define "lesson"

more from the Times:

Last month I asked my students to take a pro or con position on the topic of genetic engineering; one student vociferously announced, “I completely disagree with it. We don’t know how it’s going to affect us in the long run!” The boy next to her replied, “Ally, they use genetic engineering to manipulate bacteria to make human insulin. I’m diabetic; that insulin keeps me alive.” Other students thrust their hands into the air anxious to share their point of view, while others simply blurted out their ideas – our classroom was intellectually alive and I was the moderator.

When asked why they were so engaged in the lesson one student replied, “It affects our future. We want to help build it.”

The other students clapped in agreement.
What is a good teacher worth?

Ok, number one: No actual teenaged person ever spoke the words "They use genetic engineering to manipulate bacteria to make human insulin."

No adult ever spoke the words "They use genetic engineering to manipulate bacteria to make human insulin." Not unless the adult was giving a lecture from notes.

"They use genetic engineering to manipulate bacteria to make human insulin" is not the way people talk.

Number two: No teenaged person ever said "It affects our future. We want to help build it," either.

Number three: Structurally speaking, the student who has diabetes is given the last word, which means s/he wins the point. I object. If this isn't a class where 'critical thinking' means 'adopt politically liberal positions re: science policy,' then both the discussion and the paragraph need to be handled differently. I.e.: some contrarian points of view need to be raised by the teacher.

Or maybe students could, you know, read something about the precautionary principle before they get so fired up they're thrusting their hands into the air and blurting out their ideas.

Which brings me to Number four: How is a bull session on genetic engineering a "lesson"?

(And, just out of curiosity, how does a bull session on genetic engineering, absent any engagement with the literature on the subject, help build the future?)

update 4:28pm: High school students preparing for a debate.

"chunk decomposition" - the matchstick problems again

Belatedly, I'm posting this abstract from the matchstick math study. It relates to the Why is SAT geometry hard? post I began writing the other day; it also relates to the two different meanings of the word "chunking" that cropped up in the matchstick comments.

When we're talking about memory, "chunking" means chunking several bits of information together into one larger chunk, allowing working memory to hold more than the 3 or 4 separate items it is capable of holding at one time. So, for instance, instead of remembering 2 - 0 - 3 as three separate numbers, you come to remember 203 as just one chunk.

When we're talking about perception, "chunking" means something closer to an automatic and entirely unconscious perceptual bias towards seeing -- visually seeing -- 'wholes' or 'chunks' instead of the parts that make up the chunk. "Visual chunking" happens instantly and naturally, whereas memory chunking requires practice over time. Crucially, visual chunking is extremely difficult to resist or to undo.

I've mentioned in a couple of comments threads, I think, that I believe autistic people (and children and animals) much more readily perceive parts instead of wholes -- something Temple Grandin absolutely believes. Temple told me once that the hidden figures in hidden figures puzzles always 'pop' at her, and I believe it. After 9/11 she and I used to talk about using high-functioning autistic people to man the carry-on scanners at the airport. 

Here's the abstract:

Constraint relaxation and chunk decomposition in insight problem solving.
By Knoblich, Günther; Ohlsson, Stellan; Haider, Hilde; Rhenius, Detlef
Journal of Experimental Psychology: Learning, Memory, and Cognition, Vol 25(6), Nov 1999, 1534-1555.
Abstract
Insight problem solving is characterized by impasses, states of mind in which the thinker does not know what to do next. The authors hypothesized that impasses are broken by changing the problem representation, and 2 hypothetical mechanisms for representational change are described: the relaxation of constraints on the solution and the decomposition of perceptual chunks. These 2 mechanisms generate specific predictions about the relative difficulty of individual problems and about differential transfer effects. The predictions were tested in 4 experiments using matchstick arithmetic problems. The results were consistent with the predictions. Representational change is a more powerful explanation for insight than alternative hypotheses, if the hypothesized change processes are specified in detail. Overcoming impasses in insight is a special case of the general need to override the imperatives of past experience in the face of novel conditions.

This study of Perceptual contributions to problem solving: Chunk decomposition of Chinese characters looks interesting.

when smart is dumb

Directions:

Make each statement true by moving just one matchstick.

In Choke, Sian Bielock discusses a study in which more than 90% of adults with normal working memory correctly answered the first problem. Roughly the same number of people with damaged working memories also got it right.

Only 43% of normal adults got the answer to the second problem, while 82% of patients with damage to the prefrontal cortex figured it out.

I believe high-functioning people with autism (or a healthy loading of autism genes) will also have a high rate of success on problem number 2, but that's just me.

Better without (lateral) frontal cortex? Insight problems solved by frontal patients
Carlo Reverberi,1,2 Alessio Toraldo,3 Serena D’Agostini4 and Miran Skrap4
Brain (2005), 128, 2882–2890

more than one way to solve it

paraphrasing Wu at msmi2010:

The idea that there is always more than one way to solve it is propaganda. Sometimes you're lucky to have one way.

cumulative practice

I've been meaning to get a post up about this article for years now. I think it's incredibly important (relates to Direct Instruction, too).

No time to write now, but here's the abstract:

THE EFFECTS OF CUMULATIVE PRACTICE ON MATHEMATICS PROBLEM SOLVING (pdf file)
KRISTIN H. MAYFIELD AND PHILIP N. CHASE
JOURNAL OF APPLIED BEHAVIOR ANALYSIS
2002, 35, 105–123
NUMBER 2 (SUMMER 2002)

This study compared three different methods of teaching five basic algebra rules to college students. All methods used the same procedures to teach the rules and included four 50-question review sessions interspersed among the training of the individual rules. The differences among methods involved the kinds of practice provided during the four review sessions. Participants who received cumulative practice answered 50 questions covering a mix of the rules learned prior to each review session. Participants who received a simple review answered 50 questions on one previously trained rule. Participants who received extra practice answered 50 extra questions on the rule they had just learned. Tests administered after each review included new questions for applying each rule (application items) and problems that required novel combinations of the rules (problem-solving items). On the final test, the cumulative group outscored the other groups on application and problem-solving items. In addition, the cumulative group solved the problem-solving items significantly faster than the other groups. These results suggest that cumulative practice of component skills is an effective method of training problem solving.

Note: the effects of cumulative practice on problem solving.

Not "procedural fluency" or "automaticity" or "mastery" etc.

Problem solving.

The path to problem solving goes through a particular form of practice - cumulative practice - not through "do the problem 3 ways" (Trailblazers) or "explain how you got your answer."

Barry G on exercises vs. problems

By way of introduction, I am neither mathematician nor mathematics teacher, but I majored in math and have used it throughout my career, especially in the last 17 years as an analyst for the U.S. Environmental Protection Agency. My love of and facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.

I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.

Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retire. I enrolled in education school about two years ago, and have one class and a 15-week student teaching requirement to go. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.

In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”—to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.

To set this in context, it is important to understand an underlying belief espoused in my school of education: i.e., there is a difference between problem solving and exercises. This view holds that “exercises” are what students do when applying algorithms or routines they know and the term can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, we future teachers are told that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.

As someone who learned math largely though mere exercises and who now creatively applies math at work, I have to question this thinking.
Discovery learning in math: Exercises versus problems
by Barry Garelick

Me, too.

how Paul teaches problem solving to middle school kids

I use an organizer that approaches this technique. I think it's adapted from an ELA template called four square. Here's what I have the kids do.

Take a piece of paper and fold into quarters. Open it up and draw a rectangle in the center. Now you've got four quadrants and a rectangle which is not exactly 'four square' but in the interests of marketing I guess four square sounds sexier.

Read the problem twice. Then in the rectangle restate the question in the form 'find blah blah blah in units of xxxxxx'.

Read the problem again and in the upper left quadrant identify and define all the symbols (variables and constants) you'll use in your solution.

Read the problem again and use the lower left quadrant for a diagram/picture.

Read the problem again and use the upper right quadrant to define your strategy. This can be words or preferably a set of equations to solve.

The lower right quadrant is where you show your arithmetic.

Finally, the back of the paper is used to justify your answer.

It works well for entry level problem solving of the kind you would encounter through maybe grade six. If kids master this it should instill some good habits for more complicated things.

Dan Dempsey

in a comment on Allison's post on problem solving, Dan writes:

From Bertrand Russell or was it Alfred North Whitehead...."the goal is to have no problems only exercises".

By expanding long term memory one can turn potential problems into exercises.

It seems that the goal of many pushing the discovery/inquiry approach is to have no exercises but only problems. ^*

I am still in shock from Dr. Ruth Parker's powerpoint given at NCTM national that the Standard Algorithm always harms conceptual understanding, which seems to be a recipe for lots of problems.

The person most everyone would like to hire is the one that has so much extensive background knowledge that most everything is an exercise.

Brilliant!

Here's Ken on the subject of struggle:

• compare & contrast

• to struggle or not to struggle

^* students must struggle

Steve H on problem solving in the real world

The problem is that educators know very little math. They don't know how it's used in real life.

One of my specialties is curved surfaces for geometric modeling. I write software for shape design and analysis. I once had to write a routine that would find the intersection of two tensor product polynomial surfaces. This has to be done algorithmically, and there are lots of methods given in the literature. It would be stupid to ignore those solutions just to discover my own. I'm not proud. I'm more than willing to copy what someone else has done, so I studied the literature. In fact, if you write a technical journal article, you better show that you have a full grasp of and reference to all other key articles and books. If you ignore the literature and (re)discover a technique, then it either won't get published or it will get trashed by your colleagues. Ignorance is not treated lightly. Prior art reigns supreme. (knowledge and skills)

However, nobody had a solution that met my need for speed of calculation. I had to create my own solution, but I don't start from scratch, and I don't use some sort of pattern recognition or critical thinking to find a solution. I use my toolbox of mastered skills. First, I have a fast way to convert each polynomial surface into a large set of triangles. All I needed next was to find a very fast way to determine if any two triangles intersect. I don't "discover" a solution. I look at the problem and see how my toolbox of mastered (rote) skills can be applied; vectors, dot products, cross products, parametric equations, different forms for defining plane equations, and matrices.

These skills don't exist in some sort of rote or out-of-context space. They have a meaning and a use. Two independent vectors define a plane. If I take the cross product of the two vectors, I have a new vector that is perpendicular to that plane. I didn't discover that. I was taught that.

Don't educators understand that being creative mathematically requires a whole lot of basic, mastered skills? The larger your tookbox of mastered math skills, the more creative you will be.

I don't know why they have this rote or script hangup. They just don't have enough understanding of math to know if anything they do is correct or incorrect.

another nail in the coffin

Is Traditional Teaching Really All that Bad?

answer: no

the tennis ball problem

A tennis ball can with radius r holds a certain number of tennis balls also with the same radius. The amount of space in the tennis ball can that is not occupied by the tennis balls equals at most the volume of one tennis ball. How many tennis balls does the can hold?

Barry sent me this problem months ago & I've been avoiding it because geometry scares me.

I finally shamed myself into attempting it just now & got an answer of 2. Unfortunately, I typed up my solution, loaded it to flickr, but flickr is on the blink so I can't post.

I solved it (assuming I did solve it) algebraically, then resorted to "logic and reasoning" to check.

Unfortunately, I'm confused by logic and reasoning at the moment.

I was thinking that because a sphere is 2/3 of a cylinder of same radius, with each ball you put inside a same-radius cylinder you end up with 1/3 of a ball's worth of empty space....which now implies to me that the answer should be 3 balls, not 2.

sigh


update (1):

I'm mixing things up

The 1/3 that's left over isn't 1/3 of a tennis ball. It's 1/3 of a cylinder with the same radius as the tennis ball.

I better forget the logic and reasoning & stick to algebra.

Assuming I didn't screw up the algebra, that is.


update (2):

OK, so in between dealing with screaming autistic youths, loading the dishwasher, & microwaving a taco for Jimmy, I realized that I don't need to know "how much of a tennis ball-sized volume is left over."

I just need to know how much empty volume is left over, period, then figure out how many multiples of that empty space add up to the volume of 1 tennis ball.

volume of tennis ball with radius r: 4/3πr^3
height of cylinder that fits just one tennis ball: 2r
volume of cylinder w/height of 2r: πr^2h = 2πr^3

vol. of cylinder - vol. of 1 tennis ball = vol. of empty space

2πr^3 - 4/3πr^3 = 2/3πr^3 empty space left over when 1 tennis ball is in cylinder

2 tennis balls leaves 2 empty spaces, each 2/3πr^3 in volume:

2/3πr^3 + 2/3πr^3 = 4/3πr^3, which is the volume of 1 tennis ball

so: 2 tennis balls

update (3):

Barry says the problem comes from Dolciani's Algebra 2! (I include the exclamation point because I'm happy to discover I am able to solve a problem from that book. cool.)

original wording:

A tennis ball can in the shape of a cylinder with a flat top and bottom of the same radius as the tennis balls is designed so that the space inside the can that is NOT occupied by the balls has volume at most equal to the volume of one ball. What is the largest nubmer of balls the can will contain?

wild goose chases

Found the quote -- it's from what is apparently a classic book on Mathematical Problem Solving by Alan Schoenfeld.

The Wild Goose Chase in Problem Solving

When solving standard mathematics problems, students normally recall and apply learned procedures in a straightforward way. However, if the problem is unfamiliar, some students simply pick a method and keep persistently on the same track for a long time without getting anywhere. Schoenfeld (1987) described this behaviour as chasing the wild mathematical goose.

That's not the meaning I remember.

I probably got it wrong.

knock on wood
true confession
wild goose chase

Fido Puzzle - Solved by Third Graders

I think I've mentioned before that I teach a high ability math group at a charter school in Phoenix. Long division took less than 2 weeks to master and we're finishing up the Primary Math 3a book this week, well ahead of my predicted schedule.

Math has always come easy to this group, and I've been working challenging word problems and logic stumpers with them. Last week, I saw a post on Thoughts on Teaching about the Fido Mind Reader on the 7up website.

I showed the site to my students, who ooohed & aaahed over it, then told them I thought that they could try to out figure how it works. We discussed methods of problem solving and I told them we would try a chart or table. The worksheet provided plenty of subtraction review. Of course, one of them found the pattern that makes the puzzle work.

Conferences and report cards went home this week so I must confess, I didn't try to figure the puzzle out before I assigned it. Working the pattern on the board with the students, I saw immediately that the student's answer was correct.

Could any KTM readers tell me how a big a challenge this puzzle presents?

Check out my excited students on our classroom blog:

Brainbusters

cats and dogs living together

ISBN 1-881317-15-3


On its In-Depth page, the Morningside Academy, in Seattle, whose students "typically score in the first and second quartiles on standardized achievement tests in reading, language and mathematics," makes parents this offer:

Morningside Academy offers a money-back guarantee for progressing 2 years in 1 in the skill of greatest deficit. In twenty-five years, Morningside Academy has returned less than one percent of school-year tuition.

from Chapter 6: Comprehension, Critical Thinking, and Self-Regulation

Morningside has wrestled with the problem of guaranteeing that skills taught in isolation truly become an integral part of the everyday activity of the learner. Two methods that we are evolving are designed to bridge typical behavioral skill instruction and useful, real-world application in the spirit of progressive education and John Dewey. When we discuss our procedures with developmental psychologists, constructivist educators, and others outside of the field of applied behavior analysis, we have found them receptive. This is in part a response to our respect for many of their philosophies, methods, and materials. Many of these colleagues maintain activite dialogue with us in our joint effort to find and develop technologies of teaching from basic skills to inquiry and project-based learning.

constructivist behaviorists!

something new under the sun

Here's what they have to say about critical thinking:

Morningside directly instructs and monitors improvement in strategic thinking, reasoning, and self-monitoring skills. Strategic thinking is the glue that allows students to employ component skills and strategies in productive problem solving.... Morningside's instructional and practice strategies build tool and component skills that are needed to solve problems. In addition, most of our students need direct and explicit instruction in "process" or "integrative" repertoires--methods that help them recruit relevant knowledge and skills to solve a particular problem. At Morningside we have found that these strategic thinking skills, characteristic of everyday intellectual activity, are not automatic by-products of learning tool and component skills.

Does this contradict the notion that teaching to fluency, as opposed to mastery, produces contingency adduction? (And see here.)

Are the authors saying that Morningside students have more problems generalizing knowledge to new contexts than students who are doing well in the public schools?

Or are they saying that fluency doesn't increase generalization after all?

I'm getting the feeling that no one really knows how, exactly, critical thinking, problem-solving, and knowledge transfer emerge. As best I can tell, the most that is known is that they don't emerge before mastery of component skills and concepts within a particular domain of knowledge.

More from the book:

There are a number of reasons why traditional efforts to promote creative thinking and problem solving have not been wholly effective. First, watching someone else solve a problem does not reliably teach the process. Second, in routine practice, problem solving behavior is private behavior that other learners can't observe. Third, cooperative problem solving often reinforces already-existing problem solving repertoires of some students in the group, but doesn't enhance the skills in others, even though everyone may come away from the group believing they have "solved the problem."

[snip]

At Morningside, we view the failure to self-monitor and reason during problem solving as a failure of instruction rather than as a failure of the learner. This perspective has provided a challenge to develop instructional strategies that turn learners into productive thinkers and problem solvers.

A note: for those of you who aren't familiar with behaviorism, this is a -- or even the -- core tenet of the field. If an instructional approach isn't working, the problem lies in the instruction, not the student. Or, rather, the problem lies in the contingencies; that might be the proper way to put it (don't know). I'm not sure you could be a behaviorist without subscribing to this principle.

This principle -- it's the teaching, not the student -- led Irene Pepperberg to her breakthrough with Alex the parrot (chapter excerpt), by the way.

Back to Morningside:

Thinking Aloud Problem Solving

To develop these strategies and to provide students with a set of self-monitoring, reasoning, and problem solving strategies, Morningside turned to an approach developed by Arthur Whimbey and Jack Lockhead in the 1970s...They developed Thining Aloud Problem Solving (TAPS) to improve analytical reasoning skills of college students. Perhaps the most impressive evidence of its effectiveness comes from its use at Xavier University in a four-week pre-college summer program for entering students. The program, Stress On Analytical Reasoning (SOAR), was replicated over several summers at this perdominantly African-American college and produced stunning results.... Participants gained 2.5 grade levels on the Nelson-Denny Reading Test and an average of 120 points on the Scholastic Achievement Test. (p. 122)

This assertion flatly contradicts the conclusion of cognitive science that critical thinking and problem solving cannot be taught directly, at least not separately from extensive teaching of content. At least, I think it does; I don't know what kind of domain knowledge was involved in this program.

Interesting.


the cog-sci position

The Mind's Journey from Novice to Expert by John T. Bruer

A few of these programs, such as the Productive Thinking Program (Covington 1985) and Instrumental Enrichment (Feuerstein et al. 1985), have undergone extensive evaluation. The evaluations consistently report that students improve on problems like those contained in the course materials but show only limited improvement on novel problems or problems unlike those in the materials (Mansfield et al. 1978; Savell et al. 1986). The programs provide extensive practice on the specific kinds of problems that their designers want children to master. Children do improve on those problems, but this is different from developing general cognitive skills. After reviewing the effectiveness of several thinking-skills programs, one group of psychologists concluded that "there is no strong evidence that students in any of these thinking-skills programs improved in tasks that were dissimilar to those already explicitly practiced" (Bransford et al. 1985, p. 202). Students in the programs don't become more intelligent generally; the general problem-solving and thinking skills they learn do not transfer to novel problems. Rather, the programs help students become experts in the domain of puzzle problems.


Critical Thinking: Why Is It So Hard to Teach? (pdf file)
by Daniel Willingham

After more than 20 years of lamentation, exhortation, and little improvement, maybe it’s time to ask a fundamental question: Can critical thinking actually
be taught? Decades of cognitive research point to a disappointing answer: not really. People who have sought to teach critical thinking have assumed that it is a skill, like riding a bicycle, and that, like other skills, once you learn it, you can apply it in any situation. Research from cognitive science shows that thinking is not that sort of skill. The processes of thinking are intertwined with the content of thought (that is, domain knowledge). Thus, if you remind a student to “look at an issue from
multiple perspectives” often enough, he will learn that he ought to do so, but if he doesn’t know much about an issue, he can’t think about it from multiple perspectives. You can teach students maxims about how they ought to think, but without background knowledge and practice, they probably will not be able to implement the advice they memorize. Just as it makes no sense to try to teach factual content without giving students opportunities to practice using it, it also makes no sense to try to teach critical thinking devoid of factual content.

I would sure like some help on this issue, seeing as how I am living in failure to transfer land. For the time being I'm going to carry on assuming fluency will help -- fluency and lots of practice with word problems:

Give me a problem which you think is not by type, and I shall invent ten similar problems which will put it into a type. In fact, I often have to do this when I teach: first I solve a problem at the board, then I give a similar problem for all to solve in class, then I give a similar problem as a homework, then I give a similar problem on a test. All these stages (often more) are necessary, otherwise many students will not grasp the method.

Between Childhood and Mathematics: Word Problems in Mathematical Education (pdf file)
by Andrei Toom

Between Childhood and Mathematics: Word Problems in Mathematical Education (pdf file)
Word Problems in Russian Mathematical Education (pdf file)
How I Teach Word Problems (pdf file)

The Executive Brain
Meet Alex

problem solving

more buried treasure --

The Cognitive Science Millenium Project
The one hundred most influential works in cognitive science from the 20th century

6. Human Problem Solving
by A. Newall and H. A. Simon

The book describes Newell and Simon's attempt to build a computerized "General Problem Solver" which actually failed. They discovered that human problem soving requires more domain knowledge than a computer can implement. The book documents the efforts they made in programming the computer to solve problems such as the "tower of hanoi" and what they found about the nature of the human problem solving process. They introduce the concept of problem solving as a search in the problem space and the use of heuristics such as means-ends analysis. The studies reported in the book pave the way for using computers as a tool and an analogy to study human cognition.

If not the most cited paper in Cognitive Science, it is surely very near the top. A critical source for many developments in the field of problem solving.

Domain knowledge is knowledge stored in long-term memory, not on the internet.


How Knowledge Helps by Daniel Willingham