# A Teacher’s Guide To Reasoning And Sense Making W

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12Student 3: Wait a minute—you just drew aright triangle, because the streetsare perpendicular.10Your house864SchoolStudent 4: So that means we could use thePythagorean theorem:Friend’s house202 62 c2, so c 1365Teacher:101520But whatif you coulduse a helicopter to fly straight to yourfriend’s house? How could youfind the distance “as the crowflies”? Work with partners to establish a coordinate-axis systemand show the path that you’d haveto drive to get to your friendshouse. Then work on calculatingthe direct distance between thehouses if you could fly.Student 1: [working with students 2, 3, and 4]:What if we use the school as theorigin? Then wouldn’t my housebe at (–2, 5) and my friend’shouse be at (8, 1)?Yeah, that sounds right. Here,let’s draw the path on the streetsconnecting the two houses, andthen draw a line segment connecting the two houses.Student 2:Student 1: Maybe we could measure thelength of a block and find the distance with a ruler.8121086YourhouseStudent 3: Shouldn’t the distance be between eleven and twelve blocks,since 121 136 144? Actually, it’s probably closer to twelveblocks, since 136 is much closer to144 than to 121.The teacher then extended the discussionto consider other examples and finally todevelop the general formula. By having thestudents approach the distance formula from theperspective of reasoning and sense making, sheincreased their understanding of the formulaand why it is true, making it more likely that theywould be able to retrieve, or quickly recreate, theformula later.The focus of every mathematics classshould be on helping students make sense of themathematics for themselves. Bringing this focusto instruction depends on— selecting worthwhile tasks that engageand develop students’ mathematicalunderstanding, skills, and reasoning; creating a classroom environment in whichserious engagement in mathematicalthinking is the norm; effectively orchestrating purposefuldiscourse aimed at encouraging students toreason and make sense of what they are4264Student 2: But how many blocks would thatbe?SchoolFriend’s house–25101520A Teacher’s Guide to Reasoning and Sense Making3

using a range of assessments to monitorand promote reasoning and sense making,both in identifying student progress and inmaking instructional decisions;(for example, choosing a model forsimulating a random experiment);— defining relevant variables and conditionscarefully, including units if appropriate;constantly reflecting on teaching practiceto be sure that the focus of the class in onreasoning and sense making (based onrecommendations in Mathematics TeachingToday [NCTM 2007]).— seeking patterns and relationships (forexample, systematically examining casesor creating displays for data);— looking for hidden structures (for example,drawing auxiliary lines in geometricfigures, finding equivalent forms ofexpressions that reveal different aspectsof the problem);The teacher in the preceding exampleperformed each of these actions with theapparent goal of helping students move beyondsimply knowing how to find the distance by usinga formula, to understanding and making senseof the formula itself.— considering special cases or simpleranalogs;— applying previously learned concepts tothe problem, adapting and extending asnecessary;What should you expect students tobe able to do?Focus in High School: Reasoning and Sense Makingdescribes reasoning habits that should becomeroutine and fully expected in all mathematicsclasses at all levels of high school. Approachingthese reasoning habits as new topics to be taughtis not likely to have the desired effect. Thecrowded high school mathematics curriculumaffords little room for introducing them inthis way. Instead, you should give attention toreasoning habits and integrate them into theexisting curriculum to ensure that your studentsboth understand and can use what you teachthem.Reasoning habits involve— Analyzing a problem, for example—— identifying relevant mathematical concepts,procedures, or representations thatreveal important information about theproblem and contribute to its solution4— making preliminary deductions andconjectures, including predicting what asolution to a problem might look like orputting constraints on solutions; and— deciding whether a statistical approach isappropriate. Implementing a strategy, for example—— making purposeful use of procedures;— organizing the solution, includingcalculations, algebraic manipulations,and data displays;— making logical deductions based oncurrent progress, verifying conjectures,and extending initial findings; and— monitoring progress toward a solution,including reviewing a chosen strategyand other possible strategies generatedby oneself or others.The National Council of Teachers of Mathematics

Seeking and using connections across differentmathematical domains, different contexts,and different representations. Reflecting on a solution to a problem, forexample—— interpreting a solution and how it answersthe problem, including making decisionsunder uncertain conditions;— considering the reasonableness of asolution, including whether any numbersare reported to an unreasonable level ofaccuracy;— revisiting initial assumptions about thenature of the solution, including beingmindful of special cases and extraneoussolutions;— justifying or validating a solution,including proof or inferential reasoning;— recognizing the scope of inference for astatistical solution;— reconciling different approaches to solvingthe problem, including those proposedby others;— refining arguments so that they can beeffectively communicated; and— generalizing a solution to a broader classof problems and looking for connectionsto other problems.Many of these reasoning habits fit in morethan one category, and students should movenaturally and flexibly among them as they solveproblems and think about mathematics. Focusin High School: Reasoning and Sense Making offersexamples of ways to promote these habits in thehigh school classroom.What can you do to help studentsunderstand the importance ofmathematics in their lives and futurecareer plans?Knowing and using mathematics in meaningfulways are important for all students, regardlessof their post–high school plans. Whetherthe students attend college and major inmathematics or go straight into the workforceafter graduation, they will need to haveconfidence in their knowledge of and ability touse mathematics.To help students realize the importance ofmathematics in their lives, you should recognizeand demonstrate the need for mathematicsreasoning habits and content knowledge asessential life skills. You must show how theseskills can ensure your students’ success for manyyears to come—not just in the next mathematicscourse that the students may take.In addition, you should demonstrate anawareness of the wide range of careers thatinvolve mathematics, including finance, realestate, marketing, advertising, forensics, and evensports journalism. Exposing students to the waysin which fields such as these use mathematicswill help them appreciate the importance ofmathematics in their own lives.Beyond showing the relevance ofmathematics in an array of careers, you shouldalso emphasize its practical value in offeringapproaches to real problems. Seek contexts inwhich your students can see that mathematicscan be a useful and important tool for makingdecisions. In doing so, you will help studentsrecognize the benefit of mathematical reasoningand its importance for their adult lives. Suchlessons can contribute to the development of aproductive disposition toward mathematics.A Teacher’s Guide to Reasoning and Sense Making5

What can you do to make yourstudents’ high school mathematicalexperiences more meaningful overall?You can be an important advocate beyondyour own classroom for more meaningful highschool mathematics. Compared with teachersof mathematics in the middle and elementarygrades—or with school administrators atany level—high school mathematics teachersgenerally have stronger, more extensivemathematics backgrounds and have taken higherlevel mathematics courses.Because of these experiences, high schoolmathematics teachers are the most likely to seemathematics as a coherent subject in which thereasons that results are true are as important asthe results themselves. You can play a vital role incommunicating that message to other decisionmakers in your school.For the experience of learning high schoolmathematics to change and become somethingthat is meaningful to your students, you mustbegin today to focus your content and instructionon reasoning and sense making. In addition,you are in a unique position to work withadministrators and policymakers to achieve thegoal of broadly restructuring the high schoolmathematics program to reflect this focus.

Sense making involves developing an understanding of a situation, context, or concept by connecting it with other knowledge. Reasoning and sense making are closely interrelated. Reasoning and sense making should occur in every mathematics classroom every day. In classrooms that encourage these activities, teachers and students ask and answer such

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