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A FINAL SYNTHESIS: REVISITING THE THREE LEARNING PRINCIPLES
569 Ir Pulling Threads M.Suzanne Donovan and John D.Bransford What ties the chapters of this volume together are the three principles from How People Learn (set forth in Chapter 1) that each chapter takes as its point of departure. The collection of chapters in a sense serves as a demon- stration of the second pnnciple: that a solid foundation of detailed knowl- edge and clarity about the core concepts around which that knowledge is organized are both required to support effective learning. The three pnn- ciples themselves are the core organizing concepts, and the chapter discus- sions that place them in information-nch contexts give those concepts greater meaning. After visiting multiple topics in history, math, and science, we are now poised to use those discussions to explore further the three principles of learning. ENGAGING RESILIENT PRECONCEPTIONS All of the chapters in this volume address common preconceptions that students bring to the topic of focus Principle one from How People Learn suggests that those preconceptions must be engaged in the learning process, and the chapters suggest strategies for doing so. Those strategies can be grouped into three approaches that are likely to be applicable across a broad range of topics. 1. Drum on knorrlodgo and oxpor``racos that students commonb bring to t60 c/ass- room but are generally not actiratcd unt6 regard to t60 topic of study.
570 HOW STUDENTS LEARN N THE CEASSFOOM This technique is employed by Lee, for example, in dealing with stu- dents' common conception that historical change happens as an event He points out that students bring to history class the everyday experience of `'nothing much happening" until an event changes things. Histonans, on the other hand, generally think of change in terms of the state of affairs. Change in this sense may include, but is not equivalent to, the occurrence of events. Yet students have many experiences in which things change gradually experiences in which "nothing happening" is, upon reflection, a mischaraaenzation. Lee suggests, as an example, students might be asked to "consider the change from a state of affairs in which a class does not trust a teacher to one in which it does. There may be no event that could be singled out as marking the change, just a long and gradual process." There are many such experiences on which a teacher could draw, such as shifting alliances among friends or a gradual change in a sports team's status with an improvement in performance Each of these experiences has characteristics that support the desired conception of history Events are certainly not irrelevant. A teacher may do particular things that encourage trust, such as going to bat for a student who is in a difficult situation or postponing a quiz because students have two other tests on the same day. Similarly, there may be an incident in a group that changes the dynamic, such as a less popular member winning a valued prize or taking the blame for an incident to prevent the whole group from being punished. But in these contexts students can see, perhaps with some guided discussion, that single events are rarely the sole explanation for the state of affairs. It is often the case that students have experiences that can support the conceptions we intend to teach, but instructional guidance is required to bring these experiences to the fore. These might be thought of as "recessive" experiences. In learning about rational number, for example, it is clear that whole-number reasoningthe subject of study in earlier grades is domi- nant for most students (see Chapter 7). Yet students typically have expen- ence with thinking about percents in the context of sale items in stores, grades in school, or loading of programs on a computer Moss's approach to teaching rational number as described in Chapter 7 uses that knowledge of percents to which most students have easy access as an alternative path to learning rational number She brings students' recessive understanding of proportion in the context of reasoning about percents to the fore and strength- ens their knowledge and skill by creating multiple contexts in which propor- tional reasoning is employed (pipes and tubes, beakers, stnngs). As with events in history, students do later work with fractions, and that work at times presents them with problems that involve dividing a pizza or a pie into discrete parts a problem in which whole-number reasoning often domi- nates. Because a facility with proportional reasoning is brought to bear,
PUEE NG THREADS 571 however, The division of a pie no longer leads students so easily into whole- number traps. Moss reinforces proportional reasoning by having students play games in which fractions (such as i/4) must be lined up in order of size with deci- mals (such as .33) and percents (such as 40 percent). A Theme that runs throughout The chapters of dais volume, in fact, is dhat students need many opportunities to work with a new or recessive concept, especially when doing so requires that powerful preconceptions be overturned or modified. Pain, for example, writes about students' tendency to see "history" and "dhe past" as The same Thing: "No one should Think that merely pointing out conceptual distinctions through a classroom activity equips students to make consistent, regular, and independent use of These distinctions. Students' hab- its of seeing history and The past as The same do not disappear overnight." Bain's equivalent of repeated camper sons of fractions, decimals, and per- cents is The ever-present question regarding descriptions and matenals: is dais "history-as-event"The description of a past occurrenceor "history-as- account"an explanation of a past occurrence. Supporting conceptual change in students requires repeated efforts to strengthen the new conception so that it becomes dominant. 2 Provide o~ortu~i ties for stadrrats to oxpononco discrepant chats that affou' than to came to terms until t60 sbortw~nings in their everyday ~nodols. Relying on students' existing knowledge and experiences can be diff~- cult in some instances because everyday experiences provide little if any opportunity to become familiar with The phenomenon of interest. This is often true in science, for example, where The subject of study may require specialized tools or controlled environmental conditions that students do not commonly encounter In The study of gravity, for example, students do not come to The class- room with experiences that easily support conceptual change because grav- ity is a constant in their world. Moreover, experiences They have with o her forces often support misconceptions about gravity. For example, students can experience variation in friction because most have opportunities to walk or run an object over such surfaces as ice, polished wood, carpeting, and gravel. Likewise, movement in water or heavy winds provide experiences with resistance that many students can easily access. Minstrell found his students believed that These forces with which they had experience explained why they did not float off into space (see Chapter 11). Ideas about buoyancy and air pressure, generally not covered in units on gravity, influenced These students' thinking about gravity. Television images of astronauts floating in space reinforced for the students the idea Slat, widhout air to hold things down, they would simply float off.
572 HOW STUDENTS LEARN N THE CEASSFOOM I\linstrell posed to his students a question dhat would draw out their thinking. He showed them a large frame from which a spring scale hung and placed an object on The scale that weighed 10 pounds. He Then asked the students to consider a situation in which a large glass dome would be placed over The scale and all the air forced out with a vacuum pump. He asked the students to predict (imprecisely) what would happen to The scale reading, Half of Minstrell's students predicted that The scale reading would drop to zero without air; about a Third Thought There would be no effect at all on the scale reading; and The remainder thought There would be a small change. That students made a prediction and The predictions differed stimulated en- gagement. When The experiment was carried out, The ideas of many students were directly challenged by The results They observed. In teaching evolution, Stewart and colleagues found that students' ev- eryday observations led them to underestimate the amount of variation in common species. In such cases, student observations are not so much "wrong'' as They are insufficiently ~efinecl Scientists are more aware of variation be- cause They engage in careful measurement and attend to differences at a level of detail not commonly noticed by the lay person. Stewart and col- leagues had students count and sort sunflower seeds by Their number of stripes as an easy route to a discrepant event of sorts. The students discov- ered There is far more variation among seeds than they had noticed. Unless students understand This point, it will be difficult for them to grasp that natural selection working on natural variation can support evolutionary change. While discrepant events are perhaps used most commonly in science, Bain suggests they can be used productively in history as well (see Chapter 4), To dislodge the common belief dhat history is simply factual accounts of events, Bain asked students to predict how people living in The colonies (and later in The United States) would have marked The anniversary of Columbus's voyage 100 years after his landing in 1492 and then each hun- dred years after that drcough 1992. Students wrote their predictions in jour- nals and were the ~ given historical information about The changing Columbian story over the 500-year peliod. That information suggests that The fret two anniversaries were not really marked at all, that The view of Columbus's "discovery of The new world" as important had emerged by 1792 among former colonists and new citizens of the United States, and that by 1992 The Smidhsonian museum was making no mention of "discovery" but referred to its exhibit as The "Columbian Exchange." If students regard history as The reporting of facts, the question posed by Bain will lead Them to Think about how people might have celebrated Columbus's important discovery, and not u.~ht Her people would have considered the voyage a cause for celebration at all. The discrepancy between students' expectation regarding the answer to The question and The historical accounts They are given in The classroom
PUEE NG THREADS 573 lecture cannot help but jar The conception dhat history books simply report events as They occurred in the past. 3 Provide students lo t6 narratiro accounts of t60 discovery of (targeted) foul. edge or t60 dorolopnzont of (targotod) tools, What we teach in schools draws on our cultural heritage a heritage of scientific discovery, madhematical invention, and historical reconstruction, Narrative accounts of how tills work was done provide a window into change dhat can serve as a ready source of support for students who are being asked to undergo dhat very change Themselves. How is it dhat The earth was discov- ered to be round when nothing we casually observe tells us dhat it is? What is place value anyway? Is it, like The round earth, a natural phenomenon that was discovered? Is it truHh, like e = mc2, to be unlocked? There was a time, of course, when everyday notions prevailed, or everyday problems required a solution, If students can witness major changes Through nar ative, They will be provided an opportunity to undergo conceptual change as well. Stewart and colleagues describe The use of such an approach in teach- ing about evolution (see Chapter 12). IJarwin's dheory of natural selection operating on random variation can be difficult for students to grasp. The beliefs dhat all change represents an advance toward greater complexity and sophistication and that changes happen in response to use (dhe giraffe's neck stretching because it reaches for high leaves, for example) are wide- spread and resilient. And The scientific dheory of evolution is challenged today, as it was in t)arwin's time, by those who believe in intelligent de- signthat all organisms were made perfectly for their function by an intelli- gent creator To allow students to differentiate among These views and un- derstand why t)arwin's dheory is the one that is accepted scientifically, students work widh dlree opposing dheones as they were developed, supported, and argued in t)arwin's day: William Paley's model of intelligent design, Jean Baptiste de Lamarck's model of acquired characteristics based on use, and t)arwin's dheory of natural selection. Students' own preconceptions are gen- erally represented somewhere in the dlree dheones. By consicle ing in some depth The arguments made for each dheory, The evidence that each theolist relied upon to support his argument, and finally the course of events dhat led to The scientific community's eventually embracing Darwin's dheory, stu- dents have an opportunity to see their own ideas argued, challenged, and subjected to tests of evidence. Every scientific dheory has a history that can be used to The same end. And every scientific dheory was formulated by particular people in particular circumstances. These people had hopes, fears, and passions that drove Their work. Sometimes students can understand theolies more readily if They learn about Them in The context of Those hopes, fears, and passions. A narrative
574 HOW STUDENTS LEARN N THE CEASSFOOM that places theory in its human context need not sacrifice any of the techni- cal material to be learned, but can make that material more engaging and meaningful for students. The pnnciple, of course, does not apply only to science and is not restricted to discovery. In mathematics, for example, while some patterns and relationships were discovered, conventions that form our system of counting were invented. As the mathematics chapters suggest, the use of mathematics with understandingthe engagement with problem solving and strategy use displayed by the best mathematics studentsis undermined when students think of math as a rigid application of given algorithms to problems and look for surface hints as to which algorithm applies. If stu- dents can see the nature of the problems that mathematical conventions were designed to solve, their conceptions of what mathematics is can be influenced productively. Historical accounts of the development of mathematical conventions may not always be available. For purposes of supporting conceptual change, however, f crional story telling may do just as well as history. In Teaching as Story Telling, Egan' relates a tale that can support students' understanding of place value: A king wanted to count his army He had fire dveless covuse- lors and one ingeniov s con nselor Each of the d Recess five tried to work out a way of counting the soldiers, but came up with metb- ods that were hopeless One, for example, tried using tally sticks to make a con nt, be t the soldiers kept moving arov nd, and the con nt was confused The ingenious courtselor told the king to have the Jewess counselors pick up ten pebbles each He then bad them stand behind a table that was set up where the army was to ma rcb past. In front of each Jewess coz(rzselor a bowl was placed The army then began to ma rdl past the end of the table. As eacb soldier went by, the first coz(rzselor put one pebble into his bowl Once be had put all ten pebbles into the bowl, be scooped them zip and then corztnzuezl to put one pebble down for each sol- dier marching by the table He bad a very busy afternoon, putting down his pebbles one by one and then scooping them zip when all were in the bowl Each time be scooped zip the ten pebbles, the Jewess covnselor to his left put one pebble into her bowl Gender equity/. When her ten pebbles were in her bowl, she too scooped them out again, and corztnzz(ed to put one back into the bowl each time the dveless coz(rzselor to her right pu:ked his up. I be dveless coz(rzselor to her feel bad to watch her through the aflerrzoon, and be put one pebble into his bowl each time she picked
PUEE NG THREADS 575 hers up.And so on for the remaining counselors.At the end of the Afternoon, the counselor on the far left bad only one pebble in his bowl, the next counselor bad two, the next bad seven, the next bad sax and the counselor at the other end of the table, where the sol- diers bad marched by, bad three pebbles in his bowl So we know that the army bad 12, 763 soldiers. The king was Gel igbted that his ingeniotzs counselor bad counters the whole army with just fifty pebbles 2 When this story is used in elementary school classrooms, Egan encourages the teacher to follow up by having the students count the class or some other, more numerous objects using this method. The story illustrates nicely for students how the place-value system al- lows the complex problem of counting large numbers to be made simpler. Place value is portrayed not as a truth but as an invention. Students can then change the base from 10 to other numbers to appreciate that base 10 is not a "truth" but a "choice." This activity supports students in understanding that what they are learning is designed to make number problems raised in the course of human activity manageable. That imaginative stories can, if effectively designed, support conceptual change as well as historical accounts is worth noting for another reason: the fact that an historical account is an account might be viewed as cause for excluding it from a curriculum in which the nature of the account is not the sub ect of study. Histoncal accounts of Galileo, Newton, or t)arwin written for elementary and secondary students can be contested. One would hope that students who study history will come to understand these as accounts, and that they will be presented to students as such. But the purpose of the accounts, in this case, is to allow students to experience a time when ideas that they themselves may hold were challenged and changed, and that pur- pose can be served even if the accounts are somewhat simplified and their contested aspects not treated fully. ORGANL~NGKNOW7EDGE AROUND CORE CONCEPTS In the Fish is Fish story discussed in Chapter 1, we understand quite easily that when the description of a human generates an image of an up- nght fish wearing clothing, there are some key missing concepts: adapta- tion, warm-blooded versus cold-blooded species, and the difference in mo- bility challenges in and out of water How do we know which concepts are "core?" Is it always obvious? The work of the chapter authors, as well as the committee/author dis- cussions that supported the volume's development, provides numerous in-
576 HOW STUDENTS LEARN N THE CEASSFOOM sights about the identification of core concepts. The first is observed most explicitly in The work of Peter Lee (see Chapter 2): dhat two distinct types of core concepts must be brought to The fore simultaneously. These are con- cepts about The nature of The discipline (what it means to engage in doing history, madh, or science) and concepts that are central to The understanding of The subject mater (exploration of The new world, madhematical functions, or gravity). Lee refers to these as first-order (dhe discipline) and second- order (dhe subject) concepts. And he demonstrates very persuasively in his work dhat students bring preconceptions about The discipline that are just as powerful and difficult to change as Those they bring about The specific sub- ject mater For teachers, knowing the core concepts of The discipline itselfthe standards of evidence, what constitutes proof and disproof, and modes of reasoning and engaging in inquiryis clearly required. This requirement is undoubtedly at the root of arguments in support of teachers course work in the discipline in which They will teach. But dhat course work will be a blunt instrument if it focuses only on second-order knowledge (of subject) but not on first-order knowledge (of the discipline). Clality about the core concepts of the discipline is required if students are to grasp what the discipline history, madh, or scienceis about. For identifying both first- and second-order concepts, The obvious place to turn initially is to those widh deep expertise in the discipline. The con- cepts dhat organize experts' knowledge, structure what they see, and guide Their problem solving are clearly core. But in many cases, exploring expert knowledge directly will not be sufficient. Often experts have such facility with a concept dhat it does not even enter Their consciousness. These "expert blind spots" require dhat "knowledge packages"3sets of related concepts and skills that support expert knowledgebecome a mater for study. A striking example can be found in Chapter 7 on elementary madhemat- ics. For Those widh expertise in madhematics, There may appear to be no "core concept" in whole-number counting because it is done so automati- cally. How one first masters dhat ability may not be accessible to Those who did so long ago. Building on The work of numerous researchers on how children come to acquire whole-number knowledge, Griffin and Case's4 research conducted over many years suggests a core conceptual structure that supports the development of The critical concept of quantity Similar work has been done by Moss and Case' (on The core conceptual structure for rational number) and by Kalchman, Moss, and Cases (on The core con- ceptual structure for functions). The work of Case and his colleagues sug- gests The important role cognitive and developmental psychologists can play in extending understanding of the network of concepts that are "core" and might be framed in less detail by mathematicians (and other disciplinary expelts).
PUEE NG THREADS 577 The work of Stewart and his colleagues described in Chapter 12 is an- odher case in which observations of student efforts to learn help reshape understanding of The package of related core concepts. The clitical role of natural selection in understanding evolution would certainly be identif ed as a core concept by any expert in biology. But in The course of teaching about natural selection, these researchers realization that students underestimated the variation in populations led them to recognize the importance of this concept that they had not previously identified as core. Again, experts in evolutionary biology may not identify population variation as an important concept because They understand and use The concept routinelyperhaps without conscious attention to it. Knowledge gleaned from classroom teach- ing, then, can be critical in defining The connected concepts that help sup- port core understandings. But just as concepts defined by disciplinary experts can be incomplete without The study of student thinking and learning, so, too, The concepts as defined by teachers can fall short if The mastery of disciplinary concepts is shallow. Liping Ma's study of teachers understanding of the madhematics of subtraction widh regrouping provides a compelling example. Some teachers had little conceptual understanding, emphasizing procedure only. But as Box 13-1 suggests, odhers attempted to provide conceptual understanding without adequate mastery of The core concepts Themselves. Ma's work pro- vides many examples (in The teaching of multidigit multiplication, division of fractions, and calculation of perimeter and area) in which efforts to teach for understanding widhout a solid grasp of disciplinary concepts falls short. SUPPORTING METACOGNITION A prominent feature of all of The chapters in dlis volume is The extent to which The teaching described emphasizes The development of metacognitive skills in students. Strengthening metacognitive skills, as discussed in Chapter 1, improves The performance of all students, but has a particularly large impact on students who are lower-achieving.7 Perhaps The most striking consistency in pedagogical approach across The chapters is The ample use of classroom discussion. At times students discuss in small groups and at times as a whole class; at times The teacher leads The discussion; and at times The students take responsibility for ques- tioning. A primary goal of classroom discussion is That by observing and engaging in questioning, students become better at monitoring and ques- tioning their own Thinking. In Chapter 5 by Fuson, Kalchman, and Bransford, for example, students solve problems on The board and Then discuss alternative approaches to solving The same problem. The classroom dialogue, reproduced in Box 13-2, supports the kind of careful thinking about why a particular problem-solv-
575 HOW STUDENTS LEARN N THE CEASSFOOM BOX13-] Conceptual Explanation Without Conceptual Understanding Liping Ma explored approaches to teaching subtraction with regrouping (problems like 52 - 25, in which subtraction of the 5 ones from the 2 ones requires that the number be regrouped). She found that some teachers took a very procedural ap- proach that emphasized the order of the steps, while others emphasized the con- cept of composing a number (in this case into 5 tens and 2 ones) and decomposing a number (into 4 tens and 12 ones). Between these two approaches, however, were those of teachers whose intentions were to go beyond procedural teaching, but who did not themselves fully grasp the concepts at issue. MA describes one such teacher as follows: Tr. Barry, another experienced teacher in the procedurally directed group, mentioned using manipulatives to get across the idea that You need to borrow something. ~ He said he would bring in quarters and let students change a quarter into two dimes and one nickel: 7 good idea might be coins, using money because kids like money.... The idea of taking a quarter even, and changing it to two dimes and a nickel so you can borrow a dime, getting across that idea that you need to borrow something. ~ There are two difficulties with this idea. First of all, the mathemati- cal problem in Tr. Barry's representation was 25 - 10, which is not a subtraction with regrouping. Second Tr. Barry confused borrowing in everyday lifeborrowing a dime from a person who has a quarterwith the ~borrowing" process in subtraction with regroup- ingto regroup the minuend by rearranging within place values. In fact, Tr. Barry's manipulative would not convey any conceptual understanding of the mathematical topic he was supposed to teach. Another teacher who grasps the core concept comments on the idea of "bor- rowing" as follows:9 Some of my students may have learned from their parents that you Borrow one unit form the tens and regard it as 10 ones C . . . I will explain to them that we are not borrowing a 10, but decomposing a 10. "Borrowing ~ can't explain why you can take a 10 to the ones place. But ~decomposing" can. When you say decomposing, it implies that the digits in higher places are actually composed of those at lower places. They are exchangeable . . . borrowing one unit and turning it into 10 sounds arbitrary. My students may ask me how can we borrow from the tens 7 if we borrow something, we should return it later on.
PUEE NG THREADS 579 ing strategy does or does not work, as well as the relative benefits of differ- ent strategies, that can support skilled mathematics performance. Similarly, in the science chapters students typically work in groups, and the groups question each other and explain their reasoning. Box 13-3 repro- duces a dialogue at the high school level that is a more sophisticated version of that among young mathematics students just described. One group of students explains to another not only what they concluded about the evolu- tionary purpose of different coloration, but also the thinking that led them to that conclusion and the background knowledge from an earlier example that supported their thinking. The practice of bringing other knowledge to bear in the reasoning process is at the heart of effective problem solving, but can be diffclllt to teach directly. It involves a search through one's mental files for what is relevant. If teachers simply give students the knowledge to incorporate, the practice and skill development of doing one's own mental search is shortchanged. Group work and discussions encourage students to engage actively in the mental search; they also provide examples from other students' thinking of different searches and search results. The monitoring of consistency between explanation and theory that we see in this group dis- cussion (e.g., even if the male dies, the genes have already been passed along) is preparation for the kind of self-monitonng that biologists do rou- tinely. Having emphasized the benefits of classroom discussion, however, we offer two cautionary notes. First, the discussion cited in the chapters is guided by teachers to achieve the desired learning. Using classroom discussion well places a substantial burden on the teacher to support skilled discussion, respond flexibly to the direction the discussion is taking, and steer it produc- tively. Guiding discussion can be a challenging instructional task. Not all questions are good ones, and the art of questioning requires learning on the part of both students and teachers.'° Even at the high school level, Bain (see Chapter 4) notes the challenge a teacher faces in supporting good student questioning: Sa rena Sa rena Does anyone notice the years that these were written7 About how old are these accounts7 An d rew7 Andrew They were written in 1889 and 1836 So some of them are about 112 years old and others are about 165 years old. Teacher Why did you ask, Sarena7 I'm supposed to ask questions about when the source was written and who wrote it So, I'm just doing my job.
550 HOW STUDENTS LEARN N THE CEASSFOOM BOX13-2 Supporting Skilled Questioning and Explaining in Mathematics Problem Solving In the dialogue below, young children are learning to explain their thinking and to ask questions of each otherskills that help students guide their own learning when those skills are eventually internalized as self-ques- tioning and self-explaining. Maria Teacher Maria, can you please explain to you r f heads in the class how you solved the problems Six is bigger than 4, so l can't subtract here Ipointingl in the ones. So I have to get more ones. But I have to be fairwhen I get more ones, so l add ten to both my numbers. I add a ten here in the top [pointingl to change the 4 to a 14, and I add a ten here in the bottom in the tens place, so l write another ten by my 5. So now I count up from 6 to 14, and I get 8 ones (demonstrating by counting ~6, 7, 8, 9, 10, 11, 12, 13, 14" while raising a finger for each word from 7 to 14). And I know my doubles, so 6 plus 6 is 12, so I have 6 tens left. [She thought, "1 + 5=6and6+7=12tens.0h,1know6+6=12, so my answer is 6 tens."| I don't see the other 6 in yourtens. I only see one 6 in your answer The other 6 is from adding my 1 ten to the 5 tens to get 6 tens. I didn't write it down. But you're changing the problem. How do you get the right answer7 If I make both numbers bigger by the same amount, the difference will stay the same. Remember we looked at that on drawings last week and on the meter stick. Michelle Why did you count up7 Jorge Maria Andy Maria Palincsar" has documented the progress of students as they move be- yond early, unskilled efforts at questioning. Initially, students often parrot the questions of a teacher regardless of their appropriateness or develop questions from a written text that repeat a line of the text verbatim, leaving a blank to be filled in. With expenence, however, students become produc- tive questioners, teaming to attend to content and ask genuine questions.
PUEE NG THREADS 551 Maria Counting down is too hard, and my mother taught me to count up to subtract in first grade. Teacher How many of you remember how confused we were when we first saw Maria's method last weeks Some of us could not figure out what she was doing even though Elena and Juan and Elba did it the same way. What did we don We made drawings with our ten-sticks and dots to see what those numbers meant. And we figu red out they were both tens. Even though the 5 looked like a 15, it was really just 6. And we went home to see if any of our parents could explain it to us, but we had to figure it out ourselves and it took us 2 days. Yes, I was asking other teachers, too. We worked on other methods too, but we kept trying to understand what this method was and why it worked. And Elena and Juan decided it was clearer if they crossed out the 5 and wrote a 6, but Elba and Maria liked to do it the way they learned at home. Any other questions or comments for Maria7 No7 Ok, Peter, can you explain your method 7 Yes, I like to ungroup my top number when I don't have enough to subtract everywhere. So here I ungrouped 1 ten and gave it to the 4 ones to make 14 ones, so l had 1 ten left here. So 6 up to 10 is 4 and 4 more up to 14 is 8, so 14 minus 6 is 8 ones. And 5 tens up to 11 tens is 6 tens. So my answer is 68. Carmen How did you know it was 11 tens7 Peter Because it is 1 hundred and 1 ten and that is 1 1 tens. Rafael Peter Similarly, students' answers often cannot serve the purpose of clarifying their thinking for classmates, teachers, or themselves without substantial support from teachers. The dialogue in Box 13-4 provides an example of a student becoming clearer about the meaning of what he observed as the teacher helped structure the articulation.
552 HOW STUDENTS LEARN N THE CEASSFOOM BOX13-3 Questioning and Explaining in High School Science The teacher passes out eight pages of case materials and asks the stu- dents to get to work. Each group receives a f lie folder containing the task description and information about the natural history of the ring-necked pheasant. There are color pictures that show adult males, adult females, and young. Some of the pages contain information about predators, mat- ing behavior, and mating success. The three students spend the remain- der of the period looking over and discussing various aspects of the case. By the middle of the period on Tuesday, this group is just finalizing their explanation when Casey, a member of another group, asks if she can talk to them. Casey G race What have you guys come up with7 Our group was wondering if we could talk over our ideas with you. Sure, come over and we can each read our explanations. These two groups have very different explanations. Hillary's group is th in ki ng that the males' bright coloration distracts predators f rom the nest, while Casey's group has decided that the bright coloration confers an advantage on the males by helping them attract more mates. A lively discussion ensues. Ed But wait, I don't understand. How can dying be a good things Jerome Well, you have to think beyond just survival of the male himself. We think that the key is the survival of the kids. If the male can protect his Group work and group or classroom discussions have another potential pitfall that requires teacher attention: some students may dominate dhe dis- cussion and She group decisions, while odhers may participate little if at all. Having a classmate take charge is no more effective at promoting metacognitive development or supporting conceptual changeThan hav- ing a teacher take charge. In eidher case, active engagement becomes unnec- essary. One approach to tackling this problem is to have students rate Their group effort in terms not only of Their product, but also of their group dy-
PUEE NG THREADS 533 Claire Grace Casey Grace Hillary Grace Jerome Casey young and give them a better chance of surviving then he has an advantage. Even if he dies doing it7 Yeah, because he will have already passed on his genes and stuff to his kids before he dies. How did you come up with this7 Did you see something in the packets that we didn't see 7 One reason we thought of it had to do with the last case with the monarchs and viceroy. Yeah, we were thinking that the advantage isn't always obvious and sometimes what is good for the whole group might not seem like it is good for one bird or butterfly or whatever We also looked at the data in our packets on the number of offspring fathered by brighter versus duller males. We saw that the brighter males had a longer bar See, look on page 5, right here. So they had more kids, right7 We saw that table too, but we thought that it could back up our idea that the brighter males were able to attract more females as mates. The groups agree to disagree on their interpretation of this piece of data and continue to compare their explanations on other points. While it may take the involvement of a teacher to consider further merits of each explanation given the data, the students' group work and dialogue pro- vide the opportunity for constructing, articulating, and questioning a sci- entific hypothesis. namics.'2 Another approach, suggested by Pain (Chapter 4), is to have stu- dents pause dunng class discussion to think and write individually. As stu- dents discussed the kind of person Columbus was, Bain asked them to wnte a 2-minute essay before discussing further Such an exercise ensures that students who do not engage in the public discussion nonetheless formulate their ideas. Group work is certainly not the only approach to supporting the devel- opment of metacognitive skills. And given the potential hazard of group
584 HOW STUDENTS LEARN N THE CEASSFOOM BOX13-4 Guiding Student Observation and Articulation In an elementary classroom in which students were studying the behav- ior of light, one group of students observed that light could be both re- flected and transmitted by a single object. But students needed consider- able support from teachers to be able to articulate this observation in a way that was meaningful to them and to others in the class: Ms. Lacey Kevin Ms. Lacey Kevin Derek Kevin Derek Kevin Derek I'm wondering. I know you have a lot of see- through things, a lot of reflect things. I'm wondering howyou knew it was see-through. It would shine just, straight th rough it. What did you see happenings We saw light going th rough the . . . Like if we put light. . . Wherever we tried the flashlight, like right here, it would show on the board. And then I looked at the screen [in front of and to the side of the objectl, and then it showed a light on the screen. Then he said, come here, and look at the back. And I saw the back, and it had another [spotl. Ms. Lacey Did you see anything else happening at the materials We saw sort of a little reflection, but we, it had mostly just see-through. We put, on our paper we put reflect, but we had to decide which one to put it in. Because it had more of this than more of that. Oh. So you're saying that some materials . . . Had more than others . . . Ms. Lacey Derek dynamics, using some individual approaches to supporting seU-monitonng and evaluation may be important. For example, in two experiments with students using a cognitive tutor, Aleven and Koedinger'3 asked one group to explain the problem-solving steps to themselves as they worked. They found that students who were asked to self-explain outperformed those who spent the same amount of time on task but did not engage in seU-explanation on transfer problems. This was true even though the common time limitation meant that the self-explainers solved fewer problems.
PUEE NG THREADS 555 Derek Ms. Lacey . . . are doing, could be in two different categories. Yeah, because some th rough were really reflection and see-th rough together, but we had to decide which. intervening discussion takes place about other data presented by this group that had to do with seeing light reflected or transmitted as a particular color, and how that color com- pared with the color of the objects [at the end of this group's reporting, and after the students had been encou raged to identify seve ra I cla i ms th at the r data su ppo rted among those that had been presented previ- ously by other groups of studentsl Ms. Lacey There was something else I was kinds con- vinced of. And that was that light can do two differentthings. Didn'tyou tell me it went both see-through and reflected7 Kevin & Derek Yeah. M m-h mm Ms. Lacey So do you think you might have another claim there7 Derek Yeah. Kevin Light can do two things with one object. Ms. Lacey More than one things Kevi n Yea h. Ms. Lacey Okay What did you say7 Kevin & Derek Light can do two things with one object. See Chapter 10 for the context of this dialogue. Another individual approach to supporting metacognition is suggested by Stewart (Chapter 12). Students record their thinking early in the treatment of a new topic and refer back to it at the unit's end to see how it has changed. This brings conscious attention to the change in a student s own thinking. Similarly, the reflective assessment aspect of the ThinkerTools cur- nculum described in Chapter 1 shifts students from group inquiry work to evaluating their group's inquiry individually. The results in the ThinkerTools case suggest that the combination of group work and individual reflective
556 HOW STUDENTS LEARN N THE CEASSFOOM assessment is more powerful that The group work alone (see Box 9-5 in Chapter 9). PRINCIPLES OF LEARNING ~ CLASSROOM ENVIRONMENTS The principles dhat shaped These chapters are based on efforts by re- searchers to uncover the rules of The learning game. Those rules as we understand Them today do not tell us how to play the best instructional game. They can, however, point to The strengdhs and weakness of instruc- tional strategies and the classroom environments dhat support those strate- gies. In Chapter 1, we describe effective classroom environments as learner- centered, knowledge-centered, assessment-centered, and community- centered Each of These characteristics suggests a somewhat different focus. But at the same time They are interrelated, and the balance among Them will help determine the effectiveness of instruction. A community-centered classroom dhat relies extensively on classroom discussion, for example, can facilitate learning for several reasons (in addi- tion to supporting metacognition as discussed above): . It allows students' thinking to be made transparentan outcome dhat is critical to a learner-centered classroom. Teachers can become familiar with student ideas for example, The idea in Chapter 7 dhat two thirds of a pie is about The same as dhree-fourths of a pie because bodh are missing one piece. Teachers can also monitor The change in those ideas with learning opportunities, the pace at which students are prepared to move, and The ideas that require further workkey features of an assessment-centered class- room. . It requires dhat students explain Their thinking to others. In the course of explanation, students develop a disposition toward productive interchange with odhers (community-centered) and develop their thinking more fully (learner-centered), in many of The examples of student discussion dhrough- out tills volumefor example, The discussion in Chapter 2 of students exam- ining the role of Hider in World War 11one sees individual students becom- ing clearer about their own thinking as the discussion develops. · Conceptual change can be supported when students' thinking is chal- lenged, as when one group points out a phenomenon that another group's model cannot explain (knowledge-centered). This happens, for example, in a dialogue in Chapter 12 when t)elia explains to Scott dhat a flap might prevent more detergent from pouring out, but cannot explain why the amount of detergent would always be The same.
PUEE NG THREADS 557 At the same time, emphasizing the benefits of classroom discussion in supporting effective learning does not imply that lectures cannot be excel- lent pedagogical devices. Who among us have not been witness to a lecture from which we have come away having learned somedhing new and impor- tant~ The Feynman lectures on introductory physics mentioned in Chapter 1, for example, are well designed to support teaming. That design incorpo- rates a strategy for accomplishing The learning goals described Throughout dlis volume " Feynman anticipates and addresses The points at which stu- dentst preconceptions may be a problem. Knowing dhat students will likely have had no experiences that support grasping The size of an atom, he spends time on dlis issue, using familiar references for relative size dhat allow students to envision just how tiny an atom is. But to achieve effective learning by means of lectures alone places a major burden on The teacher to anticipate student Thinking and address prob- lems effectively. To be applied well, This approach is likely to require both a great deal of insight and much experience on The part of the teacher With- out such insight and expenence, it will be difficult for teachers to anticipate The full range of conceptions students bring and the points at which They may stumble.'sWhile one can see dhat Feynman made deliberate efforts to anticipate student misconceptions, he himself commented dhat The major difficulty in The lecture series was The lack of opportunity for student ques- tions and discussion, so that he had no way of really knowing how effective The lectures were. In a learner-centered classroom, discussion is a powerful tool for elicit ng and monitoring student thinking and teaming, In a knowledge-centered classroom, however, lectures can be an impor- tant accompaniment to classroom discussionan efficient means of consoli- dating learning or presenting a set of concepts coherendy. In Chapter 4, for example, Bain describes how, once students have spent some time working on compet ng accounts of the significance of Columbus's voyage and struggled with The question of how The anniversaries of The voyage were celebrated, he delivers a lecture dhat presents students with a description of current thinking on The topic among histonans. At The point at which This lecture is delivered, student conceptions have already been elicited and explored. Because lectures can play an important role in instruction, we stress once again Blat The emphasis in This volume on The use of discussion to elicit students' Thinking, monitor understanding, and support metacognitive de- velopmentall critical elements of effective teachingshould not be mis- taken for a pedagogical recommender on of a single approach to instruction, Indeed, inquiry-based learning may fall short of its target of providing stu- dents widh deep conceptual understanding if The teacher places The full bur- den of learning on The activities. As Box 1-3 in Chapter 1 suggests, a lecture that consolidates The lessons of an activity and places The activity in The
555 HOW STUDENTS LEARN N THE CEASSFOOM conceptual framework of The discipline explicitly can play a critical role in supporting student understanding. How The balance is struck in creating a classroom that functions as a learning community attentive to The leamers' needs, the knowledge to be mastered, and assessments That support and guide inst uction will certain vary from one teacher and classroom to The next. Our hope for tills volume, Then, is that its presentations of instructional approaches to addressing The key principles from How Pt ople Learn will support The efforts of teachers to play Their own instructional game well. This volume is a first effort to elabo- rate Those findings with regard to specific topics, but we hope it is The first of many such efforts. As teachers and researchers become more familiar widh some common aspects of student thinking about a topic, their attention may begin to shift to o h e- aspects That have previously attracted lithe notice. And as insights about one topic become commonplace, They may be applied to new topics. Beyond extending The reach of The treatment of The learning principles of How Pt ople Learn within and across topics, we hope that efforts to incor- porate those principles into teaching and learning will help st¢ngdhen and reshape our understanding of The rules of The learning game. Widh physics as his topic of concern, Feynman'6 talks about just such a process: "For a long time we will have a rule That works excellendy in an overall way, even when we cannot follow The details, and Then some time we may discover a new rule From The point of view of basic physics, The most interesting phenomena are of course in The new places, The places where The rules do not worknot The places where they do work! That is The way in which we discover new rules." We look forward to The opportunities created for The evolution of The science of learning and The professional practice of teaching as The pnn- ciples of learning on which This volume focuses are incorporated into class- room teaching. NOTES I Egan, 1986. 2 Story sumr arrived by Kieran Egan, personal communication, March 7, 2003. 3 Liping Mass work, described in Chapter I, refers to the set of core concepts and the connected concepts and knowledge that support them as "knowledge packages." 4 Griffin and Case, 1995 5. Moss and Case, 1999. 6. Kalchr an et al., 2001. 7 Palincsar, 1986; White and Erednek60n, 1998 8. Ma, 1999, p. 5 9 Ma, 1999, p 9
PUEE NG THREADS 559 10 1'~1in~i a. 198 ~ 11. Palincsar, 1986. 12. National Research Council, 2005 (Stewart et al., 2005, Chapter 12). 13. A even and Koedinger, 2002. 14. For example, he high ights core concepts conspicuously. In his first leclune, he asks, "If, in some catac ysm, all of scientific knowledge were to be destroyed, and on y one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I be love it is the atomic hypothesis that a t sings are made of atoms litt e partic es that move a ound m perpetua motion, att acting each other when they am a litt e distance apart, but ¢pellmg upon being squeezed into one another 15. Even with experience, the thinking of individual students may be unamtici- pated by the teacher 16. Feynman, 1995, p. 25. REFERENCES Aleven, V., and Koedinger, K. (2002). An effective metacogmitive st ategy: Leaming by doing and explaining with a computer-based cognitive tutor Cognitive Sci- ence, 26, 147-179. Egan, K. (1986). Teaci ing as story telling An alterllatil e approach to teaching and curriculum in the ~1~7wnta~ school (vol. iii). Chicago, IL: University of Chicago Pass. Feynman, R. P. (1995). Six easy pieces Essentials of physics explained by its most bnl- liantteacber Heading, MA Perseus Books. Griffin, s., and Case, R. (1995) Re-thirking the pnmary school math curnculum: An approach based on cognitive science. Issues in Education, 3 1), 1-49. Kale man, M., Moss, J., and Case, R. (2001) Psychological models for the develop ment of mathematical understanding Rational numbers and functions. In s. Carver and D. K ahr (Eds.), Cognition and instruction Twentyf he years of progress (pp. 1-38). Mahwah, NJ: Lawrence Erlbaum Associates. Ma, L. (1999). KnowSngand teaching elementary mathematics Mahwah, NJ: Lawrence Erlba~m Associates Moss, J., and Case, R. (1999). Developing child¢n's understanding of rational num- bers: A new model and experimental curriculum. JournalforReseareb in Matb- ematicsEducation, 3tXZ). Palincsar, A.S. (1986). Reciprocal teaching Teaching reading as thinking Oak Bmok, IL: North Central Regional Educational Laboratory. Stewart, J. , Ca tier, J.L., and Passmore, C.M. (2005). Developing understanding through model-based inquiry. In National Research Council, How students learn His- tory, mathematics, and science in the classroom. Committee on How People Leam, A Targeted Report for Teachers, M.S. Donovan and J.D. Btansford (Eds.). Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies P¢ss. White, B., and Fred ickson, J. (1998). Inqui y, modeling and metacogmition: Making science accessible to all students. Cognition and instruction, 6(1), 3-117.
590 HOW STUDENTS LEARN N THE CEASSFOOM Oll=R RESOURCES National Academy of Sciences. (1998). Teaching about evolution and the nature of science Working Group on Teaching Evolution. Washington, DC: National Acad- emy Press: Available: http:/ books.nap.edu/catalog/5787.html. National Academy of Sciences. (2004). Evolution in Hawaii: A supplement to teacb- ing about evolution and the nature of scienceby Steve Olson. Washington, DC: The National Academies Press. Available: http://www.nap.edu/books/ 030gO89913/htm /.