How Students Learn History, Mathematics, and Science in the Classroom (2005) / Chapter Skim
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351 8 Teaching and Learning Functions Mindy Kalchman and Kenneth R Koedinger This chapter focuses on teaching and learning mathematical functions.1 Functions are all around us, though students do not always realize this.
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The second principle of How People Learn argues that students need a strong conceptual understanding of function as well as procedural fluency. The new and very central concept introduced with functions is that of a dependent relationship: the value of one thing depends on, is determined by, or is a function of another.
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If students lack a conceptual understanding of linear function, what errors might they make? Figure 8-1b shows an example student solution.
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For instance, some students used a generate-and-test strategy: They estimated a value for the hourly rate (e.g., $4/hour) , computed the corre sponding pay (e.g., $90)
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TEACHINGAND LEARNING FUNCTIONS 355 and repeated as needed. Other students used a more efficient unwind or working backwards strategy.
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356 HOW STUDENTS LEARN: MATHEMATICSIN THECLASSROOM a b FIGURE 8-1 student did not recognize the inconsistency between the positive slope of the line and the negative slope in the equation. Even good mathematicians could make such a mistake, but they would likely monitor their work as they went along or reflect on the plausibility of the answer and detect the incon sistency.
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. In general, the student's work on this problem reflects an incomplete conceptual framework for linear functions, one that does not provide a solid foundation for fluid and flexible movement among a function's representations.
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358 HOW STUDENTS LEARN: MATHEMATICSIN THECLASSROOM c FIGURE 8-1 Problem: Make a table of values that would produce the function seen above. This student identified a possible y-intercept based on a reasonable scale for the y-axis.
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ADDRESSING THE THREE PRINCIPLES Principle #1: Building on Prior Knowledge Principle 1 emphasizes the importance of students and teachers continually making links between students' experiences outside the mathematics classroom and their school learning experiences. The understandings students bring to the classroom can be viewed in two ways: as their everyday, informal, experiential, out-of-school knowledge, and as their schoolbased or "instructional" knowledge.
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One dollar for every kilometer walked. So you have one dollar for each kilometer [writing "$1.00 for each kilometer" on the board while saying it]
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TEACHING AND LEARNING FUNCTIONS 361 FIGURE 8-2b The teacher and students construct the table and graph point by point, and a line is then drawn. [Students continue to provide the dollar amounts for each of the successive kilometer values.
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They can often solve problems in ways we do not teach them or expect if, and this is an important qualification, the problems are described using words, drawings, or nota tions they understand. For example, the topic of slope is typically reserved for ninth-grade mathematics, and is a part of students' introduction to rela tions and functions in general and to linear functions in particular.
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Our point, instead, is that using student language is one way of first assessing what knowledge students are bringing to a particular topic at hand, and then linking to and building on what they already know to guide them toward a deeper understanding of formal mathematical terms, algorithms, and symbols. In sum, students' prior knowledge acts as a building block for the development of more sophisticated ways of thinking mathematically.
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Level 0 Level 0 characterizes the kinds of numeric/symbolic and spatial under standings students typically bring to learning functions. Initially, the numeric and spatial understandings are separate.
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1 Spatial and numeric understandings are elaborated and integrated, forming a central conceptual structure. · Elaboration of numeric understanding: Multiply each number in the sequence -- Iteratively apply a single operation 0, 1, 2, ...
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not? · Integrate understanding of y = x and y = x + b to form a mental structure for linear functions.
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Level 1 At level 1, students begin to elaborate and integrate their initial numeric and spatial understandings of functions. They elaborate their numeric understanding in two steps.
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Students learn more complex content during levels 2 and 3. Level 2 As students progress to level 2, they begin to elaborate their initial inte grated numeric and spatial understandings.
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This integration helps students begin to understand and organize their knowledge in ways that facilitate the retrieval and application of relevant mathematical concepts and procedures. If students' numeric and spatial understandings are not integrated, they may not notice when a conclusion drawn from one understanding is inconsistent with a conclusion drawn from another.
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370 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM BOX 8-2 The Devil's in the Details: The 3-Slot Schema for Graphing a Line What students glean from instruction is often very different from what we as teachers intend. This observation is nicely illustrated by the re search of Schoenfeld and colleagues.9 They detail the surprising misun derstandings of a 16-year old advanced algebra student who is grappling with conceptual questions about equations and graphs of linear functions.
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She did not understand how using the y-intercept and slope, in particular, facilitate an efficient graphing strategy because they can be read immediately off the standard form of an equation. Schoenfeld and colleagues' fine-grained analysis of learning nicely illustrates how subtle and easily overlooked misconceptions can be -- even among our best students.13 skill and conceptual understanding (Principle 2)
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In other words, we want to encourage students to think about problems not only in
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Below we describe a unit of instruction, based on the developmental model described above, that has been shown experimentally to be more effective than traditional instruction in increasing understanding of functions for eighth and tenth graders.14 In fact, sixth-grade students taught with this instructional approach were more successful on a functions test than eighth and tenth graders who had learned functions through conventional instruction. At the secondary level, tenth graders learning with this approach demonstrated a deeper understanding of complex nonlinear functions.
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Example lessons are provided in the next section. The curricular sequence we suggest has been used effectively with stu dents in sixth, eighth, tenth, and eleventh grades.
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The three lessons and their companion activities illustrate the principles of How People Learn in three key topic areas: slope, y-intercept, and quadratic functions. Example lesson 1, "Learning Slope," illustrates principle 1, building on students' prior knowledge.
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be found in its table, graph, and equation. Curving Nonlinear functions are introduced Students are asked to decide which functions as those having up-by amounts of four functions expressed in that increase (or decrease)
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TEACHING AND LEARNING FUNCTIONS 377 TABLE 8-2 Continued Topic Description Activities derived by multiplying the asked to write an equation for and kilometers (x) by itself at least .
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So at zero kilometers how much am I going to have? Katya At zero kilometers you'll have zero.
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When introducing this up-by idea to students, we suggest beginning with the graph and the table for the rule of earning one dollar for every kilometer walked ($ = 1 x km) and having students see that in each of these representations, the dollar amount goes up by one for each kilometer walked.
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Second, stu dents' prior knowledge of natural language, such as "up-by," can be used to
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Students have termed this starting amount the "starter offer" or "starter upper." These phrases have repeatedly been shown to be simple for students to understand first in the walkathon context and then in more abstract situations. We again begin this lesson with a sponsorship arrangement of earning one dollar for every kilometer walked.
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We then connect the "starting points" of the graphs and tables with the structure of the equations to show that the starting bonus is indeed added to each x value. Emphasizing that the only effect of changing the starter offer is a vertical shift in a function is crucial because a number of researchers have found that students regularly confuse the values for slope and y-intercept in equations.
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Teacher Ya, we're going up by 5 so as we go across 1 we go up by 5. In the lessons on nonlinear functions, the starter offer idea is also applied.
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The idea of a "starter offer" gives students a reasonably familiar situation that provides a context for learning y-intercept -- ordinarily a relatively abstract and difficult mathematical topic that is often confused with slope in stu dents' understanding of linear function. In our approach, students still learn the notations, symbols, words, and methods necessary for identifying the y intercept of a function (linear or nonlinear)
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TEACHING ANDLEARNING FUNCTIONS 385 BOX 8-4 Two Different Student Solutions to an Open-Ended Problem a value larger than 1, smaller than 1 but greater than 0, and less than 0. They are then asked to compare the tables and graphs for y = x2, y = 2 *
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386 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM FIGURE 8-3 Sample computer screen. In this configuration, students can change the value of a, n, or b to effect immediate and automatic changes in the graph and the table.
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. This exchange illustrates the use of metacognitive prompting to help students supervise their own learning by suggesting the coordination of conclusions drawn from one representation (e.g., slope in linear functions)
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388 HOW STUDENTS LEARN: MATHEMATICSIN THECLASSROOM FIGURE 8-4 that two-point rule for graphing straight lines to the graphing of curved-line functions. In the example shown in Figure 8-4, an eleventh-grade advanced mathematics student who had been learning functions primarily from a textbook unit decided to calculate and plot only two points of the function y = x2 +1 and then to join them incorrectly with a straight line.
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On the other hand, the student displays a lack of coordinated conceptual understanding of linear functions and how they appear in graphical, tabular, and symbolic representations. In particular, he does not appear to be able to extract qualitative features such as linearity and the sign of the slope and to check that all three representations share these qualitative features.
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He did not see or "encode" the fact that because the graph is linear, equal changes in x must yield equal changes in y, and the values in the table must represent this critical characteristic of linearity. The curriculum presented in this chapter attempts to focus limited in structional time on core conceptual understanding by using multiple repre sentations and generalizing from variations on just a few familiar contexts.
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2 + 4 in their study of nonlinear functions across lessons 4 to 8. We do not mean to suggest that this is the only curriculum that promotes a deep conceptual understanding of functions or that illustrates the principles of How People Learn.
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Journal for Research in Mathematics Education, 31(2)
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, CBMS issues in mathematics education (vol.
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Part III SCIENCE
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