Showing posts with label fractions. Show all posts
Showing posts with label fractions. Show all posts

RYSK: Shepard's The Role of Assessment in a Learning Culture (2000)

This is the 13th in a series describing "Research You Should Know" (RYSK).

In her presidential address at the 2000 AERA conference, Lorrie Shepard revealed a vision for the future of educational assessment. That message turned into an article titled The Role of Assessment in a Learning Culture, and its message is still very much worth hearing today. Lorrie Shepard remains a globally-respected expert in assessment, psychometrics, and their misuses, and I'd think she was totally awesome even if she wasn't my boss.

Shepard is often present for debates about large-scale testing, but this paper focuses on classroom assessment -- the kind, says Shepard, "that can be used as a part of instruction to support and enhance learning" (p. 4). Shepard does this by first explaining a historical perspective, then describing a modern view of learning theories, then envisioning how new assessment practices could support those theories. Impressively, she does this all in just 11 well-written pages. (In fact, given that the paper is available on the web, I wouldn't blame you at all for skipping this summary and just reading the article for yourself.)

History

Shepard highlights several major themes from history that have continued to drive our assessment practices. One is the social efficiency movement, which "grew out of the belief that science could be used to solve the problems of industrialization and urbanization" (p. 4). While this movement might have helped our economic and educational systems scale rapidly (think about Ford and the assembly line), social efficiency carries with it a belief that people have a certain innate (and largely fixed) set of capabilities, and our society operates its most efficiently when we measure people and match their capabilities to appropriate education and employment. For example, students were often given IQ tests to determine if their future path should lie on a particular academic or vocational track.

The dominant learning theories of the early and mid-1900s were associationism and behaviorism, both of which promoted the idea that learning was an accumulation of knowledge that could be broken into very small pieces. Behaviorism was also tied closely to theories of motivation, as it was believed learning was promoted when knowledge was made smaller and opportunities for positive reinforcement for learning were made greater. Much of the assessment work related to these beliefs can be traced back to Edward Thorndike, considered to be the father of scientific measurement and earliest promoter of "objective" testing. It's been 100 years since Thorndike was elected president of the American Psychological Association, and decades since his ideas seriously influenced the leading edges of learning theory. Still, as most anyone who works in schools or experienced a traditional education can attest, ideas of social efficiency and behaviorism are still evident in schools -- especially in our assessment practices.

Together, the theories of social efficiency, scientific measurement, and beliefs about intelligence and learning form what Shepard sees as the dominant 20th-century paradigm. (See page 6 of the paper for a diagram.) It's important to begin our discussion here, says Shepard, because "any attempt to change the form and purpose of classroom assessment to make it more fundamentally a part of the learning process must acknowledge the power of these enduring and hidden beliefs" (p. 6).

Modern Theories

In the next section, Shepard describes a "social-constructivist" framework that guides modern thought on learning:

The cognitive revolution reintroduced the concept of mind. In contrast to past, mechanistic theories of knowledge acquisition, we now understand that learning is an active process of mental construction and sense making. From cognitive theory we have also learned that existing knowledge structures and beliefs work to enable or impede new learning, that intelligent thought involves self-monitoring and awareness about when and how to use skills, and that "expertise" develops in a field of study as a principled and coherent way of thinking and representing problems, not just as an accumulation of information. (pp. 6-7)

These ideas about cognition are complimented by Vygotskian realizations that the knowledge we construct "is socially and culturally determined" (p. 7). Unlike Piaget's view that development preceded learning, this modern view sees how development and learning interact as social processes. While academic debates remain about the details of cognitive vs. social (and vs. situative vs. sociocultural vs. social constructivist vs. ...), for practical purposes these theories can coexist and are already helping teachers view student learning in ways that improve upon behaviorism. However, Shepard says, since about the 1980s this has left us in an awkward state of using new theories to inform classroom instruction, while still depending on old theories to guide our assessments.

Improving Assessment

If we wish to make our theories of assessment compatible with our theories of learning, Shepard says we need to (a) change the form and content of assessments and (b) change the way we use and regard assessment in classrooms. Some of the potential changes in form are already familiar to most teachers, such as a greater use of open-ended performance tasks and setting assessment tasks in real-world contexts. Furthermore, Shepard suggests that classroom routines and related assessments should reflect the need to socialize students "into the discourse and practices of academic disciplines" (p. 8) as well as foster metacognition and important dispositions. Shepard does not go into much more detail here because others have already given attention to these ideas, but gives us this simple yet powerful idea (p. 8):

"Good assessment tasks are interchangeable
with good instructional tasks."

Next Shepard pays special attention to negative effects of high-stakes testing. Shepard could be called a believer in standards-based education, but recognizes how "the standards movement has been corrupted, in many instances, into a heavy-handed system of rewards and punishments without the capacity building and professional development originally proposed as part of the vision (McLaughlin & Shepard, 1995)" (p. 9). Unfortunately, Shepard's predictions have held true over the past 12 years: we've seen test scores distorted under political pressure, a corruption of "teaching to the test," and a trend towards the "de-skilling and de-professionalization of teachers" (p. 9). What's worse might be a decade of new teachers who've learned to "hate standardized testing and at the same time reproduce it faithfully in their own pre-post testing routines" (p. 10) because they've had such little exposure to better forms of assessment.

For the rest of the article, Shepard focuses on how assessment can and should be used to support student learning. First, classrooms need to support a learning culture where "students and teachers would have a shared expectation that finding out what makes sense and what doesn't is a joint and worthwhile project" (p. 10). This means assessment that is more informative and reflective of student learning, one where "students and teachers look to assessment as a source of insight and help instead of an occasion for meting out rewards and punishments" (p. 10). To do this, Shepard describes a set of specific strategies teachers should use in combination in their classrooms.

Dynamic Assessment

When Shepard wrote this article, formal ideas and theories about formative assessment were still emerging and the field had yet to settle on some of the language we now use. But if you're at all familiar with formative assessment, Shepard's description of "dynamic" assessment will sound familiar: teacher-student interactions continuing through the learning process rather than delayed until the end, with the goal of gaining insight about what students understand and can do both on their own and with assistance from classmates or the teacher.

Prior Knowledge

The idea of a pre-test to see what students know before instruction begins is not new, but Shepard says we should recognize that traditional pretests don't usually take account of social and cultural contexts. Because students are unfamiliar with a teacher's conceptualization of the content prior to instruction (and vice versa), scores might not accurately reflect students' knowledge as well as, say, a conversation or activity designed to elicit the understandings students bring to the classroom. Also, as Shepard has frequently observed, traditional pre-testing often doesn't significantly affect teachers' instruction. So why do it? Instead, why not focus on building a learning culture of assessment: "What safer time to admit what you don't know than at the start of an instructional activity?" (p. 11)

Feedback

The contrast in feedback under old, behaviorist theories and newer, social-constructivist theories is clear. Feedback under old theories generally consisted of labeling answers right or wrong. Feedback under new theories takes greater skill: teachers need to know how to ignore student errors that aren't immediately relevant to the learning at hand, while crafting questions and comments that force the student to question themselves and any false knowledge they might be constructing. (See Lepper, Drake, and O'Donnell-Johnson, 1997, for more on this.)

Transfer

While it is our hope that our students will be able to generalize the specific knowledge they have learned and apply it to other situations, our ability to accurately research and make claims about knowledge transfer turns out to be a pretty tricky business. Under a strict behaviorist perspective, it was appropriate to believe that each application of knowledge should be taught separately. Many of our current theories support an idea of transfer, and evidence shows that we can help students by giving them opportunities to see how their knowledge reliably works in multiple applications and contexts. So while some students might not agree, Shepard says teachers should not "agree to a contract with our students which says that the only fair test is one with familiar and well-rehearsed problems" (p. 11).

Explicit Criteria

If students are to perform well, they need to have clear guidance about what good performances look like. "In fact, the features of excellent performance should be so transparent that students can learn to evaluate their own work in the same way their teachers would" (p. 11). This reinforces ideas of metacognition and, perhaps more importantly, fairness.

Self-Assessment

There are cognitive reasons to have students self-assess, but other goals are to increase student self-responsibility and make teacher-student relationships more collaborative. Students who self-evaluate become more interested in feedback from others, are more aware of standards of excellence, and take more ownership over the learning process.

Evaluation of Teaching

This is another idea now heavily intertwined with formative assessment, but Shepard takes it one step farther than I normally see it. Instead of just using assessment to improve one's teaching, Shepard recommends that teachers be transparent about this process and "make their investigations of teaching visible to students, for example, by discussing with them decisions to redirect instruction, stop for a mini-lesson, and so-forth" (p. 12). This, Shepard says, is critical to cultural change in the classroom:

If we want to develop a community of learners -- where students naturally seek feedback and critique of their own work -- then it is reasonable that teachers would model this same commitment to using data systematically as it applies to their own role in the teaching and learning process. (p. 12)

Conclusion

Shepard admits that describing this new assessment paradigm is far easier than it is to implement in practice. It relies on a great deal of teacher ability and confronting some long-held beliefs. Shepard recommended a program of research accompanied by a public education campaign to help citizens and policymakers understand the different goals of large-scale and classroom assessments. Neither the research or educating the public is easy, because both are built upon a history of theories and practice that a new paradigm needs to discard. Perhaps we haven't taken on this challenge with the effort and seriousness we've needed, and I worry that now we're more apt to talk about "learning in an assessment culture" rather than the other way around, as Shepard titled this article. I sometimes wonder if she's considered writing a follow-up with that title, or if she's hoping she'll never have to. I guess the next time it comes up I'll have to ask her.

Math note: This is an article about assessment and not specific to mathematics, but I'd be remiss if I didn't share Shepard's inclusion of one of my all-time favorite fraction problems:


References

Lepper, M. R., Drake, M. F., O'Donnell-Johnson, T (1997). Scaffolding techniques of expert human tutors. In K. Hogan & M. Presley (eds.), Scaffolding student learning: Instructional approaches & issues. Cambridge, MA: Brookline Books.

McLaughlin, M. W., & Shepard, L.A. (1995). Improving education through standards-based reform: A report of the National Academy of Education panel on standards-based educational reform. Stanford, CA: National Academy of Education.

Shepard, L. A. (2000). The role of assessment in a learning culture. Educational Researcher, 29(7), 4–14. doi:10.2307/1176145

Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 199-221). New York: State University of New York Press.

Modeling Dimensional Analysis

I generally ask myself two questions when I examine the design of a mathematical task:
  1. What is the context?
  2. How can we model the mathematics?
Mathematical concepts with tasks for which these two questions can be answered easily tend to be easier to learn, while teaching and learning generally becomes more difficult when one or both of those questions can't be answered. For dimensional analysis (sometimes called the unit factor method or the factor-label method), the first question is easy to answer. It doesn't take much of an imagination to design a measurement conversion task that is set in a real-world context. A model, however -- whether visual, mental, or a concrete manipulative -- is generally absent. Typical dimensional analysis problems look like this:

Q: What is 60 miles per hour in meters per second?

A: \( \frac{60 \mbox{mi}}{1 \mbox{hr}} \times \frac{5280 \mbox{ft}}{1 \mbox{mi}} \times \frac{12 \mbox{in}}{1 \mbox{ft}} \times \frac{2.54 \mbox{cm}}{1 \mbox{in}} \times \frac{1 \mbox{m}}{100 \mbox{cm}} \times \frac{1 \mbox{hr}}{60 \mbox{min}} \times \frac{1 \mbox{min}}{60 \mbox{sec}} = \frac{9656064 \mbox{m}}{360000 \mbox{sec}} = \frac{26.8224 \mbox{m}}{\mbox{sec}} \)

For those who successfully learn dimensional analysis this way, there's a certain beauty to how the units drive the problem and how the conversion factors are nothing more than cleverly written values of one, the multiplicative identity. Unfortunately, many students struggle with this method. Some are intimidated by the fractions, some can't get the labels in the right place, and some just can't get the problem started.

What we need is a model. Let's start with the most basic of unit conversion models, a ruler with both inches and centimeters:

(Yes, I'm still using the same ruler I got as a 7th grader in a regional MathCounts competition.)
With only simple visual inspection, students should be able to use a ruler to estimate conversions between inches and centimeters. This is an informal model, one students can literally get their hands on. We can assist the learning by making the models progressively more formal. Here we model a trivial conversion from one inch to centimeters with a double number line:
(Yes, you still have to know your conversion factors!)
Such a simple example looks almost too easy to be useful, but we can add number lines for more complex conversions. We can even abstract the model further and go beyond conversions of distance. Suppose we wanted to convert 3 gallons to liters. I could model that conversion with number lines this way:
(I could have used any number of transition units, but I knew 1 quart was roughly 946 milliliters.)
Filling in the question marks from top to bottom, I'll see that 3 gallons, 12 quarts, 11,352 milliliters, and 11.352 liters are all the same volume. It's easy to see they're the same because on each number line those values are the same distance from zero. Because we're only converting one kind of unit (volume), we only need one dimension.

In our initial example we were converting 60 miles per hour to meters per second. That's two kinds of units, distance and time, so our model needs two dimensions. Furthermore, it can help to think of 60 miles per hour as a line, not just a point. After all, we often travel at a speed of 60 miles per hour without actually traveling a distance of 60 miles in exactly one hour.
Can you guess where our double (or however many are necessary) number lines will go in this model? The following video will demonstrate what I would call the graphing model or two dimensional model for performing conversions.
With the work shown in the video, we haven't just done one conversion. In fact, we're prepared to write 60 miles per hour 15 different ways, not that we'll ever be asked to do that. If we needed 60 miles per hour in centimeters per minute or feet per second, all the work is done. Just choose the appropriate quantity from the vertical and divide by the appropriate quantity from the horizontal. Of course, if we're in a hurry, we won't find all those intermediate figures and instead just proceed from miles to meters and hours to seconds as quickly as possible. Will that be quicker than the traditional method shown above? Probably not, but the purpose of using a model is understanding, not speed. Once the understanding is established, students can move on to a formal method or use technology when appropriate.

RYSK: Erlwanger's Benny's Conception of Rules and Answers in IPI Mathematics (1973)

This is the second in a series of posts describing "Research You Should Know" (RYSK).

In 1973, Stanley Erlwanger was a doctoral student at the University of Illinois at Urbana studying under Robert Davis (who taught many of us math as an advisor for Sesame Street) and Jack Easley when he published his landmark "Benny" article in Davis's new Journal of Children's Mathematical Behavior. (Now simply the Journal of Mathematical Behavior.) This and other Erlwanger articles became known as disaster studies (Spieser & Walter, 2004, p. 33) because they painfully reveal learning gone wrong, and they continue to impact the way we think about learning math and how we do research in mathematics education.

During the back-to-basics movement of the 1970s there was a push for programs that supported individualized instruction. One such program was Individually Prescribed Instruction, or IPI. IPI was designed for students to "proceed through sequences of objectives that are arranged in a hierarchical order so that what a student studies in any given lesson is based on prerequisite abilities that he has mastered in preceding lessons" (Lindvall and Cox, as cited in Erlwanger, 1973, p. 51). To measure that mastery, IPI relied heavily on assessments that were checked by the teacher or an aide, who would then have the opportunity to conference with the student and check for understanding. Erlwanger, however, saw a conflict inherent in the program: while the goals of IPI were "pupil independence, self-direction, and self-study" (Erlwanger, 1973, p. 52), teachers were supposed to have "continuing day-by-day exposure to the study habits, the interests, the learning styles, and the relevant personal qualities of individual students" (Lindvall and Cox, as cited in Erlwanger, 1973, p. 52). So is a teacher, with a class of students each working at their own pace, supposed to continuously monitor each individual student? How? The logical way to do this is to monitor assessment results and focus attention on strugging students. After all, if a student is passing the assessments and "mastering" objectives, how much could go wrong?

Benny was a twelve-year-old boy with an IQ of 110-115 in a 6th grade IPI classroom. Benny had been in the IPI program since 2nd grade, and the teacher identified Benny as one of her best students. By sitting down and talking to Benny about the math he was learning, Erlwanger discovered that Benny's conception of math was not only very rule based, but in many cases Benny's rules yielded wrong answers. For example:

  • Benny believed that the fraction \(\frac{5}{10} = 1.5\) and \(\frac{400}{400} = 8.00\) because he believed the rule was to add the numerator and denominator and then divide by the number represented by the highest place value. Benny was consistent and confident with this rule and it led him to believe things like \(\frac{4}{11} = \frac{11}{4} = 1.5\).
  • Benny converted decimals to fractions with the inverse of his fraction-to-decimal rule. If he needed to write 0.5 as a fraction, "it will be like this ... \(\frac{3}{2}\) or \(\frac{2}{3}\) or anything as long as it comes out with the answer 5, because you're adding them" (Erlwanger, 1973, p. 50).
  • When Benny adds decimals, he adds the number and moves the decimal point the total number of places he sees in the problem. So \(0.3 + 0.4 = 0.07\) and \(0.44 + 0.44 = 0.0088\). Benny's rule for multiplication is very similar: \(0.7 \times 0.5 = 0.35\), \(0.2 \times 0.3 \times 0.4 = 0.024\), and \(8 \times 0.4 = 3.2\). Because these are correct answers, that only served to reinforce Benny's rules about the addition of decimals.
  • Benny thinks different kinds of numbers should yield different answers: "2 + 3, that's 5. If I did 2 + .3, that will give me a decimal; that will be .5. If I did it in pictures [i.e., physical models] that will give me 2.3. If I did it in fractions like this [i.e., \(2 + \frac{3}{10}\)] that will give me \(2\frac{3}{10}\)" (Erlwanger, 1973, p. 53).

As you might guess, Benny got a lot of wrong answers and sometimes failed to achieve the 80% mastery mark on his assessments. It's clear that Benny isn't simply guessing and getting wrong answers -- his methods are consistent and he can confidently explain his reasoning. When Benny is wrong, he tries to change his answers until he gets ones that match the answer key, a process he called a "wild goose chase" (Erlwanger, 1973, p. 53). Because Benny's teacher/aide is only looking for answers that match the key (and trying to do so quickly), the emphasis is on the answer, not the reasoning. It was only Benny's persistence that resulted in him mastering more objectives than most of his classmates.

This style of learning led Benny to believe that math is little more than a collection of arbitrary rules and singularly correct answers: "In fractions, we have 100 different kinds of rules" (Erlwanger, 1973, p. 54). Erlwanger asked Benny where he thought the rules came from. "By a man or someone who was very smart. ... It must have took this guy a long time ... about 50 years ... because to get the rules he had to work all of the problems out like that..." (Erlwanger, 1973, p. 54). For both reasons of scholarship and concern for Benny, Erlwanger returned to the school twice a week for 8 weeks to work with Benny one-on-one. Unfortunately, despite Benny's eagerness to learn, Erlwanger found this to be too little time to change Benny's firmly-established view of mathematics and little progress was made.

What Benny Means to Theory, Research, and to Khan Academy

(It might be helpful to read yesterday's post about constructivism and the Khan Academy before reading this section.)

Erlwanger summed up the theoretical aspect in his conclusion:
Benny's misconceptions indicate that the weakness of IPI stems from its behaviorist approach to mathematics, its mode of instruction, and its concept of individualization. The insistence in IPI that the objectives in mathematics be defined in precise behavioral terms has produced a narrowly prescribed mathematics program that rewards correct answers only regardless of how they were obtained, thus allowing undesirable concepts to develop. (1973, p. 57)
Looking back at Benny in 1994, Steffe and Kieren summarized that
Erlwanger was able to demonstrate how Benny's understanding of mathematics conflicted with any "common sense" understanding of what would be regarded as "good mathematics." This was a crucial part of Erlwanger's work, because by demonstrating what a "common sense" view of mathematics should not be, Erlwanger was able to falsify (naively) the behavioristic movement in mathematics education at that very place where behaviorism has its greatest appeal -- at the level of common sense. (p. 72)
Prior to Benny, the large majority of research in mathematics education depended on quantitative methods -- using statistics to summarize and compare the performance of treatment and control groups. Erlwanger had opened the door to qualitative research, which essentially meant that researchers could now see the value of interviews, case studies, and similar methods. In other words, Benny showed researchers that they can, and should, talk to children.

Although we're approaching the 40th anniversary of the Benny study, anyone who has been paying attention to the debates regarding Khan Academy should be able to draw parallels between it and IPI and realize we're retreading a lot of the same water. In a recent Wired Magazine article about Khan, stories are told of students working individually, at their own pace, with their progress measured by a computer that judges answers right or wrong. The article highlights Matthew Carpenter, a fifth grader who has completed "an insane 642 inverse trig problems" (Thompson, para. 2). Carpenter has earned many Khan Academy badges, a sign of progress that pleases his teacher and amazes his classmates. Unfortunately, the article provides no evidence that Matthew Carpenter is not Benny. I, and hopefully everyone, sincerely hope he is not Benny. I hope he's developing a proper view of the nature of mathematics and developing solid mathematical reasoning and understanding. But I can't be sure, and maybe Carpenter's teacher can't be sure, either. While we sometimes can and do use behaviorist programs of instruction to learn, we can't rely on them to be sure that learning is happening the right way. That's Benny's lesson, and that's why we need to be critical (but not necessarily dismissive) of Khan Academy. People who fail to do so might be surprised with the results they get for all the wrong reasons.

References

Erlwanger, S. H. (1973/2004). Bennyʼs conception of rules and answers in IPI Mathematics. In T. P. Carpenter, J.A. Dossey, & J. L. Koehler (Eds.), Classics in mathematics education research (pp. 48-58). Reston, VA: NCTM.

Speiser, B., & Walter, C. (2004). Remembering Stanley Erlwanger. For the Learning of Mathematics, 24(3), 33-39. Retrieved from http://www.jstor.org/stable/40248471.

Steffe, L. P., & Kieren, T. (1994/2004). Radical constructivism and mathematics education. In T. P. Carpenter, J. A. Dossey, & J. L. Koehler (Eds.), Classics in Mathematics Education Research (pp. 68-82). Reston, VA: NCTM.

Thompson, C. (2011, July). How Khan Academy is changing the rules of education. Wired. Retrieved from http://www.wired.com/magazine/2011/07/ff_khan/all/1.

A Pretty Short Explanation of "Invert and Multiply"

I can't remember if I've seen this one before, but I thought it was worth sharing. I found it in a forthcoming book by Pamela Harris, at the end of an extended discussion about how to develop proportional reasoning.

We all have had students who know one rule for dividing fractions: invert and multiply. I learned it as "multiply by the reciprocal," while some of my students annoyed me with the vocabulary-loose "copy dot flip" (or flop). But why does it work?

There are many ways to explain it, including some that get into the gritty details of partitive and quotative division, but I think this explanation would satisfy most students and teachers. Let's start with a problem: \(\frac{3}{4} \div \frac{2}{3} \). I find this explanation is easiest to see if we write this as a compound fraction:
\[ \frac{\frac{3}{4}}{\frac{2}{3}} \]
The reasoning from here is simple: dividing by a fraction is hard, but dividing by one is really easy. But how do we turn \( \frac{2}{3} \) into one? By multiplying by it's reciprocal, of course. But we can't just multiply part of our problem by \( \frac{3}{2} \) without changing its value. The only thing we can multiply by without changing our value is one, and we can write one as something over itself, like this:
\[ \frac{\frac{3}{2}}{\frac{3}{2}} \]
So that's just a fancy way of writing one, and when we multiply the denominators we make a one:
\[ \frac{\frac{3}{4}}{\frac{2}{3}} \cdot \frac{\frac{3}{2}}{\frac{3}{2}} \rightarrow \frac{\frac{3}{4} \cdot \frac{3}{2}}{1} \rightarrow \frac{3}{4} \cdot \frac{3}{2} \]
Which explains "invert and multiply." It's just the result of wanting to divide by one instead of a fraction.

References

Harris, P. W. (in press). Building Powerful Numeracy for Middle and High School Students. Portsmith, NH: Heinemann.

Dividing by Fractions

In his recent post "Math is a dangerous subject to teach," Joe Bower discusses the ability to learn the procedures of math without understanding the conceptual foundations. As an example, Joe humbly admits that he has "absolutely no idea why" (emphasis his) dividing by a fraction is replaced by multiplying by the fraction's reciprocal. He can get the answer right without proper understanding, and therein lies the danger.

One of the things I've enjoyed most about teaching is finding new and deeper ways of understanding so-called "simple" math that I thought I had already mastered. Most of my mathematical upbringing focused more on procedure than understanding, so I occasionally find myself in the same position as Joe. Using Joe's post as inspiration, I've given more thought to dividing by fractions and finally have a model and a description that I hope explains what's really happening when you divide by a fraction.

First, let's look at a simple fraction:

The top number, the numerator, simply counts "how many." The bottom number, the denominator, tells us "how big." We read this fraction properly as "two-thirds," and usually think of that as two objects, each one-third the size of the whole (however big that is).

Now let's look at division. Some students are led astray with early beliefs that "multiplying makes bigger" and "division makes smaller." That kind of misguided number sense can be frighteningly persistent. Multiplication is better thought of as a "scaling" operation, and division can be thought of as a "grouping" operation. To see what I mean, let me explain using whole numbers:
The model I imagine for this problem looks like this:
Using the "grouping" concept of division, I've made four groups. Because each group contains two, the answer is two. No surprise. Let's try another:
The model:
I've made two groups. Because each group contains four, the answer is four. Still no surprises. Let's try one more with whole numbers:
The model:
I've made one group, which is as trivial as it gets. Because the group contains eight, the answer is eight. Now that we've established the pattern with a "grouping" definition, it should be easy to see why you can't divide by zero. I can't possibly make zero groups and still have the eight squares.

Okay, now let's try an easy division problem with fractions:
Remember, the numerator tells us "how many" and the denominator tells us "how big." The model:
It's still one group (how many), but a group of halves (how big). If you count objects, you get the answer: sixteen. But we haven't added or taken away anything -- the eight is still there. Got it? Let's try another:
The model:
Eight divided by two thirds, translated into "grouping-speak," is "eight grouped into two groups of thirds." Because each group contains twelve, the answer is twelve, even though you can still imagine the original eight.

So is this the same as multiplying by the reciprocal? Breaking the wholes into thirds gave us three times as many pieces (24, same as 8 times 3), and grouping into two groups gave us half of the pieces in each group (12, same as 24 divided by 2). More concisely, we multiplied 8 by 3 and divided by two. So doesn't that mean dividing by two thirds is the same as multiplying by three halves? Not exactly. We get the same answer, but for a different reason. To me, the model for eight times three halves would mean scaling eight to be three times bigger (24), then scaled back down to half of that (12). It's a different picture, even if we still get the same answer.

You can choose to accept this as a complication or a convenience; either way, I hope you have a better understanding of dividing by fractions. As always, feel free to offer criticisms in the comments below. (There has to be many a sixth grade teacher who could teach me a thing or two about this topic!)

Update 4/18/2010: Gary Davis co-authored a great guide on the division of fractions. It provides more strategies, more examples, and more detail than my post did. Thanks for sharing, Gary!