Today’s post title sounds a little like the “…walked into a bar” series of jokes: “A MOOC and my math lab class walked into a bar…”. However, it’s neither about jokes nor bars, unless one considers the bar (“‾”) , aka overbar, above a decimal number denoting infinite repetition, above the letter x denoting the mean value, or above a letter in boolean algebra denoting the logic operation “not.” [1] I titled it as such due to the confluence of a passage in an assigned reading from a Massive Open Online Course (MOOC), Introduction to Mathematical Thinking, offered by Prof. Keith Devlin of Stanford, and a comment made by a special ed teacher who often helps out in my algebra math lab / intervention course. Devlin’s passage follows.

“These days, mathematics books tend to be awash with symbols, but mathematical notation no more is mathematics than musical notation is music. A page of sheet music represents a piece of music; the music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience; the music exists not on the printed page but in our minds. The same is true for mathematics; the symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive—the mathematics lives and breathes in the mind of the reader like some abstract symphony.” [2]

I was happy to read Devlin’s comparison of mathematical notation to music notes. On occasion, I use the same example with my students, mostly my algebra 1 and math lab students, to help them understand the use of symbols in language, and in specialized applications such as mathematics and music. The fact that many musicians are very mathematical, and vice versa, accompanies my broad brush discussion on symbols and notation; I even like to contrast cymbal with symbol as a way to expand student vocabulary, and have a little linguistic fun at the same time.

Merging with Devlin’s passage in my mind, a comment made by the special ed teacher after a review for an end of marking period test inspired me to write this post. After watching me lead an interactive review of number types in which I reinforced how to distinguish between an irrational and rational number, along with knowing the reason behind the letter ** Q** as used to represent rational numbers, the special ed teacher exclaimed that I was instilling these concepts in my students to prepare them for calculus. As soon as he mentioned it, I gave a resounding “Yes!” since I do hope many of my lower level students, whether in math lab or algebra 1, eventually make it to calculus, even though the likelihood in high school is very remote. While it might even be remote for many of them in college, I nonetheless hope to inspire each of them to give more effort to learning mathematics, which requires much investment of time and effort on their part.

It also turns out that the special education teacher learned something in that lesson, which he explained to me after class as well. It helped me understand why he raised his hand from the back of the room when I asked the class if they learned a bit more about number types, even though we covered it earlier in the marking period. After I asked, most students simply sat and stared, which I rarely allow so I followed up with something like “Come on, now. Did I just waste everyone’s time reviewing something you already knew for tomorrow’s test?” That seemed to get their attention, and most smiled, nodded their heads, said something in the affirmative, or raised their hands, which I took as confirmation of not only their interest, but of their understanding. Time will tell on the latter.

Further explanation from the special ed teacher led to him mentioning “The Big Bang Theory,” which apparently is a TV show; I have not watched broadcast TV in nearly a decade, and cancelled our cable TV subscription once I decided to become a teacher as I could not afford the $100 per month any longer. I indicated that while I knew of the scientific aspect of the big bang, I knew nothing of the show. He smiled and said that in a recent episode he watched, one of the characters used some of the notation that depicts number type, or sets, and he now knew what the “physicist” was discussing. That made me happy, as not only did I help my students, but I helped a colleague, which was a great way to end the week. I say end, since I headed down to Huntington Beach later that same day to attend my brother-in-law’s wedding while all of my students took their end of marking period tests. Now, I simply have to go pick them up tomorrow, Sunday, and grade all 150 or so of them!

[1] Does anyone know an easy way to use Latex in WordPress? Previously, I used Equation Editor in MS Word to create an equation then inserted an image of it in WordPress. That way seems very tedious. Suggestions??

[2] Devlin, Keith. “What is mathematics?” *Introduction to Mathematical Thinking*. Web. Fall 2012.

Just create the equation in word, use SNIP to take a picture of it, save it and upload.

I can’t help but think you are spending too much time explaining concepts they don’t need, and will forget. It seems to me it’s better to focus on a few big ideas for low ability kids.

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This is one of the big ideas I want them to understand. And they seem to get it. I’ll have some stats after I score the tests AND find time to write about them…

And on the equation front, that’s what I do now…seems like it could be smoother…but that’s just me. 🙂

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Math is awash with symbols.

I hear you saying that the symbols stand for really wunnerful and awesome stuff like music (which is much more wonderful than its symbols, indeed).

However, I don’t see you addressing how to approach the students who, analogically speaking, don’t “read music.”

Let’s stretch the metaphor a bit. If I’m a student in a music class who can’t read music, and you tell me about music, and oh, lots of people in the class are enraptured by the music they can make… but I can’t read music so it’s not communicating to me… but oh, gosh, it’s wonderful!

… I’m likely to believe that since this course is required, I’d better smile nicely too and work with those symbols as best I can with the rules you give me, and hope I get through. I join the legions of “yea, I’m not a music person” — now, change that to math and you’ve got a reality we see every day.

Okay, the devil’s advocate in me is whining up a storm…

I’d want some evidence that students are actually understanding. From what you describe, they *affirmed* that yes, you had just wasted their time by reviewing something that was already on the test. If, in fact, they nodded that oh, yea, they had learned something… welp, perhaps it was because you were obviously not allowing the staring silently thing, and they were just goin’ with the flow and doing what they had to do to get through the period. (I wasn’t there — and neither was that devil’s advocate… but the DA says it needs to be said…)

On the other hand, please tell me the sped teacher said that stuff about calculus in front of everybody – I *love* letting students run into high expectations…

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I’m confused, Sue. I did not mean to imply that simply drawing parallels between music and math and their common use of symbols magically makes students get math. If that is how it reads, my apologies. At the same time, your dripping sarcasm does little to encourage me to think deeply about your comments. Since you linked to your site on another comment, I’ll write this comment off to having a bad day, or similar.

BTW, I do not believe I hold the cure to all that ails education, and look forward to learning from others, but when PLC commentary is caustically cloaked, the PLC benefit is lost.

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I didn’t mean to be caustic but struggled to find a way to expressing things without it coming off as such… I mean this completely seriously: where was the sarcasm?

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Your comment is dripping in it…”symbols stand for really wunnerful and awesome stuff like music,” “but I can’t read music so it’s not communicating to me… but oh, gosh, it’s wonderful!”, and ” I *love* letting students run into high expectations…” to list the top ones.

If they reflect your frustration with how math is taught, I get it. I do not like how much is taught either. I do not subscribe to bromides, or blindly accept motherhood statements. I believe in helping every student attain their highest potential, but know that the logistics of doing so works against me. At the same time, I am working to find the best way to get the majority of my intervention students to understand operations with integers so we can move on to order of operations, rational numbers, and beyond, at whatever pace it takes. Not in a “well, we have to move on now even if you do not get it” way.

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Okay, “I *love* letting students run into high expectations ”

… was not sarcastic. I meant it.

Nor was I being sarcastic about the symbols. They *do* stand for wonderful and awesome things and the math teacher obviously believes it. So the student sits there wondering what’s wrong with him/her…

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OK. I misread your comments. And so we’re on the same page, I do not let my students sit there wondering, or worse think its them. I explain my own experience struggling with math at times, how others also struggle, and how we learn through our mistakes.

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(to express things… I wish there were a way to edit …)

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Yes! Totally agree.

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… but seriously… one of the hugest frustrations of my job is doing repair work on students who are convinced that math is a procedural nightmare they must endure. Some of them describe being in classes where they *do* smile and nod in all the right places as a survival strategy; they do fool the teachers… they’re honestly afraid of what will happen if they admit how lost they are (or at least, how lost they think they are — sometimes they’re closer than they think).

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I get that the “smile and wave, boys” thing. And I’ve allowed myself to fall into that trap, which I created, due to poor timing on my part. However, much more often than not, say at the 90% level, I do not let my students “hide in plain sight” in my class. I believe it is my job, and purpose for becoming a teacher, to help ALL students, especially the lowest of the low. You have to see it (er, me) to believe it.

This issue exists in all of my classes, intervention and AP Calc. Its all relative to the individual, and I try to instill confidence in each and every one of my students that they can overcome their challenges if they trust in me and work with me. Its showing results. Whether the content understanding persists or not is unknown. However, the memory of overcoming confusion will remain with them forever.

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Amen 🙂 There is *nothing* like watching the light go on.

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A couple of my AP students created an index card with a glowing light bulb which they hold up when they get something they previously did not…At the outset of the course, I told them these moments would come, and when they did happen, these students made their cards without any urging on my part.

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I’ve used http://www.codecogs.com/latex/eqneditor.php to help with the Latex some. I’m in Devlin’s MOOC … it’s okay but not as good I was hoping so far – I’ve not been able to share as much of it with my students (Lecture 1, the next two I would go over the basics so that maybe I could show them the 4th lecture) … I like the analogy to sheet music; I also think it’s valuable to share the first few paragraphs from Mathematician’s Lament by Lockhart with students. I want them to better understand and appreciate what mathematics “really is” …

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