Battleground Schools - Thoughts

While reading, I paused when I saw that Dewey’s ideas were implemented in more progressive teacher colleges and in some classrooms, though most kept with traditional methods. Upon seeing this, I had to wonder if this was at a time of decent teacher autonomy in classrooms, which allowed for an individual selection of methods? And if so, were those who employed it still “hitting targets" or getting students to the curriculum-mandated level of understanding of facts or skill? I was interested in this aspect because it would make it harder to see why other teachers, perhaps at the same schools as these progressive ones, would not have this method spread to them by sheer proof of efficacy. I wonder if this suggests that it was often an issue of one's ideals or principles that made them stand against implementing them, even if it was in the face of proven success.


I also later paused at the first introduction of the ideas and methods of New Math. I think I found it shocking that it became such a dominant movement given how much it seems to lie in contrast to the ideas of Dewey and the progressivists, without necessarily being born out of explicit dissatisfaction with a part of the progressivists movement, but rather by the outside impact of the Cold War. While the New Math movement still seemed interested in targeting understanding, the intent of targeting university level math topics seemed like a recipe for failure given the number of students that would find now personal value in that learning unless it was already a field of interest for their future career plans. I wonder if this also speaks to how the paranoias of the state can lead to ineffective practices in spaces like schools, and even other fields like health care systems or food industries etc, as a result of not acting on more well-founded or research backed reasons to modify their operations.


Lastly, when I read about how right-wing evangelistic Christian religious lobby groups tended to align themselves with the traditionalists, hoping to bring in alongside them views on antigay rights or pro school prayer, I found this significantly highlighted the purposes school serves beyond learning course content, similar to our previous reading on the three curricula in schools. Additionally, I feel that their support for this method of school only helped clarify the essential 'con' of this model, in that the goal of the traditionalists was the serve up information to children and expect total acceptance of the 'facts,' including ones related to social or religious values if these were to become part of the curriculum. I feel that there was an understanding by these groups that progressivist models that aimed to develop problem solving skills and have more student-centered learning would necessarily output students that were more equipped to question such ideas in classrooms.

Math/Art Project Group Description + Reflection

Group Members: Sarah Dicastri, Tiffany Gong, Kia Prezeau

Original Artwork: Sudoku without Numbers by Dru Horne and Shannon McKillip

We decided to remake and extend the original art piece using cardstock and markers instead of fabric, as none of us had experience with quilting. This proved challenging in that it was quite time-consuming to cut out every individual piece and glue them all together. Additionally, we needed to glue all background squares together in a way that was structurally sound, taking us 6 hours to make just the extended art piece. Coming up with a unique concept for each layer was also difficult to decide on, as we wanted to select distinct features that layer well on one another and still show the other elements beneath. We ultimately settled on adding features like coloured borders, coloured squares, and hand-drawn icons that could fit around our larger icons without blocking the background.

We first focused on extending the piece mathematically by scaling the canvas to a 7x7 matrix and layering 6 different mutually orthogonal Latin squares. We arrived at this number of layers by using theorem 7 as described by Ballif (2008), which produces 6 as the maximal number of mutually orthogonal Latin squares that can be determined from this matrix, given that 7 is a power of a prime. We also used this theorem to construct a full matrix of six layers in a numbered sequence, by assigning a number of 0 through 6 to each element in each Latin square to construct the full picture that ensured all entries were distinct. This part of the process required some care and could be a point of challenge for others interested in recreating, as any mistakes could result in duplicate entries.

We then extended the art by focusing on how we could tie it into learning from place, specifically by having one of the Latin squares be icons from Coast Salish symbols for different local animals (sources were cited in our presentation). This was beneficial both for ourselves to explore more local Indigenous art and artists, as well as to tie in the BC curriculum’s goal of incorporating Indigenous ways of knowing.

We designed our interactive activity to be a smaller/simpler version of the mutually orthogonal Latin squares that we constructed. Specifically, we created a 4x4 matrix with two fixed elements and a movable third that students can use to layer their own Latin square. We hope this can cement an understanding of orthogonality between Latin squares, as well as spark discussion on how many possible arrangements exist when they compare with other students. We decided on a simpler matrix to have confidence it can be completed during the short allotted time during lecture, while still giving everyone a chance to have hands-on experience with these concepts. It also brought the content difficulty closer to the high school level – as combinatorics is only introduced in Foundations of Math 12 – instead of the university-level math the original piece employs.

Our digital mock-up and recreation of original art piece:


Our progress pictures:

Our completed art piece:

Ballif, S. (2008). Mutually orthogonal Latin squares. [Lecture notes]. Department of Mathematics and Statistics, Dalhousie University. https://www.mscs.dal.ca/~janssen/4370/Orthogonal_Latin_Squares_text.pdf

Eisner on Three Curricula that all Schools Teach - Response

While reading Eisner’s discussion of the three curricula, one point that stood out to me was the idea that school prepares students for the kinds of jobs they will likely take on as adults, those which are often not stimulating or of any significant interest to the worker. I paused here because I felt a key point had been made which was then left uncovered for the rest of the chapter (justifiably so, as the focus is on the education system not society at large). In particular, I believe this touches on the larger issue that critiques of the school system and its “hidden” curricula do not lead us to many feasible solutions, as long as the general population’s material conditions remain poor. While we may feel that schools are wrongly churning out subservient workers, the real issue lies in this being a necessity as long as our societies are structured around needing workers who are complacent in being alienated from their production as they attempt to sustain themselves, given the current state of our economy and the ways in which we distribute goods or access to food, healthcare, and the like. 

Another point I found significant was Eisner’s comment on the lack of soft surfaces in schools for students. This made me stop and think back to my own time in grade school and I realized how accurate this statement was, beyond perhaps sitting on carpets in very early elementary years. Beyond Eisner’s suggestion that this is part of a curriculum that primes youth to one day unquestioningly work in highly sterile environments oriented only for maximum efficiency,  like offices or factories, I also found this realization quite appalling from a standpoint focused on wellbeing. Students spend so much of their life in the classroom, enough that the space should be designed specifically to be comfortable and safe, allowing their focus to be on learning and engaging. 

The current BC curriculum connects with Eisner’s ideas in that it attempts to outline parts of these previously undisclosed curricula by listing the core competencies students are meant to develop, beyond only learning subject-specific content. Skills like communication, creative and critical thinking, or social awareness, are now all explicitly listed as the general targets of the curriculum at large, which are meant to ensure students can be successful in their future discipline/industry or in social life/global societies in general. However, we could now further expand our conceptualization of ‘curriculum’ beyond the facts and skills being literally taught. We could begin to view how we structure the physical space students are in or how we structure their timetables and subject areas as another method of passing on “lessons” to students, as per Eisner’s ideas. For example, this could prompt productive conversations around how we set up physical classroom spaces to be less streamlined for maximum ‘efficiency’ or sterility and instead help students feel welcomed or open to learning and collaborating. 

 

My favourite and least favourite math teachers

When I think about a favourite math teacher, one of my university professors comes to mind first, due to their effort to build relationships with students and reframe our perception of math as a discipline. For example, this professor was very dedicated to making time for student questions and one-on-one or small group assistance on content or assignments, such as in office hours. This was tied to their foundational belief that math should be a collaborative discipline, yet often in grade school we are trained to believe it is a solo subject/task. They had thus noticed a hesitancy in students of previous terms to fully utilize office hours, especially if they felt they were intruding upon a prof's time, leading to unanswered questions or leaving the office before fully understanding. The prof noticed this happened if the physical space was not inviting (door pulled up, prof deep in work on their computer) or even due to a misunderstanding of "office hours" as simply referencing a time when a prof can be found in their office. As such, they had reframed this time as "student hours," making it clear it was time they had specifically portioned out to assist us, and left the door open and inviting for students to drop in. 

The precautions the prof took to ensure students were feeling supported in the class and were gaining a feel for math as a collaborative subject was very inspiring to me, even at that time. It made me think about ways I could bring in similar small but impactful details, like using "student hours" or doing more check-ins and group work to get a picture of students' comfort levels with certain topics and model doing math in teams. I believe it is particularly important for being a successful math teacher to have ways for students to reach out and discuss math, rather than treating math as a highly independent subject where confusion can feel isolating, as you wonder if everyone else knows how to do a question except you. 

A math teacher I did not prefer as much was from my time in high school. This was often a surprising opinion amongst friends because on paper this teacher was viewed as "easier" and thus preferred by students. Tests in this class were often exact replicas of study questions, simply with numbers changed, and each class followed a cookie cutter method of lecture, note-packs. and assigned workbook exercises. The issue for me was the ensuing sense that I was not really learning anything deeply or of any importance, nor was there any challenge that engaged you or helped you grow. It felt like rote memorization was what was being evaluated, where each unit had information told to us, without activities or student involvement, that could then be forgotten after the test. This was in contrast to teachers who tried to convey the underlying principles and skills that connect math topics across the term and which allow for applications of math to real-world scenarios to add value or context to the abstract. 

This did not mean the teacher was bad or unkind, but is simply a critique on how this style of teaching and testing reduced math to a flat and boring topic, disconnected from real world skills or concepts. When thinking about my own future classroom, it inspires me to ensure math class includes engaging activities, that challenge students to apply general principles to novel concepts to access a deeper level of understanding and experience math as something you do, not something you are simply told about. 

The Locker Problem - my notes/solution

I intuitively begin these sorts of problems by considering a smaller case, as this will likely reveal patterns that apply to the larger quantity in the question. I also tried writing out the same 10 case in different ways, as my first attempt did not capture/reveal the pattern as clearly as the second more visual attempt. In my notes I wrote out the case for both 5 lockers/students, as well as 10. The 5 case revealed little, and so expanding to 10 was necessary to find a possible pattern. When I saw that 1 , 4 and 9 were the only ones closed in the 10 case, I asked myself what these numbers had in common and immediately theorized that the pattern was locker numbers which aligned with perfect squares would be the only ones closed by the end of the procedure. 

However, this was technically only conjecture. I then had to consider an explanation for why perfect squares would essentially be "modified," while all other numbers of lockers would just end up open again, back in their original state. To return back to an original state would require an even number of students acting upon it, such that for each student that closes a locker, there is another that un-does this action. In essence, this meant all the students could be paired up as one doing and one un-doing. From here I realized that the perfect squares would not have this even number of students to undo the last closing, as one student (aligning with the number's square root) would act on the locker's state, without another student (a factor pair student) to undo their action. 

 

Skemp on Relational Understanding and Instrumental Understanding - Response

While reading, one point that stood out was how Skemp felt the key flaw in his football and soccer analogy was how students cannot simply “refuse to play,” the way the two teams can when experiencing the understanding mismatch. For me, this highlighted the disengagement or hopelessness youth can develop in classrooms when there is both compulsory attendance but also prolonged or unresolved confusion. Skemp’s later discussion of instrumental understanding being counterintuitively less simple because of its multiplicity of rules also made me pause, as it reminded me of an “axiom” I have heard in other disciplines as well: “Simplicity, rather than complexity, is the hallmark of design.” This often refers to the goal of building or solving something effectively but with the fewest steps or parts possible. I still feel that it shares an overarching idea with Skemp’s point, as they both suggest that unnecessary complexity, rules, or steps are a detriment to solving problems, rather than a sign of sophistication or a necessary evil of more advanced math problems. One other point I stopped on was Skemp's suggestion that assessment may contribute to the difficulty of teaching relational understanding. While I agree that speaking to students would likely uncover their style of understanding quickest, I do feel that traditional math assessments, such as tests, have attempted to probe for this. For example, students with only instrumental understanding can struggle with test questions that are even slight permutations of exercises they saw while studying. In contrast, those who have grasped the fundamental principles and their applications are more likely to have success, meaning these questions would have checked for relational understanding, though perhaps not to an ideal extent.

Ultimately, I tend to agree with Skemp’s perspective that relational understanding ought to be targeted in education, due to it being a deeper and more transferable style of understanding which can apply topic to topic or even between disciplines. It seems rooted in a concept-based style of teaching that can add value to students’ learning by helping them make connections throughout their education and be motivated by the content itself. However, I feel that Skemp’s treatment of the topic implies that these two styles of understanding are more separate than they really are. In particular, I feel that the two are largely intertwined in that an instrumental understanding represents the facts or content of math, while relational understanding includes the underlying concepts, which may not even be math specific, such as proportions or change. If we can teach in this intertwined way, where topic-specific content is the tool with which to grasp larger concepts (which will be useful for the next topic or even subject), then we may be able to stimulate relational understanding without giving up the benefit and utility of math “tricks.” 

Hello World

Hi everyone! Looking forward to working with you all!

A photo I took in Monet's Gardens in Giverny.