Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at markdotmath@gmail.com. I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Openings

Posted by mark schwartz on December 18, 2016

Let me first apologize for the long delay between the last posting and this one … there was just a heap of other stuff that needed attention …

Introduction

The first day of class for a remedial/developmental at the community college level is a classroom loaded with math anxiety. These students, by definition, bring not only anxiety but also expectations about how the class will be conducted based on their previous experiences; at best, they hope to finally master some of the math that has been confounding them. Given this, rather than only the usual presentation of the course information (book, assignments, grading system, attendance, etc.), I found that an opening exercise of some kind eased their minds about how things would go in the course. Below are examples of ‘openings’ that engage the students, rather than immediately plowing into the course content.

The Story

The first thing that happens is that I assign them to groups, typically 3 in each group. I give them time to introduce themselves to each other and announce that they will be working with those in their group the whole term. Basically, share what you know and discuss within your group how to manage the material and do the work. In addition to individual work, there will be some group work. When they’re ready, they do an ‘opening’.

The where-are-you-in-math line. I draw a horizontal line on the board, marking the approximate center. On the left end, I note something like ‘math sucks’ or ‘I hate this stuff’ and on the right end I note something like ‘I get it’ or ‘math is no problem’. I then tell them that I’m going to leave the room and I want them to mark where they are on this math line … don’t use your name or initials, rather an ‘x’ or star or smiley/frowny face and when everyone is done, come and get me. Questions?

The typical picture is that there is a cluster of marks to the left of center, reflecting somewhat realistically why they are in this remedial/developmental class. I start the discussion by pointing to one of the marks and asking, “what do you suppose it would take for this person to go from here to closer to the right end?” It takes a while for the discussion to get going because they’re not quite sure what the question means, but individuals start offering things like “getting the fraction stuff”, “learning the rules for signed numbers”, “word problems”.

The point of the discussion is to identify not everything that needs to happen but rather that it may be that one (or maybe two) fundamental operations or rules can make a significant difference. I point out and emphasize that it’s not ‘math’ that they don’t get but rather some specific relationship that might be messing with their entire mastery. A good example is always operations with signed numbers. In the discussion, I make a point of doing the following: I ask that those who can finish the phrase I say, please do so out loud and I say “ a negative and a negative is a …” The response is of course “positive” but the I ask “when?” and I get some baffled looks and responses. They know the mantra but not what it really signifies. I ask for volunteers to come to the board and show me examples of when that mantra applies. Without correcting any of the statements – some of which are accurate – I simply point out that some are right and some are not and that rather than memorizing the rules, we will spend time talking about how the rules come about and how they really work.

I end this first day class at this point, unless collectively, they want to explore more about other math issues they may have. I won’t address the classic “when am I ever going to use this stuff?” but typically there are a few other issues we talk about, like “isn’t there an easier way to do fractions?”

Using an opening rather than diving right into the math content sets a different tone for the class; they realize that the class is more a dialogue than lecture; they feel comfortable asking questions; they like the idea of working in groups; they perceive math differently and this I note from questions at the beginning of the next class; they clearly have been thinking about what happened the first day and thinking about math is a very positive outcome.

Another first day opening I use once all the groups have settled down; is to ask if there are any ball players in the class – baseball, softball, basketball – and typically there are some. I ask one of them to stand and announce that I’m going to toss them an eraser and they are to catch it and throw it back. Once this is done, I ask “Was there any math done here?” This gets answers from “no” to “what do you mean?” I ask again if in the tossing and catching if any math was done and this typically gets things going. What gets focused on in being able to make judgements about trajectory, speed, acceleration, location and other quantifiable judgements which make it so that when you’re catching the eraser, you know how to place your hand to intercept the eraser in its flight and catch it. When it comes to quantifying the toss, it’s a matter of distance, energy, direction, flight path, etc. so that it makes it possible for the person to catch it.

The point of this opening and the discussion is to point out that we all do math all the time and if you ask “when am I ever going to use this stuff?”, the answer is “all the time”. I ask if anyone has any other examples of this kind of quantitative judgement. A typical response is “when I’m driving”. One student once proposed that walking up or down a set of stairs takes a lot of quantitative judgement.

The essence of this opening is that you already do math a lot and it’s a matter of realizing that a lot of stuff you will see this term are slowed-down algorithms that your brain does automatically and rapidly. This edges up to the philosophical question of “is math out there as a universal or man-made” and this sometimes comes up in discussion but the point is that it gets people – again – thinking about math. It again creates a different tone for the class and that this classroom will be different from their previous classes.

This next opening usually generates a lot of noise. First, I write an equation on the board twice, something like 2 + 3(2x ─ 1) + x = 3(x + 4). I put this equation on the left side of the board and on the right side of the board. I tell the class not to panic – they don’t have to solve it. But, what I do say is “where’s the math”? After we talk about this for a while, I make the following statement “what if I told you that numbers have nothing to do with math?” (sometimes, this question has popped up in the discussion, but if not, I state it). This really gets people going and after we talk about it for a while, I use the equations on the board to demonstrate what I mean.

I take the equation on the left side of the board and I write it without any numbers and I take the equation on the right side and write it with only the numbers.

The left side is     +   ( x ─ ) + x = ( x + )

The right side is       2 3 2 1   3 4

The question is “which statement makes the most sense?” That may not be the precise question to ask but the point is that when you compare the right side to the left side, there is an obvious difference. The left side has notation and the right side only has numbers. When we discuss this, it usually occurs that someone will say “the left side tells me things to do and I have no idea what to do with the numbers”.

This highlights the point of this opening. One can get a sense of the relationships and operations that are expressed in the equation by looking only at the notation; you get nothing by looking only at the numbers.

As we discuss this, the class reflects the importance of the notation and that the essence of math is not the numbers but – as one student said – how the numbers are connected. As in previous openings, this one again gets students thinking about math a little differently from what they had previously thought.

This last opening (I have more but 4 examples are enough for this posting) has several hidden messages; one is “read slowly and carefully” and the other is order of operations, although I don’t label this so in class. This is set up for a room that has a white board and uses markers but it could also be done with the standard chalkboard and chalk.

I give each group an envelope, in which there are brief statements, each statement on a separate piece of paper. I tell them that they are to put the statements in order and once they’ve done that, do exactly what it says to do – no more, no less. Once the instructions are clear, I watch each group sort through the statements, agree that they have the correct order and then do what it says to do. The statements, not in order are:

Walk to your seat

Write your name on the board

Cap the marker

Stand up

Uncap the marker

Sit down

Pick up a marker

Walk to the board

There is also one statement which says “choose one member of your group to do the following”. I need to note that I make sure that there are only two markers in the tray at the board because this is the core of this opening. You’ll see why in a moment.

In every class so far, every group fails the first time! When I announce this, I ask them to try it again. Sometimes someone gets it right on the second try but mostly people believe that there is the “trick” statement “write your name on the board”, so they correct themselves by writing that phrase rather than their name. No trick here.

Given that there are typically 6 or 7 groups in the class, it only takes two of them to get the exercise correct to bring out the point of the exercise. Note that in the statements, it does not tell the student to replace the marker in the tray. According to the statements, the correct thing to do is to take the marker with you back to your seat! So, once two groups get it right, the next groups can’t finish.

When this opening is done, I point out the importance of reading slowly and carefully and also of verifying what’s going on with members of your group. We talk about this when reading a text for information of when reading problems to solve.

As I said before, I believe it’s important to set a tone in these classes which signals students that this math class will be a little different from ones they’ve previously experiences. Let me conclude by quoting myself about what I consider to be the importance of an opening ….

“Using an opening rather than diving right into the math content sets a different tone for the class; they realize that the class is more a dialogue than lecture; they feel comfortable asking questions; they like the idea of working in groups; they perceive math differently and this I note from questions at the beginning of the next class; they clearly have been thinking about what happened the first day and thinking about math is a very positive outcome”

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