How Students Learn Mathematics in the Classroom (2005) / Chapter Skim
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Pages 209-230
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From page 209... ...
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.
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570 HOW STUDENTS LEARN N THE CEASSFOOM This technique is employed by Lee, for example, in dealing with students' 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.
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. 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.
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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.
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From page 213... ...
And The scientific dheory of evolution is challenged today, as it was in t) arwin's time, by those who believe in intelligent design—that all organisms were made perfectly for their function by an intelligent creator To allow students to differentiate among These views and understand why t)
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574 HOW STUDENTS LEARN N THE CEASSFOOM that places theory in its human context need not sacrifice any of the technical 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.
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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 upnght fish wearing clothing, there are some key missing concepts: adaptation, warm-blooded versus cold-blooded species, and the difference in mobility challenges in and out of water How do we know which concepts are "core? " Is it always obvious?
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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 subject mater For teachers, knowing the core concepts of The discipline itself—the standards of evidence, what constitutes proof and disproof, and modes of reasoning and engaging in inquiry—is 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.
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But as Box 13-1 suggests, odhers attempted to provide conceptual understanding without adequate mastery of The core concepts Themselves. Ma's work provides many examples (in The teaching of multidigit multiplication, division of fractions, and calculation of perimeter and area)
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From page 218... ...
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 "borrowing" 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 .
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From page 219... ...
PUEE NG THREADS 579 ing strategy does or does not work, as well as the relative benefits of different 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.
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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 other—skills that help students guide their own learning when those skills are eventually internalized as self-questioning 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.
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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.
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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 students 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.
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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.
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584 HOW STUDENTS LEARN N THE CEASSFOOM BOX13-4 Guiding Student Observation and Articulation In an elementary classroom in which students were studying the behavior of light, one group of students observed that light could be both reflected and transmitted by a single object. But students needed considerable 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.
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PUEE NG THREADS 555 Derek Ms. Lacey .
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From page 226... ...
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 work—key features of an assessment-centered classroom.
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From page 227... ...
But to achieve effective learning by means of lectures alone places a major burden on The teacher to anticipate student Thinking and address problems 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 Without 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 questions and discussion, so that he had no way of really knowing how effective The lectures were.
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From page 228... ...
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 incorporate 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 work—not The places where they do work!
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From page 229... ...
. Mahwah, NJ: Lawrence Erlbaum Associates.
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From page 230... ...
590 HOW STUDENTS LEARN N THE CEASSFOOM Oll=R RESOURCES National Academy of Sciences.
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