How to Teach Math
A very interesting article in the New York Times today, on the limitations of teaching math through real-life examples. What do you think of it?
Study suggests math teachers scrap balls and slices
By KENNETH CHANG
One train leaves Station A at 6 p.m. traveling at 40 miles per hour toward Station B. A second train leaves Station B at 7 p.m. traveling on parallel tracks at 50 m.p.h. toward Station A. The stations are 400 miles apart. When do the trains pass each other?
Entranced, perhaps, by those infamous hypothetical trains, many educators in recent years have incorporated more and more examples from the real world to teach abstract concepts. The idea is that making math more relevant makes it easier to learn.
That idea may be wrong, if researchers at Ohio State University are correct. An experiment by the researchers suggests that it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations, in this case 40 (t + 1) = 400 - 50t, where t is the travel time in hours of the second train. (The answer is below.)
“The motivation behind this research was to examine a very widespread belief about the teaching of mathematics, namely that teaching students multiple concrete examples will benefit learning,” said Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State. “It was really just that, a belief.”
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liguangdui says:
Added on April 27th, 2008 at %1:%Apr %p看了这个观点，我想到了中国有一部比较出名的电影《一个都不能少》，上面就是有一段很好的小学数学教学的例子。我建议我们的工作者可以把这个电影中的相关部分作为教研讨论的形式来看，很通俗的，却也是很深刻的，将教育教学与实际需要紧密结合，效果是出人意料的好。
Sara Lam (blog author) says:
Added on April 27th, 2008 at %1:%Apr %pThis is an obvious case of the straw man fallacy. Many people promote incorporating more real life examples in math teaching, but I’ve never in my life heard anyone argue for completely replacing teaching of abstract concepts with concrete examples, which is what they did in the study. The usual way of incorporating examples in the teaching process is to have students work on a few real life scenarios that involve the concepts you want to teach. This lets the students think about the concept in a very basic way from scratch so that they really have to work it out themselves. Then, the teacher “translates” this into the abstract concepts. This is followed by practice so that the teacher can see whether the students really got it and to reinforce students’ ability to use the concept in math problems. During the practice stage or at the end of the unit, the teacher might give students more real life problems to work on. At this stage, the purpose of the real life examples is to check if the students really understood the concept enough to apply it in real situations which is more complicated than just solving an equation. It requires the additional step of analyzing the situation in order to deduce a mathematical equation from it. Indeed, students who don’t really understand the concepts often have difficulty with this step. The real life examples serve a certain function in the learning process, but I can’t imagine teaching a math unit ONLY using real life examples without generalizing to abstract concepts. A more scientifically sound way of setting up the experiment would be to compare students who were ONLY taught the abstract rules with students who were first taught the rules through concrete examples and then were taught the abstract rules.
Steven Liu (blog author) says:
Added on April 27th, 2008 at %1:%Apr %pThe article’s main point is that by only using concrete examples students are not learning how to abstract. Without abstraction, they are unable to apply the principles learnt to other situations.
In general I agree with this. Mathematics is possibly the most abstract of subjects (with due respect to physics and philosophy), and it’s in that abstractness that its beauty and utility lies. Many problems in pure maths, thought to be no more than extremely challenging mental exercises at the time, have found uses in industry and research. Were mathematicians always to refer to real life situations as a reference point it is likely large swathes of what we know in mathematics would never have been discovered.
However, any good teacher at any level will need to use real life examples to explain new concepts. Very, very few students are good enough to do it without. Some students will need more concrete examples to learn a concept, whilst some will get it after the first. But of course in the end you have to move from concreteness to abstraction.
Speaking from the RCEF’s point of view though, the example in the article would be a poor one for many of our students. The main objects in the problem are trains, which many students have never taken or seen. Whilst you may say that in today’s television age such objects should be familiar to millions, the reality is that without first hand experience, such objects remain something of a mystery.
And so by introducing unfamiliar objects to the problem you are creating a level of abstraction within the concretization process! I think this is what we have to be careful of as rural teachers more than anything else; that many of the state textbooks are riddled with these problems, and that we need to create new examples utilize use themes and objects that they can relate to.
liguangdui says:
Added on April 27th, 2008 at %1:%Apr %p我想我们教学数学的根本目的在于应用，这也就是为什么教育具有一定的民族性和区域性（或者说是需要本土化）的依据，我理解的从具体到一般在到具体这个过程就是从具体的现象到一般的概念和规律再到指导解决具体问题的过程。当然，不能因为具体而局限于具体本身，比如只是让学生知道本地的东西，而不介绍一些他们不熟悉的东西也是片面的，影响他们对一般的掌握。哎，我这种对英文理解后的中文表达，好象连自己的中文都象是翻译的一样，太不地道了：）
Mingming Zhou says:
Added on April 30th, 2008 at %1:%Apr %pI can sort of making up a good explanation for their finding. All might come down to the sample employed– college students, who have been way beyond the stage where humans have troubles in understanding abstract concepts. For younger children who are not quite capable of understanding those concepts in the way they are set up, real-life examples/manipualtives would be very helpful. Still, it is only a way of scaffolding, not to replace other means completely.
Kiel Harell says:
Added on May 1st, 2008 at %1:%May %pI agree with Mingming Zhou. I think looking at the sample used is important. In addition to college students being more capable of understanding abtract concepts, as she pointed out, they also differ from k-12 students in their level of investment. As a 7th grade teacher, I often use real life examples to show the students the value of a given objective before moving the lesson toward the abstract. This creates a buy-in with the students that make them more interested in pursuing the abstract application. My students require a motivation beyond mastering an abstract concept.
This being said, many teachers at my school have activity-driven lesson plans that fail to teach abstract concepts because the teachers find it easier to teach kids how to play a game than to teach them an abtract concept. This, however, doesn’t suggest that using real life examples or playing games are bad teaching tools. They just need to be linked to the abstract concepts. I would like to have seen what the two lessons in the study looked like. If the latter really did not teach the sample group any abstract concepts then it’s no surprise that they didn’t master any.
My apologies if other posters have touched on these ideas. I do not read Mandarin (yet).
Umesh Sharma says:
Added on May 1st, 2008 at %1:%May %pI agree with Sara that examples in math should be given only to illustrate the concept just is an writing persuasive essays. One cannot completely do away with rules and algorithms. One cannot just give the example of an apple dropping on Newton’s head and from their try to derive Newton’s Laws.
Umesh
Wei Ji Ma (blog author) says:
Added on May 1st, 2008 at %1:%May %pI am glad this piece has elicited so much discussion. I am learning a lot from the comments, and it is great to see actual teachers commenting! I wish the readers of the NY Times article would also come to our blog
I was struck by what Kiel wrote, “Many teachers at my school have activity-driven lesson plans that fail to teach abstract concepts because the teachers find it easier to teach kids how to play a game than to teach them an abstract concept.” This seems to be an example of how a good method (linking abstract concepts to activities) is implemented wrongly (ignoring the abstract concepts) and then gets criticized based on the implementation.
This being said, when I was in school myself, I greatly enjoyed the abstract math problems and didn’t like elaborate “story problems”, where you had to distill the relevant information from a description of a real-life situation. But maybe I was an exception as a kid (I went on to become a theoretical physicist).
Zili says:
Added on October 4th, 2008 at %1:%Oct %pHere is the abstract of my colleague’s upcoming talk. I’ve never been to his talk, but will in the future. Zili
JAMES STIGLER, PH.D.
UCLA DEVELOPMENTAL PSYCHOLOGY
LESSONLAB RESEARCH INSTITUTE
Monday, October 6th
11:45 - 1:00
FRANZ 3534
“Improving Mathematics Teaching: A Journey Beyond TIMSS Video”
Videos of classroom teaching collected as part of the Third International Mathematics and Science Study reveal that teaching is a cultural activity, varying more across cultures than within. It is learned implicitly; it is largely based on hidden cultural scripts; it is embedded in wider cultural beliefs and practices; and it is difficult to change. Given these facts, how can teaching be improved?
In this presentation I will briefly describe most recent findings from the TIMSS Video Studies of mathematics and science teaching in seven countries, and discuss the implications of these findings for (a) current debates about mathematics teaching and learning in schools, and (b) efforts to improve teaching through professional development.
Zili says:
Added on October 4th, 2008 at %1:%Oct %pI learned from a colleague of mine that at UCLA’s elementary school, the rules of multiplication are not taught. Instead, students are encouraged to figure out their own ways to solve a multiplication problem. Not surprisingly, different students come up with different ways to solve the same problem. My colleague, whose background is hard science, appreciates the school’s approach. But he still tutors his child at home. The school, though considered one of the best in Los Angeles, is not so considered (by some UCLA parents) in math/science education.
I appreciate this problem solving approach because, when teaching college students (e.g., signal detection theory), they seem to be completely focused on memorizing templates for different problems (in order to do well in exams). I expressed my frustration to a friend, who is a recent UCLA graduate and the best student in one of my classes. He said that this is very common. He is a Chinese-American growing up in the Bay Area. My sense is that if this is the mindset of the college students, then it is too late trying to teach them to actually _understand_ the concepts. I am still trying though.