Heather Robinson article - Response

I always feel a little bit guilty when it comes to "trying new things" with students. I think it's partly because the students in practicum aren't "my" students, and instead these are the students for whom my sponsor teacher is ultimately responsible. Because of this, as much as I want to try out different ideas, and no matter how supportive the sponsor teacher is (and he has been lovely, for sure), I always feel as though my experimentation may prove to be unethical, or may detriment my students. So although I understand the optimism with which Heather Robinson writes about "still [having] insights into [her] aspects of pedagogy and instruction" (pp. 62) despite lesson plans not working for her, I still feel as though I am a bit of a cheat, taking "bits and pieces" from other people to create my "own" practice. I suppose that all knowledge is built on previous experiences and does not require a person to "reinvent the wheel", so to speak, but when students' success is at stake, ironically, the students' enjoyment and ability also become the ultimate reason why a teacher would try something out in the first place. Heather Robinson's article sounds like me, though; I have yet to teach with a purpose other than for teaching math, and am still somewhat detached from the lessons I teach. I love math, I really do, but I also know its applications, and that's what it makes it relevant to me. Making mathematics relevant to thirty students, times six blocks, is a huge challenge!

Although I would like to say that my detachment from math teaching is because I am not writing lesson plans for my own class, but for someone else's, I ultimately know that I need to write my lesson plans, and find a reason to teach math. I think this is largely due to an overlap of my personal opinion and the classroom mixing together. I like the idea of provoking students and discussion. I don't know how much liberty I can take to doing that in a classroom that isn't my own, though, and I've been losing quite a bit of sleep over it. IB is supposed to be interdisciplinary, yet every classroom seems like it isn't; it's just a bunch of stuff to learn. I know that there are students who will just learn the math because they're expected to, but I want them to enjoy it, as well. Where can we get questions like the "Alternative Assessments" and "Enrichment Masters" which Heather Robinson mentions in the article? What about the ones we've used in our EDCP 342A class? I'd love to expand my resources to include such materials. There's also a social justice in mathematics textbook in the Education Library which I'd like to pick up.

I love the idea of Jigsaw activities; one concern in math, however, is whether or not students will teach each other incorrectly. However, this is something that seems to work better in my math 9 class than when I taught them, so I am considering having students teach each other in that sense, because they aren't benefiting from lecture.

The other issue I have is that the evaluation of mathematics remains the same even though the methods of instruction are all different. Mathematics remains a written form. I would be interested in having an interview setting, instead of tests, for one test out of the year, to see how well students can explain their ideas. No prompting, and correct answer = top mark, slight prompting and correct answer = A- or B+, etc. That would certainly help for students who have problems with written output, but perhaps are very strong verbally. Some countries certainly had interviews as part of their evaluation. These would have to be short, however, likely spread across an entire week; perhaps they could be an option for students to complete instead of a test, if they were struggling a great deal. This sort of interview type testing is used informally in various Skills/Learning Assistance periods when students write tests under the supervision or with aid of a support worker.

Practicum Experience Response

I didn't find anything easy about teaching math at my sponsor school. It's not that I found the subject difficult, but teaching math in a way that was still exciting to students was a challenge. When it comes to relative difficulty, however, I realized that I preferred to use the whiteboard over the tablet in a classroom, which is a huge revelation for me, because I always thought I did really well with the tablet. There's still something about moving around when you teach that, according to one student, is more appealing about teaching on a whiteboard, whereas I found that the tablet keeps you locked in one place, for the most part. In general, tablets are big and bulky and can't really be held up for walking around the room. I found it easy to talk to students, though; that was nice. I think I'm still young enough to be able to talk to them (I realized I use a slight touch of slang, like "s'all good" or "no worries"; though, is that slang, or just an expression used by my expression? I definitely called a large math expression a "holy mother of a cow". This seemed confusing to some, but I had their attention, which was nice. It was pretty easy for me to connect with the students, which was nice. They weren't shy when I would ask them questions, which was nice, and I asked them for their honest feedback. Although there was slight hesitation, I pointed out that I was a student learning to be a teacher. Although that's a dangerous move when it comes to keeping authority, they were still able to tell me what they liked and what they didn't like. 

The most difficult thing was probably to make math exciting. I did a lot of activities which were very upbeat and didn't allow for practice of material as much as I think I should have. I tried the "Japanese model" of teaching we discussed in one of the articles (Hoffier?), but I quickly found that students have come to expect being fed information and practicing it, so when I gave them a problem, there was such a large amount of complaining! My goodness! And with the small number of students in the school, by the time the next grade 8 class had come in, the previous class had already informed them that the class involved "a very difficult activity that nobody got", even though the students had learned all the information they needed to solve the problem. Although I don't expect all students to be problem-solvers, I realized that for the second class, I needed to explain a little more about how to solve the problem I had assigned, and from there, some people pounced on it immediately. Also, time management! Tiiiiime management, I'm still struggling with this so much! I've finally got control of my students, which is thanks to my sponsor teacher, who suggested that I have clear transitions between activities, and that worked. Also, I realized how not to be boring. That was huge. I realized as I was boring myself to tears, that other people were probably also pretty unimpressed at the moment, and so I became a little more theatrical. It helped!

I think the most interesting thing about teaching math is how helpful it is to have a problem worked out ahead of time. A problem in surface area, for instance, is much easier to teach if you have information prepared ahead of time and pre-organized. Another lovely thing; I divided up my boards into three columns to organize my work, and that helped tremendously. Finally, no matter how you try to turn cards into a teachable moment in probability and statistics, the junior level will always want to play poker, so don't use playing cards to organize people into groups....

Reading in Mathematics - response

The article mentions four main aspects as to why students may struggle with mathematics text reading:

1) Density
2) Mixed translation
3) Changing variables, symbols, and graphics
4) Concepts are interconnected

 1) In LLED 361, we explored the idea of literacy and multiple modes of expression. One activity which was suggested for "translating" mathematical language is to create two columns, where one contains mathematical text, and the other contains the "translation" of that text. It's a little bit funny, because both are a form of English, but mathematical text is used very infrequently. This sort of activity, I would argue, is only to be used with students who have not gotten into the habit of translating mathematical text or asking themselves questions about the content or question being asked. Much like students may need to be taught how to complete a multiple choice exam efficiently, so to do they need to be taught how to "translate" mathematical language, hopefully without writing down the interpretation in the interest of time.

2) "Or", or "or". Oh, how I've missed you. "The mathematical 'or' is different from the categorical 'or'," my proofs professor used to say. For context, in the "every day world", one would say "please bring me coffee or tea", and this would be the exclusive use of "or", since you would not wish to drink a concoction of coffee and tea, call it cofftea. However, in mathematics, this would be exactly the case; asking a mathematician to bring you coffee or tea would mean that the mathematician would bring coffee, tea, or both.

3) I find that students struggle a lot of greater than and less than signs. I know in school, we were always told that the "sign has teeth on it, and the teeth eat the bigger number". But when it comes to reading an inequality, it actually takes a second to recognize which number is bigger, and whether you need to say "greater than" or "less than" as you read from left to right. I would like to suggest that the reason for the discrepancy is because as language users, we are used to reading from a certain point on a page to another point on a page, and in many languages, symbols do not have individual meeting, but must be connected before a meaning is derived. Mathematics symbols all have individual meanings, such as + <=, =/=, etc. Knowing both the meaning of the symbol as well as its collective meaning in a statement means that mathematics reading can be especially challenging.

4)  I can't get over the fact that I find myself teaching high school students about fractions more and more often. I don't mind doing so, but to me, it's appalling that they have managed without it. I imagine a calculator has something to do with it. What was the reasoning to change the math curriculum in BC from linear, compounding concepts to more scattered ones? I remember there was an issue with too much repetition; was there any other reason as to why the math curriculum changed so drastically in the last few years?

I will say, however, that I disagree with Marzano et al. in using direct instruction on words. I discussed with my inquiry class, that an inquiry science classroom may have students determine a definition together. I like this idea for mathematics; although mathematics is less fluid than science is; I would say that defining a mathematical concept in a way previously unbeknownst to a student means that the student can learn a great deal from their peers about how to express an idea that they all share but are unable to get across to other students or to the teacher.  All they would need at that time would be a label to have a definition.

Micro-teaching Retrospection #1

Here is a table outlining the feedback received for each section:

Aspects to maintain	Suggestions	Interactivity
- Walking around to help students	- Raise volume	- Encouraged group interaction
- Pace	- More clear definition of slope at beginning or end of lesson	- Asked good guiding questions
- Good intro	- Have the app available for students to practice	- Advanced group got a chance to discuss more
- Good applet and technology use		- Fun/relevant
- Addressed confusion		- If more time had been available, drawing more lines for practice would have been helpful
- Eye-contact
- Visuals
- Group work
- Positive words/ feedback/ answered questions effectively
- Relating topic to “real life”

I felt that the lesson went well, overall. I was really excited by the hook, because there was immediately a connection between slopes and skateboarding/snowboarding/skiing. There were several things which I would like to improve, in particular. Firstly, I would like to improve my teaching for intuition. Although this is a very broad goal, I will narrow it below. It is very difficult to wander from traditional teaching styles of stating definitions, demonstrating problems, then practicing problems of definitions. I think that our group may have chosen a topic which was a bit too ambitious to cover in its entirety in 15 minutes, and given the applications of a slope in calculus, it would not be wise to rush the lesson. I would stick by my choice to leave negative slopes until the end of class, however, because it has been decided, by a combination of the definition of a slope as well as the selection of which axes are positive or negative, that a line falling from the second quadrant to the fourth quadrant has a negative slope. If horizontal axes increased to the left and decreased to the right, the sign on a slope would be reversed. The magnitude of slope is really crucial here, the sign of the slope can be discussed later in the lesson. I think a first start to developing inquiry teaching would be emphasizing what is necessary and what is arbitrary in mathematics, such as the sign on a slope; I feel that this is a good guide as to what needs to be learned and what can be (self-)taught in an activity.

My second point for improvement would be the understanding of when to introduce definitions. After all the articles we've read in class, especially the Hoffman/Brahier article, I feel as though introducing a rote definition is almost a faux pas in mathematics, because if it is used incorrectly, a definition becomes not a summative statement but an introduction of something arbitrary being passed off as necessary. After all, why did we actually choose for slope to be a ratio of the change in vertical to the change in horizontal, rather than the other way around? Rather than a concept of "more steep", we would likely instead use the terminology "less flat". However, everyone who has ever encountered any wheels in their life will have an intuitive sense of what slope means, even as a matter of survival. Finding a mathematical definition is a fine balance between over-complicating an intuitive definition and over-simplifying a (often) physical concept. When is a place for definitions during a math lesson? I have always encountered definitions at the beginning of a math lesson, especially in calculus. However, could calculus also be taught intuitively, without starting from the definition? After all, this obsession with definition (I would argue) originates with the formation of proofs. The definition of a concept being proved is one of the most useful tools for verifying a mathematical statement around that concept. When, if ever, does mathematics become really intuitive in an upper level? What is the evolution of the necessity of a formal mathematical definition as a function of time in a math classroom?

Yopp - Response

Oh dear. Conjectures in a math high school classroom? Hmm. It's not that I don't think students are capable of doing it, nor do I feel guilty about temporarily frustrating my students so that they can learn something from the process. However, I don't know that the current sort of abstraction present in curriculum would serve students well, or even at all, when disproving conjectures. I would worry that students would get turned off completely from math. I imagine that the "math" that people say they "hate" is really all instrumental/arbitrary. In fact, math is so much more than that; what the general public thinks is math is just the methodology of processes used (mostly) in arithmetic. It seems to me that it is easier for a teacher to use and teach methodology than it is to effectively teach someone how to solve a problem or disprove a conjecture. And on that note, how do we wish to mold students? Do we wish for students to have a high aptitude of mathematics, logic, and abstraction, for students to be laborers capable of completing conversions and estimating necessary materials, functioning taxpayers (or both)? Given that arbitrary math is so disliked, is it worth using conjectures in a classroom? I remember rolling my eyes at any proof that came my way in a high school, because nobody ever bothered to explain why proofs were useful/necessary. My logic said, "I could see it, why did I need to prove it?". Also, at what age level would this be done? Grade 11/12? Earlier? Would the conjecture methodology be a lens or occasional teaching tool? I would like to learn more about this!

Hoffman/Brahier response

I enjoy problem solving. It's a useful tool in any classroom. Again, by saying it is a tool, I suppose that I limit it from being a way of teaching. There were two scenarios of which I was reminded in this article. I will name two.

The first scenario was when I worked at a math camp over the summer. One of the students in the class did well with rote learning and practice (e.g. learning how to simplify exponential expressions). However, he acted out often, he would distract other students, and he created a challenging learning environment. I found the "remedy" for this was to bring out University of Waterloo math contest problems for him to do. He devoured each and every one of them. He went through any puzzle books I brought in, as well. 

The other side was when I took Math 414 at UBC, where I would travel to certain high schools in Vancouver and conduct math workshops. The layout of the workshop involved students being given a certain list of problems to solve in groups, then they would present the solutions to the class. Most students would work on the set of problems they were given, but they wouldn't really pay attention to those presenting alternate solutions. The students presenting were also very hesitant to come up to the board, and in some cases, students would quietly call each other "stupid" or "slow", which I found absolutely vulgar. 

I love any puzzle, it's always a delight for me to complete puzzles, especially ones like Kenken (which, if I have my own classroom, I would love to use in a grade eight classroom on the first day in order to gauge how students understand logical thinking and how their basic math facts are). I feel like I could fill up a whole year with nothing but problems. However, would this not require a readjustment of curriculum? One of the IRPs for grade 10, for example, under "Relations and Functions" is that a student must be able to express an equation in slope-point form, slope-intercept form, an general form. So there would still be a need for "arbitrary learning" as was mentioned in the other article. I was unclear as to how these things are balanced in Japan. How is new material introduced in a Japanese classroom? Do students find additional information elsewhere (e.g. online), or do they still have an instructional portion to their classes?

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.