A chance to refelct on the first half of my first semester

I'm taking some time right now to think about the first 8 weeks of the semester. I'm doing this now, right before the big onslaught of graind that is to come next week. There are certainly a number of things I want to do differently next semester.

#1 - Homework: I'm going to try something new next semester. I'm going to make part of the homework for my algebra classes simply copying the examples out of the book. This is first to force them to read the examples and work through them. Secondly, I want to give them the opportunity to circle parts of the examples that doesn't make sense to them that they would like clearified. I also want to force them to compare the examples to their own work that follows. Hopefully, this will help them get into a good routine of using the book effectively instead of just a source of homework problems. Thirdly, I think giving students a full week to do homework is simply an excuse for them to put it off. Only a couple students ever start the homework ahead of time like I ask, and so I need to force the issue a little with them. Finally, I need to assign problems from previous sections with much more regularity than I have been. This will hopefully reinforce the necessity of not forgeting past concepts.

#2 - Grading: I made the mistake of making homework out of too many points. It really only needs to be worth 3 or 4 points. I had a student ask me why she lost 5 points out of 20 when I only marked one error on her homework. My response was that her error demonstrated that she didn't understand a particular concept very well. She argued that she did lots of other problems correctly, and I agreed with her that she did. However, the point is that she failed to grasp one of four or five concepts for that section (which she agreed didn't make sense), and that her grade reflected that reality. However, there's a psychological difference between 15/20 and 3/4 that she had a hard time getting over.

#3 - Quizzes: I might do the quizzes at the beginning of class and schedule class start time 10 minutes after the start of the class. It's a much more regular routine that way. I might also cut it to one quiz per week.

#4 - I want to buy "The Inner Game of Tennis" and read it so that I can potentially send some of that message to the students on the first day of class. Many of the students have already defeated themselves by coming in with an attitude of failure. I've heard this sports psychology book is very good at discussing how players lose before they even begin. I think I'll jump to Amazon right now to do that because I'm done posting.

Students as Clients

Link to the article

I just finished reading an extremely insightful article that was referenced in "Enhancing Scholarly Work on Teaching and Learning." The article is titled "Students as Clients in a Professional/Client Relationship" by Jeffrey J. Bailey.

I'm quite pleased that this article exists, as it reflects some of my own perspectives on teaching that I already have and enhancees it by increasing the depth of those views. What is even more exciting is that this article exists in a journal for management education, not a journal for mathematics education. One of the main points of "Enhancing Scholarly Work" is that cross-disciplinary reading in education is both valuable and necessary because there are education issues that extend beyond the boundaries of a particular subject.

There were a few poignant quotes:

The enhanced role of the professional and client in the client metaphor (compared to a sales clerk/customer metaphor) embodies additional rights, responsibilities, and expectations for both professors and students. The client rightfully has expectations that the professional operates within accepted standards and ethical guidelines and will fulfill responsibilities associated with being a member of the profession.

This statement expresses something that I've already felt and even codified in my "Standard Syllabus" in the contract that I have at the end. I feel less odd about the contractual nature of that document now.

As Franz notes, the attainment of physical fitness cannot be given to the client but must be accomplished by the client. A trainer can, however, show a client what to do, encourage him or her, and provide accountability. Similarly, students need to realize the importance of their active involvement in learning. The students must work at learning just as the trainee must exercise (work) to obtain physical fitness.

This helps to put the classroom experience in the right perspective from the students' side. If you go to the gym but don't exercise, do you expect to derive any benefit from the experience?

The professor/student relationship has dimensions to it that parallel the accounting firm/client relationship. If a student is not satisfied with a grade, he or she does not get it changed simply because of the dissatisfaction. Just as the audited client cannot say, “I’m not satisfied with your audit so change some numbers here to make me satisfied,” so too the student is generally bound by the grade the professor has assigned.

I wish I had this article five years ago, when I started as a TA. It would have given me a much clearer explanation to students as to why it is unproductive to argue for a better grade when one it is clearly not deserved.

Stubborn Students

Here is a brief passage from Ken Bain's What the Best College Teachers Do (Chapter 2, opening pages):

In the early 1980s, two physicists at Arizona State University wanted to know whether a typical introductory physics course... changed the way the students thought about motion...

Did the couse change student thinking? Not really... They had memorized formulae and learned to plug the right numbers inoto them, but they did not change their basic conceptions...

[The professors] wanted to probe this disturbing result a little further. They conducted individual interviews with some of the people who continued to reject Newton's perspectives to see if they could dissuade them from their misguided assumptions. During those interviews, they asked the students questions about some elementary motion problems, questions that required them to rely on their theories about motion to predict what would happen in a simple physics experiment. The students made their projections, and then the researchers performed the experiment in front of them so they could see whether they got it right. Obviously, those who relied on inadequate theories about motion had faulty predictions. At that point, the physicists asked the students to explain the discrepancy between their ideas and the experiment.

What they heard astonished them: many of the students still refused to give up their mistaken ideas about motion. Instead, they argued that the experiment they had just witnessed did not exactly apply to the law of motion in question; it was a special case, or it didn't quite fit the mistaken theory or law that they held as true. "As a rule," [the professors] wrote, "students held firm to mistaken beliefs even when confronted with phenomena that contradicted those beliefs." If the researchers pointed out a contradiction or the students recognized one, "they tended at first not to question their own beliefs, but to argue that the observed instance was governed by some other law or principle and the principle they were using applied to a slightly different case." The students performed all kinds of mental gymnastics to avoid confronting and revising the fundamental underlying principles that guided their understanding of the physical universe. Perhaps most disturbing, some of these students had received high grades in the class.

I earned a B.A. in physics as an undergrad at UCSB. (I didn't earn a B.S. because I didn't take the physics labs; they were too time consuming and I wasn't intending to continue in physics.) I was fortunate enough to be in a specialized program (College of Creative Studies) where we had a very good teacher for our lower-division physics classes. He taught us in ways that forced us to think about how we thought about the subject by assigning very difficult problem sets. What was difficult about them? It wasn't only computationally difficult, but he asked us to interpret the meaning of the results. This forced us to reconcile our results with reality and expanded our ability to think physically.

Here are a couple examples of things he would do:

1. In multi-body problems, we would see what would happen to the results as one of the masses becomes infinitely big or small. Did our new results match the expectations? Why or why not?
2. Some problems resulted in two solutions. Did they both have a physical interpretation?

I would like to find ways to do this with math. I don't know how it can be done, but I expect that it can be done. The details will have to wait for a specific class, or even a specific topic or problem, because I think it's impossible to talk about real life in the abstract. It can only be done in the context of things that are actually happening.

The book continues to discuss the views of the successful teachers with respect to the development of knowledge. I'll give the bullet points here and have you buy/read the book for yourself if you want to find out more:

1. Knowledge is constructed, not received
2. Mental models change slowly
3. Questions are crucial
4. Caring is crucial

The biggest hurdle for the students who are not mathematically inclined (most of them) is that they come in with the presumption of what math is and that they aren't very good at it. They don't really focus on the processes, but the end result. Why? This is how they are trained to think about math in school from the beginning. (Read this post for more on this topic.) The biggest hurdle for me as a teacher is trying to turn away the negative light on the subject. I have some ideas, but I won't know if they will work until I get the chance to try.

The importance of thinking

I was up late talking with a friend the other night. He lived with me last year, but spent this year in China teaching English and computer skills in a rural town somewhere. We had a nice conversation about a number of topics, and one in particular was relevant to this blog.

It turns out from our collective experience that most undergraduate students don't know how to think about math at all (ourselves included). This doesn't mean that they are incompetent or stupid, it's just that they have not ever developed the skills of critical thinking and self-reflection.

Before talking about math, let me point out a specific example in real life. Over by the Mandeville center, down where the art people have a space to do their thing, there used to be a painting on the ground (someone painted over it). It was a picture of what I've always supposed was Mary and baby Jesus (woman, child, halo). Next to the figure was painted "Do not push your beliefs on others" (or something like that). I am not sure whether the person who painted that appreciates the irony. As a personal philosophy ("I will not try to force you to believe what I believe"), it works just fine. However, it cannot possibly be given as an instruction to someone else in an internally self-consistent manner ("You should not impose your beliefs on me" -- by making this statement, the speaker is trying to impose HIS beliefs on the listener, thus doing what he says one should not do). A lack of critical analysis leads to nonsense.

The same internal inconsistency exists with bland, naive relativistic statements such as "all religions are the same" (different religions have different claims to truth, and they are obviously incompatible) and "you should accept everybody" (the person doesn't accept you because you don't accept someone, but then that person is himself not accepting of everyone). Most of the people I know who talk that way haven't ever really spent time thinking and reflecting on their system of beliefs -- it's much like how Christians who never study the Bible come to false conclusions about how God "should" behave (really, how God "does" behave).

But what does this have to do with math? I'll grant that philosophical reasoning is a bit stickier than mathematical reasoning. At least with mathematical reasoning, we (the math community) have a set of mostly agreed upon fundamental beliefs (axioms) upon which we build mathematical structures. There also isn't a whole lot of room for "personal interpretation" when it comes down to the formalism of mathematics (there is actually room for personal interpretation when it comes to trying to understand math... but that's a different story).

Where was I? Oh yeah, what does this have to do with math? A lack of critical analysis leads to nonsense. The nature of mathematics for a lot of students is that you find some numbers, or a formula, and plug stuff in, and move some terms around, and get some answer. It doesn't matter where the problem came from and it doesn't matter how silly the answer is. It doesn't matter that it's a word problem computing the terminal velocity of a falling object, and that the final answer they got is "t = 10 minutes." The problems have no intrinsic meaning to the students, so their answers often have no intrinsic meaning.

Here are a few examples of relatively simple mathematical processes that a lot of people know they're supposed to do, but don't actually know why it's the right thing to do (other than they were told to do it that way):

1. Why do we "carry the 1" in addition? (Even more fundamentally, why do we start on the right side when we add? -- This one isn't about being right or wrong)
   
2. Long division: Divide, subtract, multiply, bring down, repeat... What does this process actually do?
   
3. x^n * x^m = x^(n+m) -- If you're not familiar with typing math, x^n is x to the nth power.
   

The lack of understanding of these operations is the result of how math is taught. My memory of learning math in the California public school system goes something like this:

• Kindergarten - Learn to count
• 1st Grade - Learn to add and subtract
• 2nd Grade - Learn to add and subtract in multiple columns
  
• 3rd Grade - Learn to multiply, introduction to fractions
  
• 4th Grade - Learn to multiply in columns and do long division
  
• 5th Grade - I don't even remember what I was supposed to learn...
  

The point here is that math was taught as a process and a bunch of rules that you memorize and reproduce. There were some parts where math's intrinsic value was highlighted. For example, I can remember "clock arithmetic", which was a low-level introduction to modular arithmetic, but it still played out like another set of rules to memorize.

I was fortunate that my school and my teachers were willing to put in extra effort for the bright students. There was a weekly thing in the library for the more proficient math students (I remember learning about exponential growth there by the chessboard and grains of wheat problem).

My fourth grade teacher borrowed books from the library to keep me occupied while she taught arithmetic to the rest of the class (she got me books on finance, but since I didn't have a good conception of money at the time, the real value didn't sink in until much later... for example, if I understood no-interest government student aid loans and all that stuff, I would have realized that when I started my undergraduate studies, I could have taken out a maximal student loan and earned 5% interest in a CD somewhere and make free money off the government. Even at 3.5% APY on a $10,000 student loan, because I was in school for 9 years and the loan would be without interest would have given me an extra $5500 for free. Then taking the $5500 and continuing to earn 5% APY on it from age 27 to age 65 is an extra $35000 in retirement funds for doing essentially nothing. Of course, I probably could have qualified for more at the start, and probably could have gotten better than 5% return on average (I think the average market growth rate is around 8%), but I think this makes my point about the gap between being able to compute things and being able to critically analyze what the math is saying.

Who is to blame for the inability of students to think critically about their math? It's not the fault of any individual, but it's a faulty system. Teachers teach math this way because this is basically how they learned it. Students learn math this way because they have no other model for learning math. The students grow up, and this way of thinking is never corrected, so it propagates itself to the next generation.

Fortunately, I do believe change is in the works. There are a number of educators who are looking into the failing mathematics system, and while I'm not involved there, I do hope they find true solutions and not false ones like having teachers teach to standardized testing... that just reinforces the same problems.