Using Gale’s lecture, Poirier’s article, and Bear’s article, identify at least three ways in which Inuit mathematics challenge Eurocentric ideas about the purpose of mathematics and the way we learn it.
While there were many ways in which Inuit mathematics challenged Eurocentric ideas, I feel that the three strongest challenges come in the form of; number base, contextual understanding, and communication of mathematical ideas.
Before reading these articles and watching Gale’s lecture, I was actually aware of the differences in base between different cultures. What I find so interesting about this aspect of mathematics is that at its core, base doesn’t matter. In my EMTH 300 class we have explored counting systems other than base 10, and in my MATH 221 class we have converted between bases. All of the math that you would ever have to do is still doable, its just that you need to adjust the way you think about it. In some ways, I think that base 20 could make more sense than our Western are 10 system. Counting in base 20 could give students an easier understanding of large numbers for example. While counting in base 20 feels like something that goes against all logic and should not work in the sense of Eurocentric math, its interesting to find that concepts such as √2 and π still hold! This is further proof that the base does not matter, and support the idea that having different cultures count using different number systems does not put anyone at an obvious disadvantage.
As I am furthering my mathematical education I am realizing that removing the context from math is damaging. The idea presented in Gale’s lecture, Poirier’s (2007), and Bear’s (2000) articles about mathematical context is something that I feel should be brought into all classrooms. While speaking to the cultural link that mathematics has Poirier (2007) also speaks to the need for context stating: “different cultures have developed different mathematical tools according to their needs and their environment” (p. 54). If we teach mathematics as a context free abstract thing, then we remove a student’s ability to understand math’s place in our culture and community. Taking that further, if we remove the cultural context that helped to develop the mathematical understandings we are teaching, then we aren’t teaching the whole subject. Given this reasoning it is easy to see that continuing to teach math using Eurocentric methods not only lessens a student’s ability to take in math they are being taught, but also disrupts and directly challenges any cultural understandings of math that they make have brought into the classrooms with them.
Communicating mathematical reasoning is one of the toughest parts of learning math. I have been working as a math and science tutor since starting at the University of Regina in 2018. I couldn’t tell you the amount of times that I have asked a student to explain their reasoning and then have them look back at me with blank stares. What I have noticed is that when you expect students to use ‘math language’ to explain a concept they freeze up. If you ask, are those angles complimentary, supplementary, or corresponding they have no idea. If you instead as students to explain the relationship that angles have, they will eventually fall back on the above definitions without even realizing it. Extending that thought to students of different cultural backgrounds, and taking into consideration the different ways in which those cultures explain numbers, it is easy to see that no one way of explaining a math concept is better than the other. At times, it could even be said that cultures who have greater spatial awareness will excel at math problems other students would have a hard time understanding.
Which ‘single stories’ were present in your own schooling? Whose truth mattered? What biases and lenses do you bring into the classroom? How might we unlearn/work against these biases?
Looking back at my years in school, one ‘single story’ seems to have run all the way from elementary school to high school. This single story is that of the First Nations, Inuit, and Métis (FNIM) peoples of Canada. Although this isn’t the only ‘single story’ I was presented with, it is the one I would like to speak about here. What I have come to realize is that FNIM history, when it was taken up, began with the arrival of Cabot and Cartier, and ended with the closure of residential schools. This representation is offensive in that it doesn’t acknowledge the fact that FNIM peoples inhabited North America for millennia before European contact, nor does it acknowledge contemporary FNIM issues. This lack of representation leads students (myself included) to view FNIM people as simply a portion of Canadian history, as opposed to traditional stewards of the land. This reductionist take on FNIM history also framed them as something irrelevant to the contemporary understanding of life in Canada, and what it means to be a Canadian.
In the case of this ‘single story’ I feel that settler Canadian’s truths are the ones that are deemed to matter. Unfortunately, as is the case in many schools, my school had a primarily white population, which made this injustice almost impossible to see. This situation — much like the topic of last weeks post — serves as example of how important it is for people to be introduced to counter narratives in school. If we go all those years only hearing more of the same stories that we hear at home, see on TV, and read in books, we never have to grapple with the fact that we may not be seeing the while story.
Being a middle class white cisgendered heterosexual male, I have been dealt an extremely narrow lens. In the case of biases, there are definitely things I grew up with that I am actively working to remove from my beliefs. I’ve never truly dealt with an injustice. Given this fact, my biggest challenge going into the classroom is going to be that I likely never will truly understand what students dealing with misrepresentation or under-representation are going through. I’ve never had to deal with that and so empathizing will be tough. I also know that given my narrow lens, I still miss things that are problematic. One quick example is the use of ‘guys’ when referring to a group of people. This is something that I used to do and never really think about. However, in 2018 the company that I worked for misgendered me in a few emails, and it bugged me. This little mistake pushed me to move to use the gender neutral term ‘folks’. Until I was in that position of being misgendered, I was unable to see how it could bug someone, and part of me worries that I will continue to miss things going forward.
I think that the unlearning process has quite a bit to do with the people you surround yourself with. If you are making a concerted effort to change your views, or grapple with your biases then you need to be working with people that support you, and aren’t going to persuade you to lapse back to the ways you are trying to overcome. I think we also have to be willing to put ourselves in uncomfortable positions. You have to be willing to ask yourself tough questions, and have really honest conversations with people about who you truly are. I also think that we need to realize that this is likely not a process that has an end. As you start uncovering biases in your thinking and widening your lens, you’ll realize that you have more to work on than you thought.
Given the email that Dr. Mike Cappello received, use this weeks resources to craft a response, being sure to address the following questions: What is the purpose of teaching Treaty Ed (specifically) or First Nations, Metis, and Inuit (FNIM) content and perspectives (generally) where there are few or no First Nations, Metis, Inuit peoples? What does it mean for your understanding of curriculum that “we are all treaty people”?
As much as I wish that all teachers and school staff took the topic of Treaty Education seriously, I know that in our current situation this is not the case. Unfortunately you appear to have been placed within a school that may not appreciate the full potential of Treaty Education. It may also be the case that your co-teacher and/or the school’s administration don’t understand the true intention of Treaty Education.
I would suggest that you approach your co-teacher and ask to have a conversation about what Treaty Education is meant to be about. I realize this type of conversation can be disquieting, just know that you have my support. Make sure to stress that Treaty Education is not about teaching FNIM culture, but is about introducing students to the treaties, and helping them to navigate their relationship with said treaties. It is also important to note that all people are subject to the treaties that they live within. Treaties are signed agreements between two parties stating the obligations, rights, and benefits received by members on both sides. As your co-teacher has made clear, they feel the lack of FNIM students justifies a lack of Treaty Education. Contrary to your co-teacher’s beliefs, all citizens of Canada are subject to treaties, and thus should be educated on them.
As for your Social Studies 30 lesson, I would suggest having a conversation with your class about where their ideas of standard of living come from — what influences have helped them to shape these opinions. You could create a lesson that explores the changes that standard of living have gone through in recent history. I would then draw parallels to pre-colonial history and what the standard of living was at that time (make sure to do some research yourself so as not to unwittingly propagate misinformation). From there you could take a look at the Treaty Education Outcome TPP8⁴, specifically exploring what promises were made in terms of standard of living, and whether or not those have been fulfilled. Additionally, I would suggest segueing to a discussion on standard of living under the context of the Treaty Education Outcome HC12³.
I hope that you are able to put create a lesson that fosters greater understanding in both your students, and potentially your co-teacher. Feel free to reach out if you are ever in need of advice in the future.
¹⁾ What are the three major theories of learning? Briefly describe each in your own words.
²⁾ Provide an example of these theories in a way your teachers teach (or taught) — as in, what was your experience in engaging with these theories? Which of the theories do you see yourself using most in your own classroom?
¹⁾ The three major learning theories are behaviourism, cognitivism, and constructivism. Behaviourism is built on the belief that knowledge is an attainable ‘thing’ that can be ‘given’ to you by someone else; knowledge is almost seen as a tangible physical object. Behaviourism doesn’t attempt to explain how knowledge is attained by a learner, often viewing the mind as a black box, that can be filled but ultimately not understood. These beliefs about knowledge lead to behaviourist lessons that rely on rote memorization and repetition of facts, figures, and patterns. These techniques are best used when trying to develop classroom management strategies, but often end up being employed in math and science classroom, often to the detriment of students’ full understanding. Cognitivism, while still viewing knowledge as an attainable ‘thing’, draws attention to what is happening within a learner’s mind while they are taking in that knowledge. Cognitivism is build on the belief that when learners encounter new information it is not instantly understood and assimilated into their long term memory. Information needs to be introduced multiple times, using multiple methods. There is also a strong emphasis on the scaffolding of knowledge — that is, reviewing relevant lessons from the recent past to help students better understand new lessons in the present. These techniques help students to draw connection between past knowledge, strengthening their understanding of new topics. Constructivism takes a very different view of knowledge. Instead of viewing knowledge as an attainable ‘thing’ constructivist theory views knowledge as something that is created by the learner. Constructivist learning theory takes a further step back and acknowledges the impact of social interactions on learning. Learning is not viewed as an individual act, but one that is influenced by the learner, the teacher, their peers, and the community itself. These beliefs about learning greatly alter the way that teachers who prescribe to constructivist learning theory approach lessons. While behaviourist theory would say that memorizing times tables is the best way to learn how to multiply, constructivist theory would say that playing with manipulatives in a group setting is the best way to understand multiplication.
²⁾ I saw a lot of behaviourist teaching theory when I was in elementary and high school. Behaviourist teaching techniques were especially prevalent in math and science classes which is kind of expected (though unfortunate). All of my elementary teachers had been teaching for quite some time, so I feel that the only teaching methods they really knew were behaviourist ones. The biggest example of this that I can think of is the way I was taught math. All that I remember from elementary math classes is long repetative assignments and ‘mad-minutes’. Looking back at my high school experience I definitely notice a change in the way that lessons were put together and presented. All of my high school math and science teachers were recent graduates (less than 3 years) so they were definitely taught different methods than my elementary teachers. This came across pretty well. I know that in my math classes, both teachers I had did a lot of reflecting back on weeks prior, or even referencing stuff from a year or more ago. This showed a shift to cognitivism, and looking back it did help me to understand more of the topic than just rote memorization. My high school physics teacher did a really old job of embracing constructivist theory. Every physics unit had a hands on aspect to it. Looking back, these hands on activities are the things that I remember the most vividly from school. One thing that I also remember is struggling with most of these projects. The groups I was in would often start with something really extravagant and then realize mid-build just how complicated the math involved with our project would be! But the lessons really stuck with me. As for my own teaching, I don’t feel like any one theory really is the best. I want to try and build lessons that use a blend of the three techniques. As pointed out in last weeks lecture behaviourist theory is best put into action when crating classroom management techniques, which I feel are super important. In the lesson plans that I have put together, and in the tutoring that I do privately, I try to incorporate as much scaffolding as possible. At times it is tough to know exactly what student’s have previously learned, but you can also scaffold lessons to every day experiences. I will definitely be taking a page from my physics teacher’s handbook when I start teaching. Her ability to create engaging hands on projects for each unit really made class more enjoyable, and the learning more intuitive. Her lessons are probably part of the reason that I am majoring in physics today.
¹⁾ What examples of citizen education do you remember from your K-12 schooling? What types of citizenship (e.g. which of the three types mentioned in the article were the focus? Explore what this approach to the curriculum made (im)possible in regards to citizenship?
²⁾ What does the approach we take to citizenship instruction in any in any given place tell use about that place? About what will curriculum makers value? About what kinds of citizens they want to produce?
¹⁾ My high school had a Student Leadership Committee (SLC) with several branches. As students we were able to join as many branches as we liked. I personally was part of two branches; one that dealt exclusively with volunteering within and outside of the community, and a second that dealt with planning and organizing our school’s spirit days. All SLC meetings were conducted during breaks, or after school. Although it wasn’t technically part of any one class, I feel like this is a good example of citizen education. As far as which layer of citizenship the SLC programs taught, I would say it was mostly the personally responsible, sometimes reaching into the territory of participatory, citizenship level. The truth is, while we were volunteering around the community, and even made the occasional trip to Regina (approx 2 hours one way) to volunteer in a soup kitchen on the weekend, we never went any deeper. We never researched why we were needed to volunteer, or even what was happening in the communities we were going into. While the experience is something that has stayed with me, and I have volunteered personally since, looking back I can see that a divide was definitely created between us (the volunteers) and them (the people in need of help. We didn’t take the time to look into what was actually happening and didn’t make any connections. I’m not at all saying the program should be axed. Maybe though, more due diligence could be taken when exposing students to these types of situations.
²⁾ I think that the place you practice citizenship education (i.e. the school, community, municipality, etc.) has a definite affect on the way that you have to approach certain topics. Because of that, the opposite is true. If you are teaching in a school that is deep in the heart of oil territory and your students decide that they want to investigate the policies regarding renewable energy for example, you definitely have to treat lightly. Not only are you possibly going to face backlash from the students’ families, and the community at large, but you will also end up being perceived as someone who is against the status quo, even if that isn’t necessarily a bad thing. The truth is, no matter where you are teaching there will always be hot button issues that are going to create tension. If that tension is met with push back from your administration then you know that your school doesn’t necessarily want to create justice seeking citizens, but citizens who take on more of a participatory perspective. If on the flip side the tension you create is met with support from your administration then you know they want to support students in becoming not only active participants in the community, but change makers.