"Opposites" of numbers

I'm taking a little break from writing lecture notes to complain about things that textbooks do that are really stupid. I'm going through the Math 097 textbook, and I have reached the third or fourth time where I just shake my head at the textbook and hope that somehow either the textbook or the author will feel my frustration and it will be changed in the next edition. I suppose I could write a letter to the publisher or the authors, but I'm not going to do that now.

Why does it matter? I have an expectation that the students will read the textbook. As a consequence of this expectation, I am reading the textbook as well to make sure what they read actually makes sense.

As an aside, in what classes do students pay $100+ for texts that they have no expectation of reading? As far as I know, it's only the math classes.

Anyway, something that really bothered me was the discussion of negative numbers. They talk about 1 and -1 being "opposites". This random terminology that math people would never use reminds me of some terminology I ran into in high school with the CPM program. Apparently, somewhere along the line someone felt that a "common denominator" should be called a "fraction buster". If you don't believe me, just Google "fraction buster." Anyway, I suppose the idea is that when solving an equation involving fractions , you can "bust" the fraction by multiplying by the right number... or something like that. Besides the absurdity of nonstandard terminology, it also means that when you add fractions with different denominators that you would have to introduce the "common denominator" as a separate concept, even though it's the exact same thing.

But back to "opposites". The standard terminology for this is "additive inverse" for the simple reason that the relationship between them is additive, namely that if you add them together and you get zero (4 + (-4) = 0), and zero is the special number that doesn't change the result if you add it to something else (10 + 0 = 10). Then there's a link between this idea and the "multiplicative inverse" of a number. The relationship between a number and it's multiplicative inverse is that when you multiply them together you get one (4 * (1/4) = 1), and one is the special number that doesn't change the result if you multiply it by something else (10 * 1 = 10).

It just got worse as I pushed onward into the following section. Here is a quote from the text:

A symbol like -x, which has a variable, should be read "the opposite of x" or "the additive inverse of x" and not negative x," since to do so suggests that -x represents a negative number.

Furthermore,

As you read mathematics, it is important to verbalize correctly the words and symbols to yourself. Consistently reading the expression -x as "the opposite of x" is a good step in this direction.

If you talk to anyone who does math (this includes chemists, engineers, physicists, and many others), and pointed to -x and told them to read it, they would all say "negative x." So while it would make sense to a student brought up in this way, to the rest of the world, it's nonsense.

But it gets worse. Consider the following sentence "Multiply x by the opposite of x." If given that sentence outside of the context of this subject, I would consider "opposite" to be related to "multiplication" and end up with x * (1/x) instead of x * (-x). The book will likely introduce a different word, "reciprocal" for 1/x. Unfortunately, all this does is introduce more words for students to memorize, and those words are not particularly informative to the nature of the relationship. Why is "opposite" additive and "reciprocal" multiplicative? It just is. Memorize it. (At least "reciprocal" is a common terminology to people who use math...)

I think that's enough of a rant. I need to get back to writing up lecture notes. I won't get into the "Rules for Addition of Real Numbers" table right now...

The data is what it is...

Every now and then, you think something should be pretty obviously true, but it turns out to be false... or at least not entirely true. Consider the following supposition:

Students who take their remedial math class and first college-level math classes in consecutive semesters are more likely to pass the college-level class than those who take a break between the two.

On the face of it, it's seems arguably plausible and pretty simple. And when you look at the numbers, it turns out to be true. From the data available, we found that 52% of students pass when there is a break in between and 62% of students pass when there isn't. This is a 10% difference and a relative 19% increase.

However, when you take a closer look at the data, a surprising trend arises. For the weaker students (B- to C- students), the pass rate is approximately the same regardless of whether they took the classes sequentially! The data set got a bit small for this part of the analysis, with less than 100 students, but the signs point to something lurking underneath the surface that has not yet been identified.

The mystery to understand why this is the case begins...

Looking ahead

I went to my first meeting yesterday. The discussion addressed something that I was not really aware of as an "outsider" to the department. One of the major problems of the camus is that students are not exiting with as high a level of basic mathematical competency as they would like. I didn't say much during the meeting, but there were a few things which seemed very reasonable approaches to the problem

• No gaps between math classes: Many students (for whatever reason) choose to take several semesters off between their math classes. Intuitively, this seems entirely counter-productive because math, like any other skill/knowledge, gets lost with lack of use. It seems unclear whether it is possible to mandate students to take the courses in consecutive semesters, but it is something to be pushed strongly.
  


• Tracking student data: Apparently, the Nevada System of Higher Education is using a very very old program to track their students (an upgrade is in the process, but it's a number of years off before completion). This means that while there is lots of anecdotal information to support various positions, it is hard to produce the hard evidence required to back it up. Apparently, some of the faculty figured out how to access the information and get the data they want out of it, so there's an effort to get this information and process it so see if the data supports the claims.
  


• Online class limitations: Apparently, there is an effort to make a lot of classes available online. There's even an effort to have entirely online degrees. I have some misgivings about online degrees (quality control, making sure that the name on the application actually corresponds to the person doing the work), but I can see the value of it (in principle). However, the degrees that some people are trying to push include things like physics, biology, and chemistry, which is absolutely absurd. These lab component of the physical sciences is such a large part of the degree that I cannot imagine that the degree would be given much credibility if it were earned entirely online!
  

I guess if I had thought about it for a while that I could have surmised that there would be conversations of this sort in a growing campus. But actually sitting in on such a meeting and hearing the discussion really drives home the feeling of urgency to "get it right" and to provide a quality product for the students, and to put students in the best possible position to succeed. (Of course, success rests on their shoulders, not ours -- if they don't put in the work, they shouldn't earn the degree.)

First class.... sort of.

I filled in for Jason yesterday in his precalculus class. I didn't do much lecturing, but instead spent the whole time answering questions. As a result, it felt a whole lot like teaching section all over again.

I didn't do anything different from teaching sections back at UCSD. I asked them what question from the homework they had problems with, and then I proceeded to explain the concept and solution to them. However, I did notice a distinctive "desire" to learn, mostly from the students who seemed to be non-traditional (I originally typed "older", but I feel that it might be received as derogative to some).

There was one student in particular who struggled with something involving absolute values, but could not find the right question to ask to solve her difficulty. I had to ask her a couple times to try to rephrase her question because I couldn't figure out what was tripping her up. But she politely and patiently persisted in trying to wrap her mind around it until she was finally able to put it together. I've seen that sort of thing happen before at UCSD, but I also remember the student simply say "never mind" after struggling with it for a couple minutes.

I was a bit amused by her response after she put it all together: "Thanks for humoring me." That's a new one. I hope to see more of this from my evening classes this Fall.

Learning Styles

I recently got my login information for the Blackboard Learning System at NSC and have been playing around with the different links that I found there. One of the links is the Index of Learn Styles Questionnaire. I thought it would be interesting to take the quiz and see what the results say about me.

Results for: AW

      ACT                              X                    REF
          11  9   7   5   3   1   1   3   5   7   9   11
                             <-- -->

     SEN                      X                            INT
          11  9   7   5   3   1   1   3   5   7   9   11
                             <-- -->

     VIS          X                                        VRB
          11  9   7   5   3   1   1   3   5   7   9   11
                             <-- -->

     SEQ      X                                            GLO
          11  9   7   5   3   1   1   3   5   7   9   11
                             <-- -->

The test took only about 5 minutes to complete. It's 44 questions long and I probably spent about 5-7 seconds thinking about each one. How do you read the results? You can read about the four categories on the Learning Styles and Strategies page.

I'm not very surprised by the first one. I tend to learn well in all sorts of settings and I don't really believe I'm strongly biased one way or the other. The second is also a balance that doesn't surprise me, either.

The last two are clearly dramatically skewed. I had initially thought that I would be balanced between visual and verbal, since I tend to be able to hold a lot of auditory information in my head, but after some reflecting I found that the questionnaire is probably right. Memorizing information is not the same as learning. So while I can hold audio in short term memory, it's not information that is being processed in any way. When students ask me questions in section, I tend to have to write down what they say and stare at it before I can see what they're asking.

Seeing that I'm a sequential learner is unsurprising. I've always been better at the "nuts and bolts" operations than trying to see the everything at once. I think this is part of why students tend to like my teaching style: I often break things down into small steps (or sometimes decision trees).

I'm greatly amused by this sort of thing. Perhaps this is an indication that I should really start looking into Math Ed research or at least SoTL research as a scholarly pursuit.

Spring 2007 Math 20D Evaluation Results

If you've looked at my poker blog, then you would know that I like record-keeping and data. I've compiled the results from my TA Evaulations and put them in a this document for future reference.

What must I get on the final to earn an A?

I bumped into a professor in the math department the other day on the way to the math department party and during the course of the conversation he shared the following little anecdote that I think is worth sharing:

Why is it that students ask, "How well must I do on the final to earn ...?" If you them that they must get 80% on the final to earn a B, are they somehow more capable of getting an 80%?

Non-rhetorically, why do students ask this absurd question? I don't know, but I think it's all an illusion and I dislike it. Instead of the "do the best you can" mentality I hope for students to have, the exam is seen as a game of grades. Yes, the assessment leads to a grade, but the grade is the afterthought to the educational value of the content being taught. The right question is "What can I learn?" not "What must I do?"

Thinking about this question has made me contemplate the type of response I should have to students who ask that question. I have gone through a few which I've listed here along with some commentary:

• "Your best chance of getting the best possible grade is to get a perfect score." - I'm not happy with this because it brings the grade into focus, and not the act of learning.
• "Just do the best you can." - While this is better, I don't think it demonstrates to the students that I care. Instead, this sort of sidesteps the question and doesn't give a helpful response (of course they will do the best they can).
• "Does it matter?" - It's getting closer. A question puts the ball back in their court to try to justify why they care. I'm not quite happy with it because I know the answer. It matters because they care about their grade.
  
• "Would it change anything if I told you?" - This is the best answer I have so far. It puts the responsibility on them to look at their question in a different light. I'm not denying the importance that they have put on their grade (as misplaced as I believe it is), but I am leading them to realize that this question is ultimately unhelpful to them.
  

I won't have to face this question for about 5-6 months, so I'll probably put these thoughts on the back-burner for now.