Author: mvinayag

My father’s abilities to number-crunch was my first fascination. Without any formal education, he was able to play around with numbers. This fascination led me to be curious of how mathematics functioned in a society in which most were illiterate. Later on, I started looking for patterns or noticing relationships between ideas without actually intending to do so.

For me, learning mathematics has always been frustrating, in a good sense. It was more like why didn’t I think about it this or that way to find a solution. Whenever I was stuck on problems, I kept on attacking the problems from different angles. In other words, these frustrations never led temper tantrums. These frustrations made me more curious about solving problems.

I think that I liked studying mathematics, partly because of my mathematics teachers. Initially, many of my teachers were not experts in mathematics, but their enthusiasm towards teaching us the basics was contagious. Many of us loved our teachers more than the material that they were trying to teach.

• On Becoming a Reflective Teacher
• Tact in Teaching
• UBC Orchard Garden
• My Mathematical Past
• Math Art Project
• Eisner’s Three Curricula
• Battleground Schools
• Math that matters: Beyond Pizza Party Math

Our team’s mathematical art project was based on John Critchett’s algorithmic digital artwork. We chose to explore his spirolateral artwork, partly because it appeared to be simple and elegant. Also, what caught our attention was how simple geometric objects were used to generate stunning artwork.

We employed several problem-solving strategies when we were attempting to recreate and extend Dr. Critchett’s artwork. Yes, I think that we methodically followed most of the strategies identified by the so-called “seven stages of problem-solving.” These seven stages are a) identifying the problem, b) understanding the problem, c) finding a strategy, d) sorting out the available information, e) analyzing the problem, f) keeping track of progress, and g) evaluating the findings. It was definitely exciting to recreate the original artwork. At least for me, what was even more interesting than the actual problem-solving processes was the dynamics involved in creative problem-solving activities. 

Our team went through several discomforts. These ranged anywhere from trying different techniques of solving the problem to dealing with the frustrations of not a finding a solution to thinking about abandoning the problem on hand to finally overcoming the frustrations. Despite being teaching candidates who would have solved numerous problems over the course of our mathematical lifetime, our mental discomfort was evident when we were struggling. From our past struggles with mathematics, we know and have experienced the joys of overcoming these struggles when we finally solve problems.

I often wonder how easy it would be for novice learners or chronically struggling students to abandon learning mathematics altogether, because they might not have had enough opportunities to add layers of mathematical success. This brings me to a question that has popped up now and then over the years. That is, as teachers, how do we identify and present problem solving activities that are interesting, engaging, and challenging so that it makes sense and meaningful connections to students’ lives outside the classrooms?

I think that the actual mathematics involved in recreating this artwork is trivial. What was interesting to note was how geometric objects in conjunction with simple mathematical rules generate complex geometric patterns. I think that the potential mathematical learning involved in this particular activity would be to pose questions that go well beyond just constructing spirolaterals. Some of these questions can be of the following form: 

Why don’t you devise your own rule to generate spirolaterals?
Can you create new and unusual geometric patterns?
Can you design a routine (algorithm) to generate spirolaterals?
How would you prove that your spirolaterals will form closed loops? 

• On Becoming a Reflective Teacher
• Tact in Teaching
• UBC Orchard Garden
• My Mathematical Past
• Math Art Project
• Eisner’s Three Curricula
• Battleground Schools
• Math that matters: Beyond Pizza Party Math

For me, the first ‘stop’ moment was when Eisner was talking about the culture of schooling. On page 83, Eisner says, “one of the first things a student learns–and the lesson is taught throughout his or her school career–is to provide the teacher with what the teacher wants or expects.” I think that Eisner seems to suggest that students’ learning in school is tied down to satisfying teachers’ expectations. I am not sure if I fully agree with this statement. This statement appears to be more like one of his many assumptions about schooling. Eisner has also published an article titled, “Questionable Assumptions About Schooling”, in which he lists twelve assumptions, taken for granted by education policy makers. I realize that Eisner was a leading scholar in art education, qualitative research, and curriculum reform. Also, he was a well respected and leading authority in educational policy and development, receiving numerous awards in recognition of his thirty-plus years of scholarly work and contribution towards educational reform. As an aspiring teacher, this statement seems to be troublesome. That is, to imagine that every interaction between a teacher and student in a school setting is somehow linked with the Hawthorne Effect. This may be the case in some instances, but I am skeptical about this statement partly because there appears to be no evidence to backup his claim.

Another ‘stop’ moment occurred to me when Eisner was talking about appreciation of other modes of thinking. On page 101, Eisner writes, “Learning is a humble thing compared with teaching. To teach puts one in a superordinate position, to learn in the position of a subordinate.” I also think that learning is a humble thing when taught by others. For me, the concept of learning is intangible, much like responsibility, confidence, hope, fear, motivation, etc. I do not necessarily think that teachers are the primary sources of knowledge, in this day and age, who somehow transfer their knowledge to their obedient students. I think that learning can take place anywhere with or without anyone physically teaching. Having said this, I think that being an independent learner or the process of learning independently is not at all humble. To the contrary, I think that the process of independent learning is exciting, empowering, and awe-inspiring in itself. In fact, I would argue that independent learning could be an indication of an individual’s intrinsic motivation and a growth mindset towards accomplishing something. Here, I am not limiting independent learning to the abilities of an individual learner, but can also involve groups of learners working collectively.

BC’s new curriculum states the following in their website: a) graduation is a major step from childhood to adulthood, b) our children need to be prepared to succeed as adults when they walk across the stage and get their graduation certificate, c) it is our job to prepare all children for success in whatever life path they choose, and d) aims to connect students with the skills they need to succeed in their next chapter. These statements, more like advertisements on a billboard, are not surprising. On page 88, Eisner alludes to some of these kinds of offers to the community from the schools. Eisner says, “the school offers to the community an education menu of sorts; it advertises what it is prepared to provide. From this advertised list, students have, at least in principle, an array of options from which to choose.” In BC’s case, the curriculum offers the prospect of graduation, future success of students, and skills that may be needed in the future. I am wondering how are teachers supposed to accomplish these things without significant opportunities for personal and professional development. On a positive note, as part of the core competencies, the new curriculum seems to promote deep learning and life-long learning by developing all learners’ intellectual, personal, and social and emotional proficiencies. I think that Eisner would be smiling in his grave, because of the positive changes that are taking place in schooling the young. To start with, these curriculum documents had input from practitioners. 

• On Becoming a Reflective Teacher
• Tact in Teaching
• UBC Orchard Garden
• My Mathematical Past
• Math Art Project
• Eisner’s Three Curricula
• Battleground Schools
• Math that matters: Beyond Pizza Party Math

The following quote from page 394 is interesting in several ways. “… anyone who is good at math should be able to succeed…” I don’t know what that success would look and feel like. This quote reminds me of a somewhat similar quote by George Bernard Shaw. One of his many contentious quotes that rings loud and clear even today is “those who can do, those who can’t teach.” An apparent insult, whether intended or not, to the whole profession of teaching. Although, I don’t know the context under which he would have said this. I think that quotes like these make good cannon fodder, raising further tension between conservative and progressive views in regards to mathematics education in public schools. I am not sure if we should dismiss Shaw’s thought on teaching outright. Would it be worth having a meaningful conversation around this quote? I think that we ought to have a conversation around this, partly because many of us would have found success under a system that we are trying to move away from.

Another interesting quote in the paragraph, bleeding from page 395 to page 396, is also interesting. “Inquiry could be messy, uncertain, and unsettling compared to the certainty of accepting pre-made facts based on obedience to the teacher’s authority.” It is not clear why inquiry-based learning would be messy, uncertain, or unsettling. Who are the stakeholders that would feel this way? Would it be unsettling for the students or the practitioners? I think that I would argue that it would be unsettling to the practitioners, partly because inquiry-based learning may be associated with many practical difficulties. I think that some of the difficulties involved in inquiry-based teaching and learning are time constraints, variable learners’ abilities, classroom structure, beliefs and attitudes, resources for professional development, and sometimes insufficient knowledge of the practitioners.

Within the same paragraph on page 396, Dewey calls for the “development of high quality mental processes and a scientific attitude” for developing scientific and democratic thinkers. I find several issues with this. First, who was going to develop tools and techniques for these high quality mental processes? I assume that Dewey was calling on educational researchers to take on the job of developing these resources. With this approach, I find that it would quickly be perceived as a top-down approach to addressing educational issues. In one of his articles, Eisner says that in the classrooms, “teachers are the kings and queens” and not the researchers. So top-down approaches may not be fruitful in addressing education issues. Second, I am not sure what Dewey means by “high quality mental processes.” Was Dewey referring to the higher echelons of Bloom’s Taxonomy? Finally, why does Dewey focus on developing scientific thinkers? Why not artists, philosophers, or, for that matter, teachers? Is that because the real currency of good thinkers possess scientific attitude and knowledge? Can we also think of this as, yet, another attempt to promote the null curriculum as invalid?

I will stop my blabbering now.

• On Becoming a Reflective Teacher
• Tact in Teaching
• UBC Orchard Garden
• My Mathematical Past
• Math Art Project
• Eisner’s Three Curricula
• Battleground Schools
• Math that matters: Beyond Pizza Party Math

By DAVID STOCKER

It’s very interesting that the author states that “all material carries bias of some sort” in response to a teacher’s comment. The comment states that “the material seems biased to me – sort of left leaning.” It’s interesting to me because there appears to be a segment of mathematics teachers who believe that the act of teaching mathematics is neutral. This comment reminds of a required reading on race and diversity by Benedicta Egbo for another class. Egbo states that “some teachers often make claims like ‘I do not see colour; I love all my students; I treat everyone equally; as far as I am concerned, everyone is the same’.” Egbo goes on to say that these claims may be well intentioned, but they are practically unrealistic “since race is often the first thing we tend to notice about people.” There are so many things about teaching mathematics that can be considered as neutral. However, not everything is neutral about teaching mathematics. Indeed, if we were to assume that teaching mathematics is a neutral activity, then aren’t we negating obvious differences among students’ learning in mathematics classrooms?

From my early educational experiences, I don’t believe schooling is neutral at all. And, in particular, mathematics education is far from neutral. For instance, when posing problems, in order to understand the problem, one needs to understand the underlying context of the problem. And not all learners will be able to understand the context of the problems. I remember when I was growing up, we were using photocopies of mathematics textbooks that were brought from the US or Europe. Word problems from these textbooks were some of the hardest problems to solve, because, for example, we had no idea what a pizza was or why would someone bother to divide a pizza into various pieces. At one point or another, I can also remember that those of us who able to understand mathematics better than others were treated like our mathematics teachers’ pets. So I will be very surprised if a mathematics teacher says that teaching and learning mathematics is a neutral act, unless the term ‘neutral’ is interpreted as a long continuum.

I think that the author’s intention in writing this book was to invite teachers to become learners in the classrooms. That is, the author was calling on teachers to wear multiple hats in the classrooms when engaging with the students, rather than being viewed as experts at all times.

• On Becoming a Reflective Teacher
• Tact in Teaching
• UBC Orchard Garden
• My Mathematical Past
• Math Art Project
• Eisner’s Three Curricula
• Battleground Schools
• Math that matters: Beyond Pizza Party Math