"The lecture method is a process by which the lecture notes of the professor get to the notebooks of the students without passing through the brains of either."

**Part I. Teaching Physics**

The video (Confessions of a Converted Lecturer) is over an hour; it had crappy sound quality on my laptop (not so bad on my desktop); it's mostly just one person talking and showing a few slides of his data analysis; and it's not even about math. But it was totally worth it.

Eric Mazur is a physics professor at Harvard. He teaches physics to pre-med students. They don't have an interest in physics, but they are very good students. Eric was, by conventional standards, a very good teacher. But then he got his hands on the Force Concept Inventory (FCI), which has conceptually-based physics questions with no calculations, and it changed him. His students were doing well when measured by conventional physics questions, but could not answer questions that physicists see as much easier. His students could recognize which formulas to use for which problems, and could then work their way through long calculations, but they still didn't understand some of the most basic principles of physics.

As he described this process students use, of learning just the procedures, he said:

"Some students told me that physics is boring, and I could never imagine it. But, imagine that it's been reduced to following a recipe, that you don't even understand. Yes, then it would be boring."He said that most students come in, even after getting a 5 in an AP physics course, as Aristotelian thinkers. (Aristotle codified some of the ideas the ancients had about physics. Those ideas were often wrong.) After a conventional Harvard physics course, and after scoring well on a conventional exam, most of the students are still 'Aristotelian thinkers' - "they haven't understood the material in week two, on which everything else depends."

I remember one thing like this from my college physics course. I remember a question about something rolling off a table, and what its path would be. It's hard for me to believe I didn't always imagine a parabolic path. But I do remember being surprised. I think my previous conception was that the object would go straight down. It sounds so wrong to me now... Apparently, there are lots of basic concepts in physics like that, where people have firmly entrenched wrong ideas, until they've really understood the principles involved. Then they have trouble believing they ever had the wrong picture in their heads before. We don't remember not knowing...

Is math like this too? Can you imagine not knowing that 3+11 is the same as 11+3? Kids don't know that at first, and when they get it, it's a powerful tool, since counting up 3 from 11 is way easier than counting up 11 from 3. Are there

*mis*conceptions that we all had? I can't think of any right now. (Anyone who works with little kids, are there wrong ideas that come to us naturally?)

When Mazur gave a pre-test to his Harvard students, using the Force Concept Inventory, and told them how low their initial scores were, they wanted to spend time going over each question to understand it. There wasn't time during class, so he arranged to meet with interested students in the evenings. In this voluntary, ungraded environment, perhaps they felt more comfortable expressing their confusion. One evening, he gave what he thought was a good explanation of why the forces a light car and a heavy truck exert on one another in a crash are equal (as they must be, if one understands Newtonian mechanics), but what he got was a roomful of confused looks. He tried again, adding acceleration into the picture, and saw he'd only increased the confusion. He knew that about 40% of them already understood this idea, and, giving up, asked them to explain it to each other. Out of the chaos was born Mazur's version of "peer instruction".

Nowadays, when he teaches, he explains for just a few minutes, and then asks a concept question. In a lecture hall with hundreds of students, they all begin to talk. Here's the process:

- He shows the question and explains it,
- Students silently think for 1 to 1 1/2 minutes,
- Individuals answer (using clickers, which allow immediate tallying of anonymously given answers),
- Discussion ensues among small groups of students (he doesn't try to control the groupings),
- Students answer again, after their group has come to some consensus,
- He explains the answer (and goes on to repeat the process for the next small bit of lecture).

There's more to savor in the video. If you can find the time to watch it, I think you'll be glad you did.

**Part II. Teaching Math**

So how do we use this for math? There is some work being done along similar lines. Jerome Epstein has developed a Calculus Concept Inventory, and found similar results - that conventional courses don't help the students to truly understand the concepts.* And Cornell has the Good Questions Project (which I mentioned before, here).

Good questions hook in to a student's personal experiences. This one may work at Cornell (the author's students really liked it), but would definitely not work at my college:

An article in the Wall Street Journal's Heard on the Street column Money and Investment (August 1, 2001) reported that investors often look at the change in the rate of change to help them get into the market before any big rallies. Your stock broker alerts you that the rate of change in a stock's price is increasing. As a result:My students definitely do not have stock brokers, nor do most of their teachers. As I glance through the Good Questions file I found here, I don't see others like this one. Perhaps someone has thought about the issues of privilege already. Here are a few questions from the 53-page list:

a. you can conclude the stock's price is increasing

b.you cannot determine whether the stock's price is increasing or decreasing

c. you can conclude the stock's price is decreasing

True or False. At some time since you were born your weight in pounds equaled your height in inches.

True or False. As x increases to 100, f (x) = 1/x gets closer and closer to 0, so the limit as x goes to 100 of f (x) is 0.

Your mother says “If you eat your dinner, you can have dessert.” You know this means, “If you don’t eat your dinner, you cannot have dessert.” Your calculus teacher says, “If f is differentiable at x, f is continuous at x.” You know this meansI'm hoping all these good questions will help me to teach a much better section of calc II this fall. (It sure would help if I could get a 'smart classroom' to teach in for the fall, so I could have students use those clickers...)

(a) if f is not continuous at x, f is not differentiable at x.

(b) if f is not differentiable at x, f is not continuous at x.

(c) knowing f is not continuous at x does not give us enough information to deduce anything about whether the derivative of f exists at x.

I'd like to think about how to use these same ideas at the algebra level, too. How do we ask the questions that will get students thinking about concepts rather than just focusing on the procedures? There's lots of good work being done by the bloggers I read. Jason Dyer, at Number Warrior, has a Fractions Concepts test that I'll use when I review fractions at the start of the term with my beginning algebra students. Hmm, maybe we could all work together to create an Algebra Concept Inventory...

[Hat tip once again to Dan, whose blog, Math for Love, has been very inspiring.]

______

*This is not available online, since they'd like to keep the content of the questions private, in order to retain their validity as a test instrument. I have emailed the author, asking for a copy to use in my calculus course in the fall.

Thank you for posting this. I started watching the video, and have to pause it to go to bed, but I'll definitely finish it this weekend. I'm intrigued by the ideas he brought up, and I have an inkling that if I probed my students deeply enough, they may not grasp the concepts like it seems they do.

ReplyDeleteMs. Cookie

[Matthias H. Fröhlich wrote to me by email, because he couldn't get this meaty comment in.]

ReplyDeleteHe wrote:

I started watching the video and do almost fully agree to Professor Mazur's arguments.

However, even being a physicist myself, I do highly disagree with the example of the truck and the light car. Please let me share my thoughts and I would be grateful for all comments.

When a truck and a light car are going by the same speed v then the momentum P_t = M_t * v of the truck will be larger than that of the light car P_c = M_c * v, as the mass M_t of the truck is larger than that of the light car M_c.

They collide head-on and come to rest during time t. Now the average force F_t exerted by the truck is F_t = P_t / t while the average force exerted by the car is F_c = P_c / t.

As the momentum of the truck is larger, i.e. P_t > P_c hence it follows F_t > F_c. Q.e.d.

Alas, did I just prove a Harvard professor in physics wrong? Well, let's take one thing at a time.

First of all, I think that the question is given much to vague, as to come up with a precise answer in terms of Newtonian physics. E.g. are the vehicles going by the same speed or which one is faster than the other?

The thing boils down pretty easily, when assuming, that the light car goes so much faster, that P_t = P_c. Voilá, same force exerted.

But what happens, when this is not the case? Well, let's assume (without loss of generality - ha, I always wanted to write that ;-) that the truck bears a higher momentum than the car.

Obviously, both vehicles will then *NOT* come to rest at the point of collision but the truck will "push" the car a bit backwards.

Now, while pushing it backwards, there is still some momentum to be "transferred" somewhere - remember, it can't get lost! Another one of those nice laws from Newton.

Now, what will happen is, that while both vehicle are now moving jointly as one "solid body", some frictional forces will "transfer" momentum (try to imagine where that additional momentum comes from - nice question, by the way) and bring the whole wreckage to halt.

(continued in next comment...)

Matthias H. Fröhlich wrote, Part 2:

ReplyDeleteHowever, is that force now "exerted" by the light car? It may as well partly be exerted by the blocking wheels of the truck or some parts that have been bend down in the collision and are now crashing over the asphalt or else.

However, if we "simplify" our model in such a way, that we say, both come to rest and the light car bears the same "responsibility" for coming to rest as the truck, than we can say, that the average force exerted was the same.

To me, this is much to much of a simplification as to imply it in the way the question is posed. Our physical models do always depend on the preconditions, limitations and simplificatoins that are applied. To me, there are much to many "silent" assumptions in the example as to make a sound argument.

I think the example given by Sue VanHattum with something rolling of a table is brilliant in that respect. A lot of things will *NOT* continue on a parabolic path except when that table is located on the moon without atmosphere and any friction of air.

Turn the fan of the air condition to highest setting and roll down a little paper ball from your desk. Rest assured, it will *NOT* go down a parabolic path, it might even go upwards...

IMHO it is a crucial point when teaching physics to raise an understanding, that every physical model is only valid within its own limitations and all results obtained from that model do need to be carefully scrutinized for there validity in "real" life.

In that respect the question given in the example can just not be answered by the simple choices offered.

The same does surely hold for a lot of other sciences, including mathematics - so we need to be extra careful, when coming up with those "conceptual" questions of understanding. More often than not, we "experts" might not even be consciously aware of all the assumptions, limitations, preconditions and so on, that we have silently put for granted. However, they are not - they need explanation!

To me those absolutely indispensable explanations are not sufficiently given in the example in the video and in a lot of similar "conceptual" questions. What do you think?

Mattias, what I think is ... I said it wrong. I was thinking of writing about how, even though I loved physics, and did well in my class, I don't really know much physics. I didn't take notes on that example, and I'm sure I just wrote it up wrong.

ReplyDeleteSo everyone who watches the video, your homework is to come back and explain the car-truck thing more properly. ;^)

Nice post.

ReplyDeleteThe Contexts for Learning Mathematics curriculum provides for teachers what they call the Landscape for Learning. It's a kind of anchor chart or concept map of all the related ideas in their unit. They divide the content into models, strategies and concepts. It's really the most helpful representation for content I've ever seen. When my students were tutoring on money and time this semester we made some to help organize our teaching. http://mathhombre.blogspot.com/2010/02/more-money.html

I'd be interested in pitching in on an algebra landscape.

Those materials sound exciting. I'd like to get my hands on them.

ReplyDeleteWhat would be involved in an algebra landscape? (Your money landscape and the mindmaps I've seen don't work so well for me. I like my organizational devices to be more one-dimensional, I guess.) I've been thinking about how I want to teach my beginning algebra course...

As my brain whirred last night and refused sleep, I got to thinking about fraction concepts. Here's a multiple choice problem that I thought might actually get students thinking.

ReplyDeleteIs n/d less than 1? (Choose the best answer.)

(a) Yes, because it's a fraction

(b) If n < d it is

(c) If d < n it is

(d) Impossible to determine because we don't know what n and d are.

Choice d is troubling me, since it's somewhat reasonable. But I think it might get at a trouble some students might have with variables.(We don't need particular values for n and d, just their relationship.)

I prefer projects and math circles to multiple choice questions, but my semester will start with over 40 students in each class. That may not be the hundreds Mazur developed his method for, but I think he can help me get more students involved, especially at the beginning of the term, when they haven't yet built up a sense of safety and community.

Is n/d less than 1? (Choose the best answer.)

ReplyDelete(a) Yes, because it's a fraction

(b) If n < d it is

(c) If d < n it is

(d) Impossible to determine because we don't know what n and d are.

given no context,

(d) is the only thing

that even *resembles*

a "right" answer.

A 2/1

B (-1)/1

C 2/1

give the necessary

counterexamples

to (a), (b), & (c).

Oops! I was thinking of positive numbers...

ReplyDelete(Your counterexample for b should be something like -2 / -1.)

Revised:

n and d are both positive. Is n/d less than 1? (Choose the best answer.)

(a) Yes, because it's a fraction

(b) If n < d it is

(c) If d < n it is

(d) Impossible to determine because we don't know what numbers n and d are.

What do you think?

(I'm thinking of this as a question to provoke conversation, so I'm more comfortable with the ambiguity of the last choice than I would be if it were graded. I'd still like to find a wording that makes it more clearly wrong.)

thanks for the correction...

ReplyDeletedon't know *what* i was thinking.

i'd go ahead and make the #'s

positive *integers*: extra

"vocabulary" i suppose but

it might clarify the thinking.

meanwhile i'd rather this

weren't a "multiple choice"

question at all...

"shalmaneser says

n/d < 1

for *any* numbers 'n' and 'd'

'because fractions are

parts of wholes'.

athalijah doesn't agree.

can *you* find numbers

('n' and 'd') such that

the statement 'n/d < 1'

is *false*? how?"

kinda thing.

If I had a proper class size of 15 to 20, I'd agree. But the multiple choice is a way to use clickers (or Kate's low-tech alternative), and get everyone participating.

ReplyDeleteEven with my 40 plus students in the first week, I can see what everyone thinks. Having a limited bunch of choices might also help them focus. Id rather not even be thinking of that, but my students do not like math, and anything that helps them stay with it is probably good, especially at the beginning.

Sue, thanks for raising this issue and providing a link to Mazur’s talk.

ReplyDeleteI can’t help but compare Mazur’s approach to other approaches to teaching. He is essentially incorporating a group problem solving approach into the large (auditorium) class format. But as a physicist, he took a data-based route, plus some good fortune, rather than pedagogical theory, to develop his method. And as a physicist, he is not taking the educator’s approach of trying to understand the students’ thinking. Mazur says that the experts don’t understand the student’s misconceptions because they are far removed from the time when they did not know the material, while the students who understand can relate to the thinking of the confused student. He is implying that peer instruction is to some extent a teacher-proof way of teaching because the peers, not the teacher, are clarifying the target concept to the student who doesn’t understand. The teacher doesn’t need to gain the perspective of the confused student.

In Anticipating Children’s Thinking: A Japanese Approach to Instruction Tab Watanabe describes how “The anticipation of children's thinking plays an important role before, during, and after the lesson.” Deborah Ball also includes understanding the students’ thinking and misconceptions as part of the knowledge that teachers need. It isn’t generally understood that this kind of knowledge is important for teaching, because being able to effectively apply the content knowledge is sufficient for other professions. It’s in education that we need to also understand the process of acquiring that knowledge.

Peer discussion and debate are part of problem solving approaches to teaching in several different traditions, including the constructivist, Davydovian (Vygotskian), and Japanese approaches. But it is not just that someone who understands how you are thinking can explain it to you better than an expert. It is not just a simple transfer of a ready-made concept. It is the active engagement with the issues, the deep thinking and reflection, that results in making the connections and having the insights that lead to true understanding. No matter how it is explained, to understand you have to struggle with the material and reconstruct that conceptual understanding in your own mind. Mazur’s method brings this feature into the large class format.

Creating and selecting good problems are another part of Mazur’s method. Selecting good problems is critical to any problem solving approach, including Davydov’s curriculum and Japanese lessons. For example, Schmittau and Morris said that Davydov’s curriculum “consists of nothing but a carefully developed sequence of problems, which children are expected to solve.” Takahashi discusses “Carefully selected word problems and activities, and their cohesiveness” as the first of three characteristics of Japanese lessons.

Mazur found that improving the quality of problems improved student learning. How does Mazur develop good questions? It’s by studying students’ thinking and misconceptions. To be a better teacher, Mazur couldn’t get away from understanding students’ thinking. I would think that that understanding improves his explanations, but it doesn’t eliminate the need for students to discuss and debate, to grapple with the ideas.

BTW, I would recommend the Schmittau and Morris article (link above) for anyone teaching beginning algebra.

Thanks for the thought-provoking comment, Burt.

ReplyDelete>Selecting good problems is critical to any problem solving approach...

And I'm not at all sure how well I do that. If I do it well, I'm not conscious of how I go about it. This is something I'll want to think about more when I get back in the classroom (in the fall).

Ooh~ that was me in physics! I was always great with pushing numbers but weak conceptually. I took a class recently at UCLA only to find that nothing has changed. =(

ReplyDeleteThank you for this post. It has given me much to think about in my classes as well.

And also, your fractions question, I get (d) all the time in my ninth grade classes. I think it's the multiple ways that a variable is used for. We don't always use a variable as a "variant" very often. They mostly understand it as a specific, but not yet known, number to "solve" for. I think this is why they have such a hard time graphing because we suddenly switch to a different way of using variables. I've always wanted to have such a discussion, what do you think?

Thank you. Yes, I agree. I don't know how to talk about it clearly, but I'm thinking that students have issues with analyzing a situation that doesn't have specific numbers.

ReplyDeleteI promised Sue to try to come up with some hopefully good "concept question" about mathematics. Here's my first try.

ReplyDeleteYou will probably remember the example of the quiz show were the candidate is finally presented three closed doors. Two are holding a goat, one the incredible valuable prize.

After choosing one door (still closed), the host opens "a different door with a goat behind it" asking the candidate whether s/he would like to change his/her choice.

Question is now, whether it is better to change choice or stay with the initial selection? Result is well known, that changing will double the probability of winning from 1/3 to 2/3. This will be nothing new to a lot of people with at least some mathematical background and can easily be given to a class as a reading assignment.

But now for some hopefully new "concept questions":

1.) Consider that the host will not open "a different door with a goat behind it" but randomly just one of the two doors holding a goat. It might even be the one, the candidate has initially chosen!

Still, the candidate is given the chance to change the initial selection. S/He will surely do so when his/her chosen door is opened.

In all other cases, will it now still be better to change choice or not? Try to motiviate your answer without calculations.

Will the best probability for winning be higher, smaller or equal to the usual setting? Again, how can you argue without calculations?

Can you come up with some reasonable arguments?

What a fascinating question... I'm going to begin logging the natural misconceptions my pre-schoolers have about math. I never thought to look at it from this angle (mostly I'm scratching my head and trying to guide them towards the right answer without teaching it!) But we learn more from mistakes than from answers, and this would make a fascinating observation topic!

ReplyDeleteOn another note, that quote about lecture notes is priceless and sums up my high school/college experience. I bet most people (professors included) feel this way, and yet how many people care enough to do something about it?

Thanks, Pilar, that will be exciting to hear about.

ReplyDeleteI too am interested in the idea of an Algebra concept inventory. I'm a math grad student at the Univ. of Utah, and I am teaching a college algebra class for Business majors. I've started a preliminary Algebra concept inventory. I would love your feedback on it. You can find it on my site at:

ReplyDeletehttp://www.math.utah.edu/~jasonu/aci.pdf

ps I lifted your fraction question from the comments to this post. I hope that's OK.

Lift away, Jay, I'm honored! I'll try to look at your stuff. (I'm excited to see it.) But this is a crazy week for me, and I'm headed out on a trip on Friday. If you don't hear from me, please email me. (suevanhattum on hotmail)

ReplyDelete