is covered at the time, but also their general capacity to mathematically reason.

As far as "understanding" in a general sense goes, this is a greyer issue that depends on the set student learning outcomes. Do we only want them to understand instrumentally? Do we want them to have deeper understanding on top of their instrumental mastery? Once this is decided, then the reflection loops back to the previous two paragraphs on how to assess specific types of understanding. There is also a question of how much to weigh each type, but to me that is again a question of SLOs.