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8 Teaching and Learning Functions
Pages 163-206

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From page 163...
... Algebraic tools allow us to express these functional relationships very efficiently; find the value of one thing (such as the gas puce) when we know the value of the other (the number of gallons)
From page 164...
... The second principle of How People Learn argues dhat students need a strong conceptual understanding of function as well as procedural fluency. The new and very central concept introduced widh functions is dhat of a dependent at latwnshtp The value of one Thing depends on, is detemmined by, or is a function of anodher The kinds of problems we are dealing with no longer are focused on determining a specific value (dhe cost of 5 gallons of gas)
From page 165...
... Are such skills sufficient for a correct solution? If students lack a conceptual understanding of linear function, what errors might they make?
From page 166...
... 354 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM BOX 8-1 Linking Formal Mathematical Understanding to informal Reasoning Which of these problems is most difficult for a beginning algebra student? Story Problem When Ted got home f rom his waiter job, he multiplied his hourly wage by the 6 hours he worked that day.
From page 167...
... Looking closely at students' work, the strategies they employ, and the errors they make, and even comparing their performance on similar kinds of problems, are some of the ways we can get past such blind spots and our natural tendency to think students think as we do. Such studies of student th in ki ng contributed to the creation of a technology-enhanced algebra course, originally Pump Algebra Tutor and now Cognitive Tutor Algebra.7 That course includes an intelligent tutor that provides students with individualized assistance as they use multiple representations (words, tables, graphs, and equations)
From page 168...
... 356 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM a b FIGURE 8 1 ~ 1 ~ — y >.4,~/ it. ~ a/ / x 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 inconsistency.
From page 169...
... More important, however, these surface errors reflex a deeper weakness in the student's conceptual understanding of function. The student either did not have or did not apply knowledge for interpreting key features (e.g., increasing or decreasing)
From page 170...
... She determined and recorded values that show a constant increase in y for every positive unit change in x. She also derived an equation for the function that not only corresponds to both the graph and the table, but also represents a linear relationship between x and y.
From page 171...
... 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. In the instructional approach illustrated here, students are introduced to function and its multiple representations by having their prior experiences and knowledge engaged in the context of a walkathon.
From page 172...
... One doll r for every kilometer walked. So you have one dollar for each kilometer [writing ~$1.00 for each kilometer" on the board while saying itl.
From page 173...
... 10 q , b- 5 3 . / / i ~ I ~ g I I ~ ~ llo Teacher Melissa Teacher TEACH NG AND LEARN NG FUNCT ONS 361 Kht o o 1 1 3 3 h , FIGURE 8 2b The leacher ond students construct ibe Able ond groph point by point, ond o line is t en drown IStudents continue to provide the dollar amounts for each of the successive kilometer values.
From page 174...
... 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 notations They understand. For example, The topic of slope is typically reserved for nindh-grade madhematics, and is a part of students' introduction to relations and functions in general and to linear functions in particular It is generally defined as The ratio of vertical distance to horizontal distance, or `'nse to run.
From page 175...
... 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 fommal 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.
From page 176...
... To This end, we need an instructional plan dhat deliberately builds and secures dhat knowledge. Good teaching requires not only a solid understanding of The content domain, but also specific knowledge of student development of these conceptual understandings and procedural competencies.
From page 177...
... within a string of positive whole numbers. · Initial spatial understanding: students represent the relative sizes of quantities as bars on a graph and perceive patterns of qualitative changes in amount by a left-to-right visual scan of the graph, but cannot quantify those changes.
From page 178...
... · Elaborate initial integrated numeric and spatial understandings to create more sophisticated variations. · Integrate understanding of y = x and y = x + b to form a mental structure for linear functions.
From page 179...
... Level I 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.
From page 180...
... With this new integrated mental structure for functions, students can support numeric and spatial understandings of algebraic representations such as y = 1x, Grasping why and how the line on a graph maps onto the relationship described in a word problem or an equation is a core conceptual understanding. If students' understanding is only procedural, they will not be well prepared for the next level (see Box 8-2)
From page 181...
... This integration helps students begin to understand and organize Their knowledge in ways that facilitate The retrieval and application of relevant madhematical 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 anodher.
From page 182...
... Clea rly to is student had received extensive instruction in linearfunctions. For instance, in an earlier exchange, when asked for an equation of Principle #3: Building Resourceful, Self-Regulating Problem Solvers As discussed above, teaching aimed at developing robust and fluent mathematical knowledge of functions should build on students' existing realworld and school knowledge (Principle 1)
From page 183...
... 3 skill and conceptual understanding (Pnnciple 2)
From page 184...
... In this example, good metacognitive thinking was not about checking The consistency of numeric answers obtained using different strategy es, but about checking The consistency of verbal f nterpretaffons (e.g., increasing vs. decreasing)
From page 185...
... 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 tendh graclers.'4 in fact, srxth-grade students taught with tills instructional approach were more successful on a functions test Than eighdh and tendh graders who had learned functions through conventional instruction. At the secondary level, tendh graders learning widh tills approach demonstrated a deeper understanding of complex nonlinear functions.
From page 186...
... The use of multiple representations is anodher significant feature of The suggested curriculum, one That again distinguishes it from more traditional approaches. In many traditional approaches, instruction may be focused on a single representation (e.g., equation or graph)
From page 187...
... Although we do not provide a complete description of these lessons, the example activities should be sufficient to suggest how the lessons might be used in other classrooms. The three lessons and their companion activities illustrate the principles of How Peop/e Leant in three key topic areas: slope, y-intercept, and quadratic functions Example lesson 1, "Learning Slope," illustrates plinciple 1, building on students prior knowledge Example lesson 2, "Learning y-intercept," illustrates principle 2, connecting students' factual/procedural and conceptual knowledge Example lesson 3, "Operating on y = x2," illustrates principle 3, fostering reflective thinking or metacognition in students.
From page 188...
... after each kilometer walked. They are Students invent two linear functions that allow them to earn exactly $153.00 after walking 10 kilometers.
From page 189...
... Presentations stimulate discussion and summarizing of key concepts and serve as a partial teacher assessment for evaluating students' postinstruction understanding about functions Activities asked to write an equation for and to sketch and label the graph of each function. Students are asked to come up with a curvedline function for earning $153.00 over 10 kilometers.
From page 190...
... In this interaction we can see how Katya quickly grasps the idea of slope as relative steepness, as defined by the variable relationship between two quantities (distance walked and money earned in this case) : Teacher Katya Teacher Katya Teacher Katya Teacher Katya Teacher Katya: Teacher Katya Teacher Katya Teacher Katya Teacher Katya Teacher I want to think of a way, let's see, Katya, how might you sponsor me that would make a line that is steeper than this BY = x is already drawn on graph, as in Figure 8-2b|7 Steeper7 Alright .
From page 191...
... When introducing this up-by idea to students, we suggest beginning with the graph and the table for the rule of earning one dollar for every kilometer walked ($ = 1 x km) and having students see that in each of these representations, the dollar amount goes up by one for each kilometer walked.
From page 192...
... Second, students' prior knowledge of natural language, such as "up-by," can be used to
From page 193...
... Instead of starting by formally introducing this method, this lesson begins by having students explore situations in which a nonzero starting amount is used. This approach appears to do a better job of helping students learn the formal procedure in the context of a robust conceptual understanding, The Lesson.
From page 194...
... We Then connect The "starting points" of The graphs and tables widh The structure of The equations to show dhat The starting bonus is indeed added to each xvalue. Emphasizing dhat The only effect of changing The starter offer is a vertical shift in a function is crucial because a number of researchers have found dhat students regularly confuse The values for slope and y-intercept in equations.
From page 195...
... Ya, we're going up by 5 so as we go across 1 wegoupby5. Justin Teacher in The lessons on nonlinear functions, The starter offer idea is also applied.
From page 196...
... The idea of a "starter offer" gives students a reasonably familiar situation dhat provides a context for learning y-intercept—ordinarily a relatively abstract and difficult madhematical topic dhat is often confused widh slope in students' understanding of linear function. In our approach, students still learn the notations, symbols, words, and methods necessary for identifying the yintercept of a function (linear or nonlinear)
From page 197...
... TEACH NG AND LEARN NG FUNCT ONS 335 BOX B~ Two Different Student Sol utions to an Open-Ended Problem Think of a st-aight-line function that would allow you to corn wacky that amount in a I O km waL\cathon. Consider that you might be given an initial donation ("sta ter offer',)
From page 198...
... ~ 1 -_-: _ · __ __ ~ j At, ! lo _= FIGURE 8 3 So mple computer screen In His con hgurohon, siuden is con chonge Ike volue of o, n, orb to eheciimmedioreondoubmorcchonges in ibegroph Arid the table For Chomps, if s Indents chonge the volue of K just the y iniercepiof the curve will chonge If students chonge o or n lo o positive volue other than 1, Ike degree of steepness of Ike curve will chonge If students chonge Ike volue of o lo o negonve volue, he curve will come down.
From page 199...
... . This exchange I lus z a es 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)
From page 200...
... In The example shown in Figure 8-4, an elevendh-grade advanced madhematics student who had been learning functions primarily from a textbook unit decided to calculate and plot only two points of The function y = x~ +1 and Then to join Them incorrectly widh a straight line. This student had just finished a unit that included transformations of quadratic functions and Thus presumably knew That y = ~ makes a parabola radher than a straight line.
From page 201...
... On The odher 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 dlree representations share These qualitative features.
From page 202...
... 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 lineanty. The curriculum presented in this chapter attempts to focus limited instructional time on core conceptual understanding by using multiple representations and generalizing from variations on just a few familiar contexts.
From page 203...
... All of these programs build on students' prior knowledge by using problem situations and making connections among multiple representations of function. However, whereas the Jasper Woodbury series emphasizes rich, complex, real-world contexts, the approach described in this chapter keeps the context simple to help students perceive and understand the richness and complexity of the underlying mathematical functions.
From page 204...
... . Multiple representations: A vehic e for understanding.
From page 205...
... Glaser (Ed.) , Advances Sn Instructional Psychology (vol.


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