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?