Hewitt Response

(I'm all over the place in this response, I apologize. I had a hard time maintaining a continuous stream of thought for this one.)

I appreciate that Hewitt makes a distinction between the arbitrary and the necessary. Sometimes, I have the initial reaction as the author did to say, "Because it just is" when asked about an arbitrary point in mathematics, such as positive rotations being counterclockwise. It's another convention decided upon within mathematics, and I remember one of my professors stating it as such. The one practical issue which arises from this article relates to classroom management. My understanding is that Hewitt prefers an inquiry-style of teaching over a direct-instruction one, as direct-teaching tends to create more overlap of teaching both arbitrary and necessary information, whereas inquiry-style learning tends to lean toward using arbitrary information and allows students to determine necessary information for themselves. Initially I wondered about students who find it sufficient to trust a result without proof, and how to use inquiry to engage them. However, I am learning that there is much more to inquiry than previously expected; I have also had the opportunity to see various types of inquiry work, including some IB (International Baccalaureate) assignments. One of the assignments involves calculating a side length based on sine ratio of an equilateral triangle located in a circle, followed by finding the area of the associated triangle, followed by finding the area of two other regular polygons in the circle, then generalizing to n-sided polygons. I would like to design a few assignments like that, just to get a better understanding of inquiry. I have not yet designed a lesson based on inquiry. A lot of my work tends to be teacher-focused; however, I am working on changing this.

Back to Hewitt's article; I also think that distinguishing between arbitrary and necessary avoids misuse of memory aids like, "two negatives make a positive". I've seen multiple abuses of the "rule", but if students were able to follow through with a brief proof, I feel that this could be much better understood. For example,
Dr Math provides an excellent explanation on their website:

Let a and b be any two real numbers. 
  Consider the number x defined by

      x = ab + (-a)(b) + (-a)(-b).

  We can write

      x = ab + (-a)[ (b) + (-b) ]       (factor out -a)
        = ab + (-a)(0)
        = ab + 0
        = ab.

  Also,

      x = [ a + (-a) ]b + (-a)(-b)      (factor out b)
        = 0 * b + (-a)(-b)
        = 0 + (-a)(-b)
        = (-a)(-b).

  So we have

      x = ab
  
  and
  
      x = (-a)(-b)

  Two things that are equal to the same thing are equal
  to each other, so 

      ab = (-a)(-b)

(Source: http://mathforum.org/library/drmath/view/64406.html)

Also, suppose a teacher is instructing part of the lesson, and a student asks why a certain rule applies, perhaps one which would have been covered years earlier. At what point can a teacher, if ever, say that something is "the rule"? I know as an undergrad, I was often extremely frustrated when a professor would say, "You should know this by now", because they rarely ever continued with, "But if you don't, here's where you can find how to do x, y, and z." Because of such experiences as an undergraduate student (whether in math courses or others), I tend to over-explain things when I teach, and I was flagged for it by my school advisor when I conducted my two-week practicum last year. I remember a sea of blank stares in a classroom when teaching part of the rational expressions unit; I had questions arising over why I needed to multiply both the denominator and numerator of a fraction in order to add it to another one (in a grade 11 precalculus class). Although I explained it on the spot, I also wonder about the remainder of my students; if I had waited to answer the question, I imagine the student would've understood how to simplify the remaining part of the question, except for that one line of finding common denominators. What about one student who looked absolutely bored as I was going through the explanation of how to add fractions? Next time, can I open that question to the class? Would that have been a better thing to do? Perhaps it would've maintained attention a bit better. 

Finally, Hewitt states, "A teacher taking a stance of deliberately not informing students of anything which is necessary is aware that developing as a mathematician is about developing awareness rather than collecting and retaining memories. Furthermore, this stance clarifies for the students the way of working which is appropriate for any particular aspect of the curriculum - the arbitrary has to memorized, but what is necessary is about educating their awareness." This makes me wonder: How many students know a single of the (over four hundred) proofs available for the Pythagorean Theorem? Yet how many can use it effectively throughout their high school mathematics career, onwards? Do students need to be able to explain every math result they encounter? I wonder, is there time to do so for every particular topic of study? How early on can you begin with inquiry in mathematics, or to acknowledge conventions? Wouldn't it be confusing to students to say in kindergarten when first learning numbers to say, "We use the Arabic number system because it is the most useful to use in calculations and we can establish place value based on the order of the numbers, but had we been in another country, our numbers would've been based on a placement of horizontal and vertical bars"? 

Locker Problem - Response

A school has 1000 lockers and 1000 students. On the first morning:

*The first student walks along and opens every locker.

*The second student walks along and closes every second locker.

*The third student walks along and changes the position of every third locker door.

And so on.

Once all 1000 students have completed this process, which lockers are open?

a) Briefly describe how you solved it?

In order to solve this problem, I created a chart on which I wrote "student #" and "locker #", and simulated a student running through the hallway and closing lockers over 10 trials and 10 lockers. After realizing that I couldn't draw any proper conclusion from this, I decided to simulate this over 20 trials and 20 lockers. I still didn't see the pattern, but upon recording the number of times each locker had been opened and closed, I realized that every so often, a locker had been opened/closed an odd number of times. In fact, these lockers remained open, and I found that their corresponding locker numbers were perfect squares. I found that the largest perfect square less than 1000 is 961, or 31^2. Hence, of the 1000 lockers, 31 lockers remained open.

b) Where do you think a student might get stuck?

I feel a student would get stuck right at the start of the problem, and have a hard time wrapping their head around what the question was asking. I know that I originally didn't read the question correctly, and it took me a second to understand that the number of the student corresponded to the lockers the student was closing. It was also very easy, when doing a test trial, to make mistakes of correctly marking the lockers which were open and which remained closed. It was also overwhelming, in general, that there were 1000 lockers in the problem; I was worried that I wouldn't find the pattern and be stuck for much longer creating more trials or abandoning the trial version altogether. I think they would likely get frustrated if they made mistakes in their logical deduction, because there would be no pattern emerging from the tiles. 

c) How might you assist that stuck student?

In this example, I really like the idea of manipulating algebra tiles; I might suggest using black for closed lockers, and white for open lockers. Because the problem is also initially intimidating, I would have the students work in pairs or even groups of three, especially because it is easy to make a mistake in preliminary reasoning. 

While algebra tiles work for kinesthaetic learners, it would make more sense for others to simply write down symbols to represent closed and lockers (e.g. X for closed and O for open), and this is the method I preferred. I found that it was much easier to catch my mistakes by writing down the trials than it would have been to simulate them with algebra tiles. This is especially true if there is no opportunity to work in a group, because a person may even forget the number of the trial they're finishing, and may feel an overwhelming urge to re-start the problem. Finally, if a student does not recognize perfect squares, then they may not identify that the lockers remaining open end up having a perfect square as their number; working in partners would help amend this. Finally, I would have the student write down the number of times a locker is being opened/closed in a table. It was only then that I noticed the pattern which existed in this activity.

d) What extensions could you offer students ready for a challenge?

In order to provide more of a challenge for any students who were willing, I would ask about what would happen if the students, after the 1000th student, repeated the process and ran down the hallway in the opposite direction. I could also have the student prove that the order of the students running through the hallway and which multiple the students represent doesn't matter, and that the same 1000 lockers would remain open after 1000 trials. 

Most Memorable Teachers

The teacher I remember most fondly was X.Y., my vector calculus instructor. He was different from most professors I had had before, because he put his heart entirely into the class, and his fun personality came through every lesson. Armed with blue Converse high tops, he always came into the class with cool, clean and efficient lessons. He rarely trailed off topic (except mathematically, once in a blue moon), he delivered the information effectively, and he insisted that everyone call him by his first name. People felt comfortable within the class (with sixty people in the room) to say his name in the middle of lecture and ask a question.

There are three things in particular I remember best. One of these was his odd sense of humour; he came into the classroom one afternoon and said that he had consumed old creamer in his coffee, and that he may need to run out of the classroom at a moment’s notice. At various parts throughout the lecture, he would cling to the table at the front of the room, with his eyes squeezed tightly shut, and mumble repeatedly “Mind over matter” while the class looked on in fascination.

The second thing I remember about X.Y. was his homework and grading policy. The standard independent work and academic honesty were expected. However, the final grade was marked on the maximum of one of two schemes:

           

1)      A combination of percentages of grades in the class, at different value for each
midterm, final exam and the quizzes (with the quiz with the lowest grade excluded from the total)

2)      The final exam mark, minus 10%

His homework policy was also interesting, and from a psychological perspective of variable-ratio reward, was brilliant. Every Thursday (homework or quiz day (mathematical “or”)), he would bring in an opaque cup and a handful of beads (black, green, yellow, and red). His collection policy, rigged in our favour, was as follows:

-          If a green bead is pulled from the cup, no homework will be collected and no

quiz will be administered

-          If a yellow bead is pulled from the cup, a quiz will be administered

-          If a red bead is pulled from the cup, homework will be collected

-          If a black bead is pulled from the cup, we must complete both the quiz and submit our homework

The last thing I remember about X.Y. was how comfortable his office hours were; some professors are absolutely intimidating. X.Y., however, was welcoming and answered any question you asked about the course. He was just a very human sort of teacher, and for this, I’m happy I was in his class!

------------------------

My other memorable teacher taught calculus at my old high school. Y.Z. did everything right; she used a variety of resources to teach her classes, including Brainpop and careful lectures. She helped us pool information, offered extra hours, and an email address where she could be reached any time. She was well organized and tackled every problem with a profound prowess. However, I didn’t remember her fondly. There was nothing in terms of her teaching style. In fact, I plan to do many things that were the same when it comes to my teaching. Some time before I saw her last, she asked me what I was hoping to study in university. I replied, “I’m interested in medical school, but I don’t know, things could change. I could even pursue a math major, who knows!” I was surprised to hear from her that I “might need to reconsider” because it was possible I “would have a hard time finishing a math degree”. I don’t remember the exact words, but that was the general tone. After changing my mind about medical school part-way through first year, I decided I would have none of her nonsense. I expect to graduate with a BSc in Mathematics in May 2014. While visiting my old high school to pick up some materials two weeks ago from a student I am currently tutoring, one of the other math teachers found out I was working on becoming a math teacher. She said, “Good for you,” and I burst out laughing when she said the next thing: “I should tell Y.Z.! I’m sure she’d want to hear.” With a bit of a giggle, I replied, “Yes, please do, that would be great!”

I discovered that day that there is more to teaching than just the tools and strategies a person uses. I can say without the least bit of hesitation, that I am very stubborn, so I will not take other people seriously when they tell me I can’t do something (outside of situations of authority, I mean). But, what if I had been more unstable in my identity? What if I couldn’t distinguish between a person who, in this case, abused her authority, and a person who was trying to give me a solid piece of advice? Perhaps she didn’t mean it in the way that she said it, and I’m sure she had some reason to say what she did.

Nevertheless, Y.Z. is a good reminder to me of the teacher I do not want to become.

Battleground Schools - Article Response

As intimidating as it is, I think that Dewey's philosophy of inquiry mathematics is an extremely useful way of learning mathematics, and the inquiry learning model is one that is gaining popularity in many other fields. Not only is a student engaging in inquiry studies experiencing mathematics in the field, but they are broadening their understanding beyond the instrumental to gain further insight in the topic. However, having been taught in an instrumental way for so long, I can't say that I have much experience with in-field, environmentally programmed, learner-focused activities. I wonder sometimes, however, how much of a balance must be struck between the two. After all, one cannot use tools they lack to solve a problem; these tools must be shown in some way. I feel that instrumental development is crucial that way; perhaps, the teacher could preface each class with an introduction and a short lesson on the topic before engaging in the inquiry activity?

It seems counter-intuitive to me that the New Math program did not use inquiry-based learning and focused so much on set theory and abstract mathematics. I would have thought that "rocket scientists", as it were, would have an excellent understanding of their understanding of astrophysics, and in this would have developed more than an instrumental learning style. Although there is a great deal of abstraction in upper level mathematics (as in set theory, which New Math embraced), at the level of importance at which students were being placed in the 1960s, it doesn't make sense that their overall, in-depth understanding as not being embraced. Perhaps it was believed to have taken too much time.

Image: scratch.mit.edu

I still don't understand where the dislike of mathematics actually originated, but I would suggest after reading this article, that it originated sometime during/after the "New Math" was implemented. After all, a mandatory mathematics curriculum which makes no sense to anyone, not even the teacher, would surely cause parents to dislike and even hate mathematics. Is it possible, then, that a child would grow up believing that mathematics is an evil to be abolished and an unnecessary skill, because their parents/guardians treated it as such? The one thing I would like to verify is the implementation of the New Math program in Canada. Was it implemented at all, or was it a reworking of curriculum kept only within the United States? No matter the case, it seems that if there are constantly battles within the math curriculum and development groups, then it seems as though the reform of math will not come from curriculum groups, and by my speculation will likely not come from the homes of students. The New Math argument that students will not necessarily be following a career in math or physics and hence needn't focus on mathematics is a predominant one still. The most recent articles in the Province and 24Hours newspaper further contribute to the mentality that theoretical higher education is not what will lead students to be hired and become "successful", whatever that means.

Squares on a Chessboard Challenge

On Monday, we were asked to find the number of squares in a standard 8x8 chessboard.

How many squares are there?

Consider a standard 8x8 chessboard. The maximum dimension of a square found on the chessboard is 8x8 squares, and the minimum dimension is 1x1 squares. Beginning with maximum dimensions and working toward the smaller dimensions, we find the following:

For an 8x8 square, there is 1 possible square (the board itself)
For a 7x7 square, there are 4 possible squares (the 7x7 squares originating on each side of the board).

All remaining squares are identified by finding the number of squares of a given dimension which can fit in each row and in each column. For a 6x6 square, for instance, it can fit into the upper-left corner, in the same corner but moved one square to the right, then another moved one square right from the previous, hence filling up all "horizontal" possibilities. The same can be done vertically. The number of possible placements are 3 horizontally and 3 vertically, so 3x3 gives all 9 placements for a 6x6 square. Continuing this pattern, we find that the total number of squares in this board is:

 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 8*(8+1)*(2*8+1)/6 = 8*9*17/6 = 4*3*17*6/6 = 204

1) From a student's point of view, what did you do to solve this?

I decided that I didn't want to count the squares, and I hoped that there would be a faster way. I began with a list of all the dimensions of squares so that I didn't miss any. I also didn't want to have to count them all, so I started off with the largest dimension and worked my way down, since I realized that counting a smaller number of squares (that is, finding a square with larger dimensions) would be easier than counting/finding the squares making up a square with smaller dimensions.

2) From a teacher's point of view,

     a) Where do you think students get stuck?

I think that without a systematic approach, it would be very easy to make mistakes in this question. It's not a difficult task, necessarily, to find squares, but to find all the squares would require a precise algorithm. I feel that students may get stuck if they don't realize that they can find squares with dimensions greater than 1x1. I also think that some students may want to find every square just to be sure of their answer, or that they can simply draw the squares that they think are the only squares present. They may be overwhelmed by the potential combinations and their placements on the board.

     b) What would you do to assist them?

In order to help students complete this question, it would make sense to remind them that there are squares with various dimensions, not just 1x1 squares. I would also create a chart for them to fill out of each dimension of the squares, such as:

Dimensions	Number of Squares
8x8
7x7
6x6
5x5
4x4
3x3
2x2
1x1
TOTAL

Dimensions	Number of Squares
8x8
7x7
6x6
5x5
4x4
3x3
2x2
1x1
TOTAL

I feel that it is easier to approach the problem if you start with the squares of larger dimension first, and then work to find the squares with the next dimension. It's also much less frustrating to find larger quares, since identifying the pattern (if it happens) will likely occur sooner.

Dimensions	Number of Squares
8x8
7x7
6x6
5x5
4x4
3x3
2x2
1x1
TOTAL

    c) How would you extend this task for students ready for more of a challenge?

For students who were not feeling challenged, I liked Andrew's suggestion of asking them to find the number of rectangles in a chessboard. One of the questions from graph theory which I encountered involved finding whether or not a path (a manner of visiting every desired location and coming back to the starting point) can be drawn on a chessboard. It turns out that this depends on the board, which would be an investigative activity.

Response to Thurston article - "On Proof and Progress in Mathematics"

In the field of teaching, I feel it is critical to communicate your ideas, work, and development. I would agree with Thurston that this is a shortfall of many teachers and professors. This paper shines a light on why students get frustrated and insist that they dislike mathematics. In particular, the issue of communication reminds me of being lost in Montreal looking for La Banquise. Although I knew my way around, it was difficult to get to my final destination, because the language being used by those around me was not well defined in an already foreign space. I was, certainly, frustrated (and very hungry). It was impossible to find, until someone spoke my language and directed me in a way I could understand. This was the ninth person I asked. My weak foundation of speaking French (European) was insufficient in a substantially different, but still French (Quebecois), environment.

Now suppose this is translated into teaching; if one in nine teachers can actually explain a math concept to a struggling student with a weak foundation in mathematics and in unfamiliar territory, then of course the student will insist they dislike math. It's confusing, and they're only being shown one way to get there, if they can understand the instructions at all! Introducing proofs to such a student, is by far very frustrating.

For this reason, I appreciate that Thurston showed the multiple ways of thinking used to understand a proof, especially visual and kinaesthetic. I think that as a math teacher candidate, I forget that my teaching is not meant for other people in my field, and that I may cause my students frustration in the future. Of course, this does not permit me to call factoring of quadratic trinomials "plus-ing and times-ing", as one of my tutorees decided to call it, but it does emphasize a need for clarity in math teaching. I think that re-inforcing definitions and using them often can be helpful. I feel like a personal word-bank assignment at the end of each chapter could also be useful, where students need to express a definition in two ways, e.g. as words or using a picture. Perhaps it could be virtual, such as in a video, etc, and hence allow them to use multiple intelligences? The one risk with this is that students would take much too long to complete it, but I imagine that creating an example and setting a suggested time frame would be useful.

When it comes to something more complicated, however, like a proof, I would say that it is up for debate whether or not proofs are critical in a high school classroom. I find it interesting that they have made their way into Foundations 11, followed by logic and set theory in Foundations 12. Are proofs part of the curriculum for any other course, or would this become difficult for teachers teaching math without a math background?

Response to Erlwanger (1973) article

Alexandra Bella
#74823097

September 9, 2013

Response to Benny’s Rules article

While reading this article, I was absolutely fascinated by his rules. He’s consistent in what he does, he initially seems to understand place value when working with numbers, and yet each number can exist as its own entity. I like that although most of his work isn’t correct, he still seems excited to show the rules that he is using. If this article was written in 1973, I can’t imagine the number of times he was probably singled out for being incorrect, perhaps even ridiculed. Many of these mistakes remind me of mistakes I constantly see when reviewing assignments, etc, for students I tutor, for instance:

2/3 + 4/5 = 8/15

This seems like a serious case of over-use of instrumental learning led to Benny’s operations in his math work. I wonder if technology today could help remedy that. After all, not having correctly explained concepts would lead to a misuse of instruments to demonstrate those concepts. Therefore, if Benny had more support so as to be able to re-see the operations being done(say, in an online video, etc.) and the explanations behind them, I wonder if he would have been able to recall better how to work with the math as well as the explanation behind the work he was doing. I can’t imagine how frustrating it would have to be to work with a student who had made up so many different rules. Is it ethical to make him un-learn them? Also, it seems absurd to me that he said he could show (I think it was)

2 + 0.3 = 0.5

using a visual aid. I can’t imagine how he might’ve misused the visual aids to do such a thing. Whatever happened to this student?

Response to Skemp (1976) article

Faux Amis reminded me that the word “gift” in German actually means “poison”, as I learned in my German reading class! From reading this article, though (more on topic), my entire high school math education experience was based on instrumental understanding. I did very well in math classes in high school, but it was moreso because I knew what formulas to use, what “type” of question was being asked and the methods to answer it, and the style of questions and what was expected of me. It wasn’t until I started working students with poor fundamental understanding of math (or, at least, I thought so – it was more the operational understanding of math) that I realized how difficult it was to explain the simplest concepts; sometimes, I still do. Most of the math websites I visit offer an operational explanation and method to solve problems, as opposed to relational. I’ve searched for some time for an explanation of the origins of the sine, tangent, and cosine ratio that stems beyond the ratios of a right triangle, but have not yet come up with a book that explains how they are derived.

Reading the section where the author describes transcription of music and instrumental learning…. No wonder so many students dislike math. It’s not to say that there aren’t more issues underfoot, like a weak basis for math and a lack of confidence in using the tools that they do have, but if all the math is for them is “a bunch of formulas” and “all those word problems”, then of course we have students who dislike it and vow never to take it again, cheering at the end of grade 11 because “they don’t have to take math anymore”. Relational mathematics, it seems, is difficult to teach, but it sounds more rewarding. I didn’t see the author mention it, but I feel as though relational mathematics stays in a person’s memory longer. This is nice, because I’ve heard the argument that “we don’t need to learn math” because “we can look it up online”. I find this disturbing.

When it comes to dislike of the topic, though, I’ve discussed this with many teachers. Some insist that, in fact, it comes from having teachers that have no formal training in math. I don’t exactly disagree, but I feel as though anyone may have a more instinctive understanding of math, not just a math teacher. Now, I do believe, that guardians/parents/family and culture can influence a dislike of math. I grew up in a family where my mum encouraged me to do math and tried to insist that math is a game. I think this helped! However, among parents of the students I tutor, I often have them say, “Oh, I could never do this, I’m so glad that you are here to help with it” or, “I was never good at math, I didn’t like it in school”. I don’t expect everyone to be mathematical geniuses. However, the attitude that they “can’t do something” and therefore “didn’t like it” or were afraid of it is likely to influence students to feel the same way. Saying that something is fun is more likely to encourage someone to do something than saying that it is difficult.