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A New Recipe: No More Cookbook LessonsAuthor(s): Suzanne R. Harper and Michael Todd EdwardsSource: The Mathematics Teacher, Vol. 105, No. 3 (October 2011), pp. 180-188Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/10.5951/mathteacher.105.3.0180 .

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180 MatheMatics teacher | Vol. 105, No. 3 • October 2011

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Mathematics educators frequently extol the virtues of inquiry-based instruction to classroom teach-ers (Brahier 2008). Indeed, visions of motivated students collaboratively investigating mathemat-ics tasks of their own design provide an appealing

instructional picture for teachers. Although research suggests that inquiry benefits learners by allowing them to make sense of the mathematics they encounter, particularly through discourse with peers (Carpenter et al. 1989; Jaworski 2007), to assume that all (or even most) teachers successfully create and imple-ment inquiry-oriented lessons in their classrooms is wishful thinking.

Teachers may incorrectly identify cookbook lessons—those that lead students through a series of procedural steps in a recipe-like fashion—as mathematical inquiry because students are active as they work through such tasks. Unfortunately, cookbook tasks give students few opportunities to develop their own methods of investigation or to realize the potential of mathematics as a creative area of study. Recognizing the unsatisfactory nature of recipe-oriented teaching materials, we share an approach we use with teachers to transform cook-book lessons into materials that more fully embrace the funda-mental tenets of mathematical inquiry.

PRINCIPLES OF INQUIRY TEACHINGInquiry-based teaching is firmly grounded in the scientific method. As such, the literature of science education provides a wealth of ideas for engaging mathematics students in inquiry-oriented experiences. For instance, in “Rethinking Laborato-ries,” Volkmann and Abell (2003) discuss strategies for trans-forming recipe-based science labs into inquiry-oriented labs. The authors encourage teachers to modify lessons according to four adaptation principles: questions, evidence, explanation, and communication. Following this recommendation, we have developed (and continue to refine) tools to assess the extent to which various teaching materials are inquiry oriented. For instance, the rubric in figure 1 allows teachers to reflect on the meaning and nature of inquiry in their classrooms as they plan daily instruction.

The four lesson criteria listed in the leftmost column of the rubric—task, analysis, revision, and presentation (TARP)—are based on Volkmann and Abell’s four adaptation principles, with explanation and communication collapsed into a single criterion: presentation. Because the notion of revision is argu-ably less familiar in secondary school mathematics classrooms than in science classes, we have amplified its importance with its own descriptor. The remaining columns of the rubric

Mathematics educators frequently extol the virtues of inquiry-based instruction to classroom teach-ers (Brahier 2008). Indeed, visions of motivated

PRINCIPLES OF INQUIRY TEACHINGInquiry-based teaching is firmly grounded in the scientific method. As such, the literature of science education provides a

A New Recipe:

No More

Cookbook Lessonssuzanne r. harper and

Michael todd edwards

Cookbook materials can be readily transformed into

lessons that refl ect a genuine inquiry approach.

Copyright © 2011 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.




The Triangle Centers TaskToo often, materials that incorporate technology in student explorations present content in a cook-book manner, with step-by-step button pushing and mouse clicking that overshadows the development of important mathematics. Exploratory materials that make use of technology often proceed in a pre-dictable “construction, conjecture, construction” pattern (Stein 2005, p. 71), with students following a suggested sequence of constructions. Consider, for instance, the investigation highlighted in figure 2.

Assessing the Triangle Centers TaskThe format of the materials shown in figure 2 is likely familiar. Students are carefully led through a series of steps, using interactive geometry soft-ware, to “discover” properties associated with the circumcenter of a triangle. At first glance, the materials may seem appealing, particularly to new teachers. After all, the questions encourage active student participation. The video screen capture available at http://tinyurl.com/recipe-math high-lights ways in which students make and test con-jectures as they progress through the construction steps. Moreover, the open-ended questions encour-age students to communicate their mathematical understanding. Isn’t this the sort of engagement advocated by various state and national mathemat-ics teaching standards?

However, as we examine the materials more criti-cally, a different picture emerges. Although the activ-ity sheet in figure 2 presents a series of tasks that is arguably more active than most traditional mathe-matics lessons, the Triangle Centers task is markedly cookbook in orientation, as figure 3 suggests.

describe three levels of inquiry, from “full inquiry” (col. 2) to “full recipe” (col. 4). Recognizing that most mathematics lessons are neither full inquiry nor full recipe, the rubric provides a vehicle for teachers as they shift instruction from a predomi-nantly recipe-based orientation to one that encour-ages students to assume more ownership of their learning.

A COOKBOOK EXAMPLEWhat is a cookbook mathematics lesson, anyway? We explore this question through careful analysis of the Triangle Centers task, a geometry investi-gation familiar to interactive geometry software enthusiasts. Using the rubric shown in figure 1 as a guide, we illustrate ways in which teaching mate-rials may be enhanced to better support inquiry-based teaching methods.

Fig. 1 this rubric enables assessment of inquiry-oriented mathematics lessons (tarP).

Fig. 2 this cookbook-style investigation leads students through a step-by-step process.



Vol. 105, No. 3 • October 2011 | MatheMatics teacher 183

Assessing teaching materials with an inquiry rubric such as the one in figure 3 is helpful for teachers in at least two ways. First, rubrics encour-age us to slow down and consider materials more carefully. For instance, what learning opportunities are afforded by the activity sheet? Do the materi-als actually support creative thinking and student autonomy? Second, the rubric we use encourages teachers to consider alternatives. For instance, what content connections could be added? In what context could the activity sheet tasks be presented?

FROM COOKBOOK TO INQUIRYUsing the Rubric to Brainstorm about RevisionsOnce teachers determine that instructional materi-als fail to support inquiry, they use criteria from the TARP rubric to inform their revisions. Next we explore possible modifications of the Triangle Centers task along the four dimensions—task, analysis, revision, and presentation—articulated in the rubric.

Revising the TaskAs teachers assess the Triangle Centers task, it is apparent that the activity sheet is little more than a checklist of procedures to be completed with technol-ogy. As such, it allows little room for multiple interpre-tations or solution strategies. Although several poten-tially interesting questions appear at the end of the activity sheet (e.g., does the circumcenter always lie within the triangle?), these are not student generated. Further, the tasks lack context, which spurs student interest as well as meaningful follow-up questions. Eliminating the checklist of procedures while framing the exploration as a problem with a meaningful con-text will lead to materials that better support inquiry.

Modifying the AnalysisOpportunities for student analysis are squelched by the checklist of procedures in the activity sheet. For instance, rather than uncovering the useful-ness of perpendicular bisectors in the task, students are told to construct them. Moreover, the explicit mention of the term circumcenter at the end of the activity inhibits student exploration. As Wanko (2008) suggests, framing questions with embedded answers may lead to trivialization of the task.

Figure 4 illustrates a result obtained by search-ing for the phrase “circumcenter inside triangle” with a popular Internet search utility. Because it is so easy to find this kind of information on the Web, specific mathematics terminology directly related to the Triangle Centers task should be removed. Further, the checklist of student procedures should be eliminated.

Fig. 3 a scored tarP rubric for the triangle centers task shows a high degree of teacher centeredness.

Fig. 4 the solution to the activity sheet task can be found through a simple internet

search (www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle).




Modifying the RevisionOpportunities for student revision are not explicitly incorporated into the Triangle Centers task activity sheet. For instance, students are not encouraged to explore variations of tasks—what, for instance, would happen if bisectors of angles rather than sides were constructed?—and they are not explic-itly encouraged to “drag” vertices of their initial triangle when crafting answers to the questions at the end of the activity sheet. Revision lies at the heart of mathematical inquiry in interactive envi-ronments (De Villiers 1999). Modifying the activity sheet explicitly to encourage student revision is a must for teachers wishing to provide students with inquiry-oriented experiences in the investigation.

Modifying the PresentationThe final products generated by students in the original Triangle Centers task are twofold: a sketch constructed with interactive geometry software and written responses to two short questions. Although the activity sheet does not specifically mention the audience, the sole beneficiary of stu-dent work is likely the classroom teacher. Because the ways in which students can solve the activ-ity sheet tasks are limited (the checklist of steps affords little variation or creativity), students have little incentive to share their problem-solving strategies with other students. Further, given the absence of a real-world context, it is unlikely that anyone other than the classroom teacher would be interested in such solutions.

As teachers encourage students’ presentation of work in a more inquiry-oriented manner, the reasons for sharing work should be compelling. Providing rich, contextual tasks with multiple solu-tions offers students an environment in which to share work.

REVISING MATERIALS WITH THE TARP RUBRICThe Choosing a House Problem Using our rubric to inform activity sheet modifica-tions, we constructed the revision of the Triangle Centers task (see fig. 5).

Revising the TaskBefore students solve the problem with tech-nology, we provide a paper copy of the written problem and a map (see fig. 5) and ask them to make a conjecture about the location of a house equidistant from the three schools. This prelimi-nary conjecture helps students keep a record of their initial construction ideas as they revise their solutions. Many students construct a point on the interior of the triangle formed by the three schools, as shown in figure 6.

Fig. 5 the triangle centers task can be reconceptualized as the choosing a house

problem.

Fig. 6 Most students will incorrectly conjecture that the house lies within the tri-

angle formed by connecting the three schools with line segments.

The Choosing a House ProblemYour cousin is planning to move to Madeira, Ohio. His parents want a house equidistant to the Madeira elementary school, middle school, and high school (marked as points A, B, and C, respectively, on the map). Locate such a site (or sites) on the map. Then write an e-mail message describing how you located where your cousin’s family should live. Be prepared to present your solution in class by showing your Construction Protocol using GeoGebra.




When students are presented with this task, they typically recognize that a triangle center is involved; however, they do not remember or are not familiar with the circumcenter construction. We have found that students examine the construction tools avail-able in the software menus to prompt their initial ideas of how to solve this problem. Students work-ing in GeoGebra sometimes begin by constructing the medians of the triangle (see fig. 7).

Not until students measure the distance from the point of intersection of the three medians (the trian-gle’s centroid) to the three vertices are they convinced that their location for the family’s new house is not correct. Many students will revise their constructions and try to solve the problem using the capabilities of the software until they find the circumcenter of the triangle (see fig. 8). Students then communicate their geometric solution as well as their construction tech-niques in a written format as an e-mail message. Stu-dents also present their solution to classmates using construction protocol tools in GeoGebra (see fig. 9).

Assessing the RevisionThe revision is more open-ended and provides a con-text for exploration. In the revision, students are not led through a series of technology steps; instead, they use their problem-solving skills to find a solution to the task. As the scored rubric in figure 10 indicates, the revised task is not completely student centered; however, it provides opportunities for students to test construction techniques, make conjectures, and communicate their understanding in multiple ways.

AN EXAMPLE FROM ALGEBRA Cookbook examples are not restricted to interac-tive geometry software explorations. For virtually

any mathematical subject and any technology, numerous examples of recipe-based teaching exist. The example in figure 11 highlights a second-year algebra lesson that encourages recipe-based work as students solve systems of equations. The activity sheet leads the student through the steps to produce

Fig. 7 this student’s solution attempt was motivated by construction

menus.

Fig. 8 One student provided this solution and e-mail communication for

the choosing a house problem.

Fig. 9 this student’s presentation made use of Construction

Protocol tools in GeoGebra.




a reduced-row echelon form (rref) of a coefficient matrix using a TI-Nspire graphing calculator.

Using the TARP rubric, we made the following notes about the activity sheet:

• Thetaskslackcontext.Whenstudentscompleteproblems 1–3 in the activity sheet, the exer-cises are solved the same way by all students—namely, by using the rref calculator function. The tasks are not sufficiently rich to support multiple solution strategies.

• Techniquesforanalyzingtheexercisesarewholly initiated by the materials. Students are instructed to use augmented matrices. This method is illustrated immediately before stu-dents are asked to solve problems 1–3.

• Thematerialsofferfewopportunitiesforcriticalthinking as students solve the exercises. Thus, students have few if any opportunities to form and revise conjectures through experimentation.

With these observations in mind, we can trans-form the materials into a more inquiry-based activity. Figure 12 illustrates a task inspired and adapted from materials presented in Mathematics in Context (Kindt et al. 2006) for use with secondary school students.

Note that the revised task corresponds to prob-lem 3 of the cookbook activity sheet. Unlike the original, the revision presents students with a real-world context for studying systems of equations—namely, determining the total cost of a food order at a sandwich shop. Further, it provides students with opportunities to explore a wider variety of solu-tion strategies. For instance, students may begin by combining orders to calculate the unit cost for top-

Fig. 10 the completed rubric for the triangle centers task indicates that this task is not completely student centered.

Fig. 11 the ti-Nspire serves as the environment for this cookbook-style activity sheet.

With most any graphing calculator, you can solve a system of linear equations using MATRIX functions. An augmented matrix contains the coefficients of each equation with an extra column containing constant terms. 2x + 1y = 60Consider the following example: 1x + 2y = 72

STEP 1: Begin by creating a 2 × 3 matrix.

STEP 2: Enter the augmented matrix into the calculator.

STEP 3: Calculate the reduced-row echelon form (rref) of the matrix. The first row represents x = 16, and the second row represents y = 28. The solution is (16, 28).

Use an augmented matrix for each system. Solve with a graphing calculator.

1. 2x + 1y = 90 2. 15x + 11y = 36 3. 2x + 4y = 5 1x + 2y = 72 4x + 3y = 18 1x + 2y + 3z = 4 3x + 3z = 4.5




pings; such an approach is illustrated in figure 13. Students taking such an approach are actu-

ally performing matrix row operations, although they may not recognize it as such. Continuing in this fashion, students may use paper-and-pencil methods to find the unit cost of meats and veg-gies, or they may use a mix of technology-oriented approaches (see figs. 14 and 15).

With the cost of each item known, calculating the cost of the entire order is a straightforward exercise. A spreadsheet-based approach is shown in figure 15.

The revised task provides students with a realis-tic context for exploring systems of equations while providing them with freedom to explore multiple solution methods in a manner more consistent with inquiry-based teaching methods. The TARP rubric was useful for framing this revision.

Fig. 12 a revised system of equations task is more student

centered.

Fig. 13 the cost of a topping can be found manually.

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students with revised tasks that better convey math-ematics as a meaningful, creative area of study.

ACKNOWLEDGMENTSFor their helpful revisions of the tasks presented in this article, the authors wish to thank Steve Phelps of Madeira City Schools, Madeira, Ohio; Amy Hud-son of Middletown City Schools, Middletown, Ohio; and Chris Rose, a recent Miami University graduate.

REFERENCESBrahier, Daniel J. 2008. Teaching Secondary and Mid-

dle School Mathematics. 3rd ed. New York: Allyn and Bacon.

Carpenter, Thomas P., Elizabeth Fennema, Penelope L. Peterson, Chi-Pang Chiang, and Megan Loef. 1989. “Using Knowledge of Children’s Mathemati-cal Thinking in Classroom Teaching: An Experi-mental Study.” American Educational Research Journal 26: 499–532.

De Villiers, Michael. 1999. Rethinking Proof with The Geometer’s Sketchpad. Emeryville, CA: Key Curricu-lum Press.

Jaworski, Barbara. 2007. “Theory and Practice in Mathematics Teaching Development: Critical Inquiry as a Mode of Learning in Teaching.” Journal of Mathematics Teacher Education 9 (2): 187–211.

Kindt, Martin, Mieke Abels, Margaret R. Meyer, Margaret A. Pligge, and Gail Burrill. 2006. “Com-paring Quantities.” In Mathematics in Context: A Connected Curriculum for Grades 5–8, edited by National Center for Research in Mathematical Sciences Education and Freudenthal Institute. Chicago: Encyclopaedia Britannica Educational Corporation.

Stein, Deborah. 2005. “Engaging Music: Essays in Music Analysis.” New York: Oxford University Press.

Volkmann, Mark, and Sandra K. Abell. 2003. “Rethinking Laboratories: Tools for Converting Cookbook Labs into Inquiry.” The Science Teacher 70 (6): 38–41.

Wanko, Jeffrey. 2008. “Internet: Friend or Foe?” Mathematics Teacher 100 (6): 402–7.

SUZANNE R. HARPER, [email protected], and MICHAEL TODD EDWARDS, [email protected], are associate professors of mathe-matics education at Miami University in Oxford, Ohio. Their professional interests include the use of technol-ogy in the teaching and learning of

school mathematics with particular emphasis on computer algebra systems, interactive geometry software, and pencasting technology.

CONCLUSION In this article, we provide a framework for trans-forming cookbook lessons into inquiry-based ones. Too often, the teaching materials that we use in our classrooms fail to allow our students to develop their own methods of investigation. In the examples given here, we used our framework—the TARP rubric—to consider materials with fresh eyes, providing

Fig. 14 Once the cost of the topping is known, the unit cost of meats and veggies

can be found.

Fig. 15 students can define cost as a formula in a ti-Nspire

spreadsheet (a), fill down the cost formula (b), and then use

the sum function to determine the total cost of all orders (c).

(a)

(b)

(c)



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