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.