EDCP 342 Group Project

Group members: Pla Sey, Leo Lee, Jarrad Fjelstad. The art piece is Little Wire Quadrics by Amanda Taylor Lipnicki.

We first needed to figure out what materials we needed. Rebuilding the pieces using a wire base felt obvious, but instead of copper wire, we used aluminum wire as we thought it would be easier to manipulate and solidify. To connect pieces together, we used magnets and silicon tape. It still took some time to get used to the materials:

Pla thought of interesting ways to spin and form the wire:

Leo eventually managed to create a hyperboloid:

Then we quickly followed that with a paraboloid and cone:

We had to pay close attention to the orientation and position of the magnets:

 

Though small, the magnets were powerful, and it was a challenge keeping them separated for use:

Taping tiny magnets to wire was tedious and often ineffective, so we abandoned them and decided to try just using the tape:

The tape was sticky enough for the pieces to hang together. 

At this point, we also discovered that none of us knew how to knit or crochet, so we decided for a more minimalist presentation, with the pieces’ frames visible.

The pieces starting taking recognizable form: 

The paraboloid-in-pieces takes its final form:

As does the saddle:

For the interactive part of a project, we are planning on presenting the pure-wire replicas and asking students to see if they can identify the functions that create the particular 3D graphs, either as one-variable functions (for cross-sections) or two-variables.

Then, we will present the 2D tape/magnet/wire pieces in a pile and see if students can rebuild the figures and identify the functions. 

Response to Eisner

Eisner’s proposed division of a school’s curriculum into three conceptual parts will be helpful for me going forward. The contrast between the explicit curriculum (what is being directly taught), the implicit curriculum (what is being taught about proper comportment indirectly through social and disciplinary forces), and the null curriculum (what is not being currently taught but could be). I’m curious about the process by which decisions are made about what belongs in the explicit curriculum and the null curriculum. Does Eisner go too far in saying that the decisions are mostly the product of status quo bias and “largely unintentional”? Or are current curriculum designers making pragmatic decisions about what are more “useful” subjects, or what stakeholders they are accountable to believe to be useful? Eisner argues convincingly that subjects in the null curriculum have much to offer students and that they can help them navigate the increasing complex and often overwhelming modern world. The challenge is of course convincing people that this is the case, and that resource- and time-constrained schools should make room for material from the null curriculum. I am reminded of the concept of opportunity cost in economics, which states that the cost of a decision isn’t just the direct costs required to enact the decision, but also the benefits of forgone decisions. The opportunity costs of swapping out one class for another, or emphasizing one and deemphasizing another, are significant, making decisions that much more challenging and the tradeoffs that much more sizeable. Do we think it is likely that stakeholders will allow, for example, reading and English to cede bandwidth to a lesser-known subject like anthropology? I think many parents would also object to this, and not necessarily because of traditionalism. Perhaps a solution is to identify the key skills that subjects in the null curriculum offer and incorporate them elsewhere in the current explicit curriculum. For example, it seems like media literacy could be taught in another course. 

I am still getting familiar with the BC Provincial Curriculum, but it does seem like there are to efforts to teach material that may have been previously consigned to the null curriculum. My understanding is that Indigenous studies plays a much more prominent role, not only as a subject, but also as a tool for gaining a deeper understanding for or appreciation of what may have been previously considered unrelated subjects. Though this example does illustrate that moving things from the null curriculum to the explicit can be politically challenging and take time.

Response to "Battleground Schools"

I still see jokes now and then on social media about new math. The scenario involves people finding bizarre notation and symbols out in the real world, taking a photo, and commenting something along the lines of “I should have learned new math”. Never underestimate the ability of a small group of French people to change the world. I found the argument that new math was neither strictly progressive nor conservative interesting and convincing, and a reminder that many reformers and reactionaries don’t fit neatly in one ideological category.

I appreciated the long history of modern education, showing how different traditions respond to new entrants that aim to displace or reform them. Motivation for changes in education policies and curricula has historically often been due to perceived threats, especially external threats. Dewey and progressive reforms worried about training workers for the increasingly complex industrial revolution economies and democracies, and the rivalry with the Soviet Union led to the space race and massive increases in educational spending and attendance. I believe the Babylonians fit this mold, as much of their advances in math were aimed to further their knowledge of astronomy, and in turn, astrology. So, in a sense, they were still responding to a “threat”, but in this case it was the menace of vengeful or unpredictable deities. 

The prospects of educational reform are dependent on capacity. Societies may lack enough trained teachers and related resources to accomplish progressive or other goals. How do societies develop the human, physical, and financial capital required to accomplish ambitious changes? Many times, the trickiest part is getting the process started. I can imagine a scenario where once a society has reached some critical level of investment, these reforms create a virtuous cycle: investments in current students produce highly educated and capable workers who then are available to become future teachers themselves. 

Favorite and least favorite math teachers

A favourite math teacher of mine was the teaching assistant for an advanced calculus course I took as an undergraduate. The class was challenging, made even more so because I had recently decided to try to understand subjects, especially math, more deeply than I had previously. This teacher was supportive of me in this, and I remember spending many hours working with him between classes, asking him deeper questions than he would generally receive during class. Probably the best part of working with this teacher is that he knew exactly when to intervene to provide help and when to allow me to struggle to find the solution myself (something I have trouble with a lot as an educator). I think by the end of that course, I really started making the connections needed for fluency in calculus.

I don’t think I had a bad teacher in high school, but I wasn’t paying a lot of attention to their quality. I only remember one name among them, and don’t have strong memories associated with it, which is likely a sign that these teachers didn’t leave impressions, either good or bad. I don’t recall a specific lesson or moment from the entire four years. Much of this is my fault as a student: math was not a passion for me at the time, and I was only interested in receiving good grades. Contrasting my experience in high school and college makes me wonder how much of finding a “good” or “bad” teacher depends on the student and where they are in their intellectual development.

Locker Problem Response

 

Initial reaction to Skemp's "Relational Understanding and Instrumental Understanding"

 The terms “relational understanding” and “instrumental understanding” describe phenomena I have recognized, and I suspect most educators have. In my math teaching, I have called what Skemp terms relational understanding as “deep” or “fundamental” understanding, and “instrumental” as “shallow” or “utilitarian”. For example, there are specific tools used to solve equations, the choice of which can depend on the relationship between the variables in these equations. These tools can feel disconnected. Teaching factoring in the context of polynomials, I like to emphasize to students that there are certain advantages to having expressions represented as products. Perhaps this is an example of relational understanding. However, often students don’t care about this.  Many want the specific tool needed to solve the problem on the page, and then the next tool to solve the next problem. Some are interested in relational understanding, or at least curious about it, but they seem much fewer in number, which makes me think the math mismatches described on page 4 will be a major issue and that the mismatch will be heavily weighted toward students who prefer the instrumental approach. 

Skemp’s argument on page 9 that relational understanding is “easier to remember” first struck me as a surprising, as students may argue that they would rather memorize tricks than spend the time required to get comfortable with broader, potentially more abstract principles. However, the latter approach--few fundamental points of knowledge and understanding how to vary them--requires less cognitive storage than memorization of highly specific tools. I found the argument number six on page ten interesting; I did not think it would be easy to measure the level to which relational and instrumental knowledge can be thought of “goals in [themselves]”. I’m curious about the studies in question, and although I don’t have any specific reasons to doubt whether it’s applicable to math teaching, I would like to see whether there have been studies conducted on that topic since the publication of the text.