Given the amount of time I've spent writing about academic issues this week, it's only fitting that the science story getting the most play is about math education. Ed Yong provides a detailed explanation, and Kenneth Chang summarizes the work in the New York Times. Here's Ed's introduction:
Except they don't really work. A new study shows that far from easily grasping mathematical concepts, students who are fed a diet of real-world problems fail to apply their knowledge to new situations. Instead, and against all expectations, they were much more likely to transfer their skills if they were taught with abstract rules and symbols.
The use of concrete, real-world examples is a deeply ingrained part of the maths classroom. Its advantages have never really been tested properly, for they appear to be straightforward. Maths is difficult because it is a largely abstract field and is both difficult to learn and to apply in new situations. The solution seems obvious: present students with many familiar examples that illustrate the concepts in question and they can make connections between their existing knowledge and the more difficult concepts they are trying to pick up.
The train problem is a classic example. Another is the teaching of probability with rolls of a die, or by asking people to pick red marbles from a bag containing both blue and red ones. The idea is that, armed with these examples, students will recognise similar problems and apply what they have learned. It's a technique deeply rooted in common sense, which is probably as good an indicator as any that it might be totally wrong.
As with any research involving subjects more complicated than a diatomic molecule, of course, there are a billion ways that the study could be going wrong. After the initial shock of seeing the findings, though, it's not actually that surprising. I wonder if this isn't related to the well-known problem of compartmentalization in science education.
Well, ok, I'm not sure that's the technical name for it, but anybody teaching in a mathematical science is probably familiar with what I'm talking about. When we present new mathematical apparatus for solving physics problems, students will often get tripped up on basic elements of calculus. When asked "Haven't you done this in your math classes?" they often respond with an answer that amounts to "Yeah, but that's math class. It doesn't have anything to do with physics." We also hit some student resistance when we ask them to do computer simulations of physical situations-- they grumble about having to write fairly trivial bits of code, despite the fact that they're taking computer programming classes at the same time.
This is a persistent source of annoyance for college science faculty-- many students seem unable or unwilling to take things learned in a class in one department and apply them to subjects studied in a class in another department. The problem with concrete examples noted in the current work seems like just a more extreme version of the same thing. If you teach algebra using the classic example of trains leaving stations and so on, that goes into the mental compartment for "math problems with trains" and when they get to a later problem about chemical reaction rates, they won't think to apply the same technique, even though the two situations may be mathematically identical. After all, that was about trains, not chemicals.
Of course, then I wonder if this isn't a confounding factor in their study-- the stuff I've read says that the students who learned a mathematical technique using abstract symbols did a better job of applying it to a new situation than students who learned using real-world examples. The "new situation" described by Ed sounds like it maps pretty easily onto the abstract symbols of the training-- they're games that fall into the same compartments. What would be interesting to see (and I don't know if it's in the actual paper) is how well the abstract symbol people did when asked to apply their knowledge to one of the concrete examples.
The issue might be that "problems with abstract symbols" is mentally closer to "games with odd objects" than "problems with measuring cups" is. In which case, it's be interesting to see how students who learned using "problems with abstract symbols" do when given "problems with measuring cups" on a test.
That may be in there-- I can't access the paper itself from home, and I'm kind of busy at work. I'll try to find time to look at it this weekend.
Regardless, there's a tricky balancing act here. Even if it's true that the best method of teaching problems is to use abstraction rather than concrete examples, you leave out the concrete examples at your own peril. The point of giving concrete examples is not just to aid learning, but also to stave off questions of the form "Why do we have to learn this, anyway?" and more importantly, student evaluation comments of the form "This class didn't have anything to do with the real world."
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I was up until midnight writing my lesson plan for next week, which has to be turned in to the principal's office this morning. Excerpt:
Professor Jonathan Vos Post
Rose City High School Room 133
CAHSEE Math Lesson Plan [Optimistic plan, may take more than 1 week; assess]
28 Apr-2 May 2008
=====================================
Monday 28 April 2008
(1) Module 2: Statistics, Data Analysis, and Probability
Review Of Median, Mean, and Mode (refresh from Thursday 24 Apr 08)
CA Content Standard 65DAP1.1: Compute the range, median, and mode of data sets.
Example: Kobe Bryant's top 5 playoff games [L.A. Times 24 Apr 08, p.D1]
2001 vs. San Antonio, Game 1: 45
2001 vs. Sacramento, Game 4: 48
2006 vs. Phoenix, Game 6: 50
2007 vs. Phoenix, Game 3: 45
2008 vs. Denver, Game 2: 49
What are the median, mean, and mode of Kobe's top 5 playoff games?
Rubric: justify procedure (comprehension) and calculate to correct answer (application). Note: on Thursday, one student had trouble with median meaning "the middle" of the data set, because she forgot that the data must first be sorted from smallest to largest. The example given is chronological, student must re-sort by score.
(2) Module 4: Measurement and Geometry
Review of Areas of Rectangles and Squares, Parallelograms
CA Content Standard 7MG2.1: Use formulas routinely for finding the perimeter and area of basic two-dimensional figures
Standard: Application: Use a concept in a new situation or unprompted use of an abstraction. Applies what was learned in the classroom into novel situations in the work place.
QUIZ: 4 questions
41. Find the area of an Olympic Swimming Pool with length L of 50 m, and width W of 25 m. This is the regulation size.
Answer: A = WL
= 50 m * 25 m = 1250 m^2
42. Handball is played on a court forty meters long by twenty meters wide. That is the Olympic regulation size. Find the area.
Answer: A = WL = 40 m * 20 m = 800 m^2
43. Find the perimeter of an Olympic Swimming Pool with length L of 50
m, and width W of 25 m.
Answer: P = 2L + 2W = 2*50 m + 2*25 m = 100 m + 50 m = 150 m
=====================================
I'll be at school today 7:30 a.m.-3:00 p.m. and then zoom home so that my wife and I can rush to Beverly Hills for the 40th Anniversary Screening of 2001: A Space Odyssey at the Academy of Motion Picture Arts and Sciences with guests Tom Hanks, Keir Dullea, and Douglas Trumbull on Friday, April 25 at 7:30 PM.
This is a persistent source of annoyance for college science faculty-- many students seem unable or unwilling to take things learned in a class in one department and apply them to subjects studied in a class in another department.
Reminds me of working as a TA for "physics for pre-meds" when I was an undergrad, and I discovered that some of the students separately memorized V = IR, I = V/R, and R = V/I. When I pointed out that they were identical formulas related by algebra, they pointed out (with a sardonic roll of the eyes) that this wasn't an algebra class. This was not an isolated incident.
I was also continually surprised when, TAing in grad school, students thought it was unfair if elements of a previous semester's physics class were re-used in the current class. A student once told me he thought each semester should be a completely self contained unit. This was at Caltech where I would have guessed that the idea of science building on itself would have been clear. (I was sufficiently annoyed by this kid that in response I pointed out that they were also going to be required to add integers, and that there was an elementary school down the street if he needed a refresher. About 80% of the class thought this was hilarious, about 20% thought I was mean.)
I actually think I had sort of the opposite problem. For a long while I was undecided between math and physics and so the math classes I took were the math major ones, not the ones designed for physicists. Proofs and all. My calculus class was taught out of Michael Spivak's Calculus, which I still consider the most wonderful textbook in any subject ever. But trying to approach physics problems with the level of rigor that you learn in the math department is futile. (It's not, as I eventually figured out, that physicists are sloppy: it's that someone has already done the rigorous work, and that the point of physics is to understand the physics, not bask in the rigor.)
As an example, if physicists have no problem multiplying and inverting differentials: if z=dy/dx, then dx/dy=1/z. This can be shown rigorously, and requires a bunch of conditions which, for the sort of functions that physicists deal with, are virtually always true, but the rigorous calculation actually obscures the matter at hand and really slows down the work.
I would agree with Chad that it's less about abstract vs real world, and more about teaching students to make those all-important connections. Personally, real-world examples work very well for me when learning new concepts -- but they still need to be extrapolated into the abstract universal principles before one can say I have truly "learned" the material.
I guess I'm taking the obvious middle road and saying, "Hey -- we need both!" :)
Scott, that kid definitely needed a whack with a clue-by-four, and you did a better job of it than I would have done. There is a reason why most science/math/engineering subjects (and some humanities subjects, most obviously foreign languages) have prerequisites or corequisites: the department is warning any would-be students that the instructor will assume that you know the material covered elsewhere and can learn to apply it to the new situations covered in this class.
This is why learning is (should be) fun: finding out that something you already knew can be applied in a different context. Students who compartmentalize are missing out on this.
Perhaps the best approach is to use two examples which are very different from each other (e.g., both a train schedule problem and a reaction rates problem, for the algebra example) and stress to the students that the underlying principles are the same. This approach has the added benefit of reaching the students who for whatever reason fail to grasp a certain type of concrete example (e.g., Americans who have never seen a passenger train and don't know anybody who routinely uses one).
I always found abstracts harder to learn than examples. But, then, I also recall that the standard way was to say "Here's the abstract, here's how we'll use the abstract, now let's run a few more abstract versions - what happens if we change this function, or that one, etc"
I'm having trouble with the jargon in the paper, so it's not clear to me if this is addressed on the second page, but -- the core of their study deals with the initial explanation of a concept. That's very different from the usual practice (as John notes) of teaching with a runthrough of the abstract concept followed by its application to the passing of trains or the emptying of conical swimming pools.
The NYT story seems to be treating it as a general attack on word problems, which isn't the case. (Unless that's what "participants learned Concrete A and Concrete B instantiations and were given the alignment of analogous elements across the learning instantiations" means.)
Fascinating result. It would certainly explain one instance I came across when advising a student: Student could work a common denominator problem with symbols (putting her in an algebra class) but could not do it with numbers (putting her in our pre-algebra arithmetic class). It is definitely something to think about with respect to how we introduce new (abstract) concepts in physics. I've always done it from abstract to concrete, but some do examples or demos first.
I don't see this issue as related to prerequisites. I'd suspect it has to do with how the brain does mathematics. But the pre-req issue is a long-standing one. My blogs on that topic are all tagged
http://doctorpion.blogspot.com/search/label/prerequisites
when I carry on about students failing to grasp the "concept of prerequisites". I attribute it to the way K-12 info is presented and tested, so I attack it by making it clear to them that (a) their ignorance of why they wasted their money paying for a class that they learned nothing from is understandable but (b) they need to stop doing it yesterday if they want to be engineers. I have e-mails from former students that are quite effective at opening some eyes.
First, a diatomic molecule is one atom too many.
Second, abstract vs. applied? Moderation in all things, Buddah says do both.
Connections is the thinking, and hard to "teach". There was a nice example in Physics Today a few years ago. Three problems shown to students. Can't remember exactly but...
1) Fly on turntable, static friction, what speed can the turntable go before fly gets tossed.
2) Take a pendulum, pull the mass up a certain angle, release, what is the velocity at the bottom
3) Elastic collision, given some info, what is velocity of block 2 after collistion.
OK, students were asked to say which were most alike, what pair.
Bad students said 1 and 2 cause crap rotated. Good students said 2 and 3 cause they are con. of energy problems really.
How do you get folks to make those connedctions? We do it all the damn time, but we have to tell them this stuff as we show them examples, but I'm not sure of great general techniques.
People are not identical like rubidium atoms, pain in the ass to research and draw conclusions! But I think that all physics education talks and papers should have the true student to teacher ratio in bold at the top of every page.
If I have 5-10 students, I can do much better no matter what techiniques I use, than say 100. And you have to count TA's that circulate, "oh I teach to 900 students, my methods are scalable......I just need 24 TA's to answer some questions......"
Eric Lund @ 5:
Perhaps the best approach is to use two examples which are very different from each other (e.g., both a train schedule problem and a reaction rates problem, for the algebra example) and stress to the students that the underlying principles are the same.
Interestingly, the study also tried precisely that approach; the results were apparently no better than with just one concrete example. When the students were explictly pushed to work out the underlying principles themselves, they did better -- but still not as good as the students taught the purely abstract method.
(All this is explained in the post by Ed Yong which Chad linked to.)
Five-Step Lesson Plan for Algebra (Draft 2)
Subject: Algebra 1C
Grade: 8-9
Theme/Topic: Solving Linear Equations in One-Variable
Subtitle: 201 Minutes of a Space Idiocy
ESL Level: 1-2
Involved: Hook about special "2001: A Space Odyssey" event
attended by teacher
============================================
Content Standards:
5.0 Students solve multi-step problems, including word problems,
involving linear equations and linear inequalities in one variable and
provide justification for each step.
Lesson Rationale-Level 1-2:
Key components in understanding mathematics include
not only comprehension of the mathematical procedures, but a solid conception of the required vocabulary as well. Students who are in the process of learning English as second languages have an extra step they need to take in gaining this latter concept and thus may have a more difficult time with fully understanding mathematics. This lesson is designed to introduce Level 1 and 2 learners to the process of solving equations. Students will be familiarized to the vocabulary
using realia (defined below) as well as by a graphic organizer.
Realia is a term used in library science and education to refer
to certain real-life objects.
The concept will be introduced to the learners utilizing a situation that they may encounter in their present lives. The class will use the same word problem to build their mathematical understanding, moving from the scenario, to values, and finally to terms, thus gradually establishing a solid foundation. In this
manner, the teacher/lesson-planner expects that students will learn not only the new mathematical knowledge but also a connection of how these mathematical concepts are important in their everyday lives.
Solving an equation is a very important concept for students to understand because the concept so often is implemented in everyday life. While many times in the form of mental math, for this
mental math to be strong, they must have a solid understanding of the underlying mathematical concepts.
Lesson Objectives-Level 1-2:
1. Students will explain in short phrases how math is applicable in real life following the discussion of a real life word problem.
2. Students will classify various parts of the word problem and place these values with the appropriate locations/illustrations on the worksheet.
3. Students will distinguish what they know in the problem from what they need to discover.
4. Students will successfully complete a worksheet which takes the problem they just solved using arithmetic into an algebra equation.
5. Students will successfully identify variable as being the unknown as well as be able to distinguish the variable in other mathematical problems.
6. Students will be able to explain in short phrases how they figured the word problem out using arithmetic.
7. Students will successfully identify inverse operations including how they are used to solve an algebra equation.
8. Students will know a little more about movies and TV from the professional perspective, and have heard examples of how Math and sophisticated analytical judgment is useful to the people who make movies as well as to the audiences.
Step 1: Anticipatory Set:
Focus: teacher holds up a movie poster of "2001: A Space Odyssey" and says "My wife and I spent over 127 dollars last night for dinner and a movie. We can't afford to spend that much very
often, so we will all work on how we can spend money on movies, food, and drinks. [see Hook below for elaboration]
Objective: The class is going to work towards being able to watch a movie in the near future. In order to watch the movie, students, after being given an allotted amount of money, will
need to figure out how to spend that money under the given specifications.
Purpose: how we spend money is an important part of life. We need to be fair about sharing money in a group, be sure that we stay in our budget, and have fun with what we buy. Math can help, and, in fact, is at the core of shopping and movie nights.
Transfer: To introduce the topic, I first present a true story about this weekend's movie and dinner event with my wife, with details of stars and dinner items. Then as a class we will discuss the word problem including what is expected of each student. We will also shortly discuss how mathematics fits into this scenario.
Given that the students are in 8th or 9th grade, and assuming they have some experience with money, they will not need to use algebra, simply figure out a solution using arithmetic. But this lays the foundation for deeper understanding of elementary algebra and mathematical problem solving in general.
"Hook":
Last night my wife and I went to see a movie. This took us 8 hours and $127.60. Why? First, what the movie was, where, and why we went. Second, where the time and money went.
What Movie, Where, and Why:
This was part of the birthday celebration for my wife. The day before, I gave her presents and 3 boxes of different dark chocolate. This evening was all about dinner, a movie, and more.
I hand around my over-sized glossy program for the event, for all students to see and, if they want to, read. They don't have to read it; they can just pass it along.
2001 in 2008: A Cinematic Odyssey. The Academy of Motion Picture Arts & Sciences celebration of the 40th anniversary of "2001: A Space Odyssey" continues with a special "behind the scenes" evening hosted
by Douglas Trumbull and Tom Hanks. "2001 in 2008: A Cinematic Odyssey" will feature never-before-seen images and clips from the
making of Stanley Kubrick's/Arthur C. Clarke's Academy Award(R) winning science fiction masterpiece. This special event was at the Samuel Goldwyn Theatre, 8949 Wilshire Boulevard, Beverly Hills, CA, on
25 April 2008 starting at 7:30 p.m. [extra, if time: who was Samuel Goldwyn? (17 August 1879 - 31 January 1974) was an Academy Award and Golden Globe Award-winning producer, also a well-known Hollywood
motion picture producer and founding contributor of several motion picture studios. Some famous Goldwynisms are given as an Appendix.]
My wife and I both consider this the greatest movie ever made. We have seen it dozens of times. In my case, that started the day it opened in New York City, where I saw the famous visionary science fiction writer earlier in that premier week. He is the man who invented the communications satellite -- in 1945! He and I eventually
did at last 4 projects together that resulted in something being published, 2 books and two technical papers. I'm working on a movie project right now based on his most famous story. Stan Kubrick was
one of the most famous movie directors in the world, and now, after he has died, other film people are still trying to figure out how he made his movies. Clarke, now Sir Arthur C. Clarke, died last month, and that was news around the world. In a sense, this movie changed my life.
Before I go on, can anyone tell us a story about a movie that changed some one's life?
Money:
(a) $30.00 cash, gas for my wife's white Camry for trip from Altadena to Beverly Hills and back. Of course, this leaves enough gas in the car for several more days, so she can get to and from her job in
Burbank.
(b) $10.00 for two tickets at $5.00 each ordered through the internet, and printed out (with bar code) on laser printer.
(c) $10.00 parking, fixed fee, public garage across the street.
(d) $77.60 for dinner, at Kate Mantilini's, 9101 Wilshire Boulevard.
---------------
....$127.60 total for evening. More than we usually spend for a night out, but we both thought it was well worth the time and price.
Dinner:
Kate Mantilini's, just two or 3 blocks along Wilshire from the theatre is a restaurant that many Hollywood stars go to. They brought really good sourdough bread, fresh from the oven, while we read the
menu. I had the (excellent) Sauteed Sand Dabs with lemon caper sauce, shoestring potatoes, cole slaw, and tartar sauce, washed down with a Sam Adams beer, which I like anyway and was reminded of by the "John Adams" TV miniseries. My wife had a lovely smoked salmon salad plate,
and sparkling water. Desert was pretty strong coffee and "The Ultimate Dark Chocolate Glazed Cheesecake on a Stick, Wonder of All Wonders." That's exactly what it said on the menu. Tasty!
My wife noticed that the star of 2001, Keir Dullea, in a tan with blue thread tartan suit, still slim, remote, and boyishly handsome, was in the restaurant. He came and stood right next to our table on his way out, talking to someone, close enough to touch. We didn't say anything to him because, in Southern California, it is more hip to notice stars but not to disturb them.
Time:
Right after I finished our Period 7 class at Rose City High School, about 3:00 p.m., and follow-up meetings and paperwork with the principal, assistant principle, and office manager, I drove home to
Altadena, getting home about 3:30 p.m.. I was already in my black pinstriped suit, tan shirt, and blue constellations tie. My wife changed into a nice dress and a silver and turquoise necklace that she
had made for herself years ago. We drove her car to Beverly Hills, leaving about 4:00 p.m.
Even though we had bought tickets a couple of weeks in advance, we knew we'd have to wait on a ticket holder's line, for first-come first-served seats (except for the seats reserved for the celebrities
and their friends who were at some special reception before hand).
There were some very big names in the audience such as Buzz Aldrin, the Apollo 11 astronaut, and 2nd man on the Moon, with whom I've worked before. We were parked by about 4:30, and finished with dinner and desert about 6 or 6:30. We were on line until just before 7:30, got pretty good seats, and were very close to where Tom Hanks waited before taking the podium to give his introduction.
Tom Hanks is currently an Academy Governor and Vice President. He's won two back-to-back Oscars ("Philadelphia" and "Forrest Gump"), starred in Apollo 13 (1995), and and produced "From the Earth to the Moon" (1998 miniseries). He had first been introduced by Sid Ganis, the current President of the Academy. Tom Hanks gave a very funny talk about the movie, from his perspective as a boy in Oakland, who didn't see the movie until after he's heard about it, and read the Mad Magazine parody called "201 Minutes of a Space Idiocy" -- for which he said "I didn't get a single one of the jokes." He gave some funny examples of what was NOT said in dialogue or narration in the film:
"These ape-men are seeing something they never saw before, which would change the very nature of humanity." "He grabs the bone; which has a handle." "So, how are things going, now that we've trained together for 9 months and been on this spaceship for nine months? Really,
trust me, the Rolling Stones don't ask each other how they're doing."
He had not heard of Stanley Kubrick, but after being totally blown away by the movie, he looked at the poster in lobby to see who made the movie. The stage was flanked by giant gold-painted Oscar statues.
Then we saw an introductory video by astronauts in the space station right now, who said that some people decided to become
astronauts after seeing 2001 when they were children. This was followed by the best archival print of the 70-mm movie, with a
10-minute intermission. Then, after the movie, was a long panel discussion between stars Keir Dullea ["David and Lisa", "Bunny Lake is Missing", "The Fox", "de Sade"), Gary Lockwood (former UCLA football
star, stuntman, star of "Splendor in the Grass", "It Happened at the World's Fair", "Firecreek", and guest star of Mission Impossible, Hawaii Five-O, MacGuyver, Murder She Wrote, and the pilot of Star
Trek), and Daniel Richter (a mime and teacher who choreographed the "Dawn of Man" ape-man sequence at the start of the film, and who wore an ape suit and mask and so I'd never seen his face before; plus Bruce
Logan, Director of Photography, and Douglas Trumbull, Visual Effects Supervisor.
Media-savvy Question to class: what do I mean by "the pilot of
Star Trek"? Not Captain Kirk or Picard, but the word "pilot" as used in TV. A television pilot is a test episode of an intended television series. It is an early step in the development of a television series, much like pilot lights or pilot studies serve as precursors to start of larger activity. Networks use pilots to discover whether an entertaining concept can be successfully realized. After seeing this sample of the proposed product, networks will then determine whether the expense of additional episodes is justified. They are best thought of as prototypes of the show that is to follow, because elements often change from pilot to series. Variety estimates that only a little over a quarter of all pilots made for American television succeed to the
series stage, although the figure may be even lower.
It was all over by 11:30, and we were home at midnight -- 9 hours after I finished teaching all day, and 8 hours after we left home for the dinner and movie event.
Now, our whole class will work on some problems about movies, meals, drinks, and how we use Math in real life.
Step 2: Instruction/Presentation-Level 1-2:
Pretest/Peel Off: not necessary; lesson is appropriate
for the entire class.
Provide Information: Following the class discussion about going to the movies, containing how mathematics fits into this idea, students will be working in groups to finish the problem
(Independent Practice) With their groups, they will be working from a worksheet. The worksheet contains the word problem and pictures.
Model: As a class we will talk about the word problem and what it tells us. The students are to place the values from the word problem, which corresponds with the correct picture. For
instance, if a soda costs $3, the students will put $3 next to the picture of a soda. Inevitably, they will be left with one picture, which they will not have a value for and will have to write a question
mark. As a group, they must figure out what this value is. To accompany them they will have money to work with. Once the group has reached a consensus on the answer, they are to come to the front of
the room where they can purchase a "bank voucher", "food voucher" and "movie ticket". They will know if they are correct if they have enough money to buy each item.
[truncated]
Note: I acknowledge that this includes and significantly adapts material from Lindsay Claseman, 23 April 2006, PLC 915B: Dr. Mora, San Diego State University, as well as the example given by Dr. Nick Doom....
Reminds me of working as a TA for "physics for pre-meds" when I was an undergrad, and I discovered that some of the students separately memorized V = IR, I = V/R, and R = V/I. <\I>
Some years ago I was approached my an Economics undergrad student who wanted help in understanding linear equations. I started saying: "a linear equation of the form ax+b=y...". He stopped me right there. "Idiot", he grumbled. "It's sx+r=y. You don't know any math at all, you f***ing moron". And he walked away, angrily.