Is There a Geometric Imperative? by Dick Tahta

This weeks reading, "Is There a Geometric Imperative?" by Dick Tahta I found fascinating and challenging. It stretched my thinking past what I have experienced and thought of as Geometry and for that I am grateful. Tahta begins by talking about "time and space as basic areas of mathematical experience," and that "perceptions of the external world are internalised as mental images and these are the 'stuff' of geometry." I had never thought of mental images being considered as geometry, which lead me to believe there was a lot I needed to learn.

The main points made throughout the article are that geometry contains three powers: imagining, construing and figuring. I will endeavour to give you a glimpse into what each is.

•  Imagining - The power of taking something that is described by someone and seeing it in your mind. Each individual is unique and therefore will have a unique form of imagining.  
• Construing - The power of seeing something that is drawn and then describing what you saw. "All seeing involves a 'seeing', an interpreting of what our eyes are conveying to the brain." Having the context of a image/scene allows for a more accurate construing of it. Tahta acknowledges that it is a skill to take a two-dimensional picture and see it with three-dimensional depth, is something that should be practiced to experience improvement. Once again, individuals are unique and therefore construe images in different ways, or through a different "point of view."
• Figuring - The power of "drawing what is seen." This is obviously something that takes practice and should not be underestimated as a difficult skill. As with the other two powers, figuring can be viewed differently by different people given where they are when observing the object or situation.

I think that breaking apart geometry into three powers is extremely powerful. It is important to know that there is not just one main umbrella that fits all of geometry but there are distinctly different parts. Having said that, Tahta acknowledges that all three of these powers contain an element of personal difference and that the person engaging in the geometric activity will bring with them their own 'point of view,' and concept of 'space.' I have never thought of this when I have taught geometry. Tahta mentions the use of our semi-circular canals in our ears as an example of symmetry, which I find fascinating. This leads me directly into relating to the question asked this week about if I know enough about geometry to bring it into what we teach when it is relevant; I would have initially said yes, but I guess I would have to say no. I feel like I know enough but I do not think I look for it enough in the other subject areas I teach. I teach a unit on sound and I love the idea of bringing symmetry and the ear together! In teaching grade 4 we do a lot of work about seeing two-dimensional and three-dimensional objects in our environment which I hope is helping the students practice the power of construing. Relating geometry to the students' relatable 'space' is going to be a goal for me moving forward, as I feel geometry is extremely important and allowing the students' to see the applicability of it to their world at any given time I feel is imperative.

Learning from Learners: Robust Counterarguments in Fifth Graders' Talk about Reasoning and Proving by Vicki Zack

This week I read "Learning from Learners: Robust Counterarguments in Fifth Graders' Talk about Reasoning and Proving" by Vicki Zack. I was initially drawn to reading this article, given it's approximate grade level to the class I teach, hoping to make some connections or gain some insights. In this article, Zack uses her fifth grade class as a subject group for her research surrounding the "Chessboard Problem." Her students are well versed in problem solving, as it is at the "core of their mathematics curriculum." Zack has her students' follow a distinct process when discussing their strategies for problem solving; firstly, they keep track of the thinking in Math Logs, they then share their solutions to a partner, a group of 4 or 5 and finally, a group of half the class (thirteen students.)
Zack then moves on to speaking directly about the "Chessboard problem." The students go through three tasks:
     Task 1: Find all the squares (in a four by four grid given as a figure). Can you prove you have             found them all?
     Task 2: What if ... this was a 5 by 5 square? How many squares would you have? Extensions were      subsequently posed.  
     Task 3: What if this was a 10 by 10 square? What if this was a 60 by 60 square? How many                square would there be?
Zack provided the class with a Ted-Ross strategy that came up with the following answers:
" The answer for the 10 by 10 grid is 385 squares. So take the answer for the 10 by 10 square (385) and 10 x 6 = 60 so multiply 385 x 6 = 2310 and you have the answer for the 60 by 60. What would you say?"
There are five counterarguments that are found as trends when students are coming up with reasons why the Ted-Ross strategy does not work. I was quite confused in the article whether the counterarguments were presented for the students to use or whether their counterarguments were merely looked at to see if they fit the previous counterarguments found.
I like the idea of the students' looking at a possible solution and trying to diagnose why it may be wrong. I think this shows true understanding of the task and may lead to possible correct solutions. A quote that resonated with me was "The students who have generalised a rule after testing it, use what they know and trust it." I believe this is so powerful; the feeling of self-belief! I do not think that if the counterarguments were provided that this would be as beneficial or meaningful.
I am also left wondering if the final task of finding the 60 by 60 squares is one that is too lofty. I would love to have a better understanding of time frame surrounding this task and if there were any students who were stuck on the 10 by 10 square for extended periods of time. My last desire is that there were work samples shown, I think it would lead to a deeper understanding of the students' thinking and process.
 

Our Culture, Geometrical Thinking and Mathematics Education by Paulus Gerdes

This week I read "Our Culture, Geometrical Thinking and Mathematics Education" by Paulus Gerdes. Gerdes identifies the need of engaging Colonized countries in mathematics through culture. He makes particular note of Africa and the fact that although there has been more attention to education in the country, there are still real concerns with regards to class sizes, qualified teachers and accessible information. A hasty curriculum has been implemented to the developing countries, leading to mathematics being seen as useless unless you intend on pursuing education past high school. He quotes the president of the Interamerican Committee on Mathematics Education saying "... mathematics has been used as a barrier to social access, reinforcing the power structure which prevails in the societies (of the Third World)." Gerdes mentions results of studies that have proved that although children learn mathematics in real world scenarios in their every day lives, when they reach school, no matter how similar the mathematics skills are that they are learning, they are confused. "D'Ambrosio concludes that 'learned matheracy eliminates the so-called spontaneous matheracy," and that "the early stages of mathematics education(offer) a very efficient way of instilling the sense of failure, of dependency in the children." D'Ambrosio furthermore was able to acknowledge "the need for incorporation of ethnomathematics into the curriculum in order to avoid a psychological blockade."
Now for the part of the article I found most fascinating and wonderful! Gerdes identifies areas of the African culture that uncover "hidden or frozen mathemathics." He identifies ways in which to see growth in mathematical confidence through cultural-mathematics. He links peasants' houses to rectangular axioms, artisans' funnels to polygon constructions, woven button patterns to pythagoras theorems, and finally, the woven fishing traps and woven baskets to the soccer ball patterns, both using "regular polyhedra." Gerdes realizes that pupils can reinvent artisan techniques, consequentially, doing and learning mathematics, "only if teachers themselves are conscious of hidden mathematics, are convinced of the cultural, educational, and scientific value of rediscovering and exploring hidden mathematics, are aware of the potential of 'unfreezing' this 'frozen mathematics.'" He acknowledges the need to educate those who are educating others.
When reading, I wondered if this has been looked at as a way to engage the First Nations children in mathematics education. In grade 4, we teach a lot about the First Nations' culture and I would love to incorporate some of their artisan work into math.
Lastly, my biggest hope is that since 1988, when this article was written, changes have been made! These significant findings I feel could widely influence the populations of many countries and have a huge impact on societies and mathematical engagement.

In Fostering Communities of Inquiry, Must it Matter That the Teacher Knows "The Answer"? by Alan H. Schoenfeld

In this article Schoenfeld compares and contrasts two different post-secondary classes that he conducts: an undergraduate mathematical problem solving course, and a graduate level mathematics education research group. Schoenfeld sets the stage by acknowledging how differently his two courses are structured. In the problem solving course, he has control over what is taught, learned, and ultimately the steps the students take to get the answers; whereas, in his research group it is more student directed, he and the students engage in joint work together to find answers to both their questions and his.

Ultimately his main point is that he builds community in both situations which allows for engaged inquiry.  This could be easily seen for me in the graduate level mathematics course, where Schoenfeld often finds himself in an area of unfamiliar research and is working with his graduate students to discover new territories. I was more intrigued with how he did it in his problem-solving course. The three main things that Schoenfeld acknowledge as crucial are:
"a. There is a common understanding that we are all seeking a particular kind of knowledge, and that while some of us know more than others, 'answers' are not generally known in advance.
b. The real 'authority' is not the Professor - it's a communally accepted standard for the quality of explanations, and our sense of what's right ...
c. There is a feeling of trust, in that we must feel free to have our ideas (and not ourselves) compete ..."  

I have always realized the need to build a problem solving environment in the classroom where the students feel it is safe to explore, experiment, and ultimately work together to find the solution to problems, but I had never thought really thought of my role as the teacher in the end when we had developed that community. I love the idea that the students ultimately don't need the teachers' authority to tell them if their answers are right or wrong, that they will be able to "internalize the standards of mathematical judgment for themselves." To do so, Schoenfeld makes reference to John Mason's framework: "First convince yourself. Then convince a friend. Finally, convince an enemy."   Schoenfeld goes into great details of his ability to teach and release the students safely into this community atmosphere. The problems he chooses are clearly purposeful and allow for further questions when the "initial answer" is found. I would love to ask the author how he would approach the student who is reluctant to find multiple solutions or defend their answer. Ultimately, I guess you are hoping that the community atmosphere will prevail, however, I have found that it can often take only one to ruin the atmosphere in the room.

Lastly, I am left excited by the fact that lots of schools are currently making a conscious switch to trying to do more mathematical problem based learning and I feel that if this inquiry starts at a young age, the sky is the limits with where their thinking may go!