A Teacher’s Guide To Reasoning And Sense Making W

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A Teacher’s Guide toReasoning and Sense MakingWof experiences should highschool mathematics offer students?The key is to provide importantmathematical opportunities centered onreasoning and sense making. How can you andother high school mathematics teachers like yougive students these kinds of experiences?Focus in High School Mathematics: Reasoningand Sense Making, a new publication from theNational Council of Teachers of Mathematics,offers guidelines to improve high schoolmathematics by refocusing it in this way.Teachers will play a crucial role in realizing thevision of this innovative publication from thenation’s leading advocate for more and bettermathematics.hat kindsWhat do reasoning and sense makingmean?Reasoning and sense making refer to students’abilities to think about and use mathematicsin meaningful ways. In any subject, simplyexposing students to topics is not enough. Noris it enough for students to know only how toperform procedures. For example, in your ownexperiences with mathematics in high school,you may have solved page after page of equationsor factored page after page of polynomials.Students today must understand more aboutalgebra than how to apply procedures. Theyneed to develop critical thinking skills to succeedin the subject—and in other areas of life andlearning.For instance, in high school literaturecourses, students often must analyze, interpret,or think critically about books that they arereading. Reasoning is important in all fields—particularly mathematics. Mathematicalreasoning involves drawing logical conclusionson the basis of assumptions and definitions.Sense making involves developing anunderstanding of a situation, context, orconcept by connecting it with other knowledge.Reasoning and sense making are closelyinterrelated.Reasoning and sense making should occurin every mathematics classroom every day. Inclassrooms that encourage these activities,teachers and students ask and answer suchquestions as “What’s going on here?” and “Whydo you think that?” Addressing reasoning andsense making does not need to be an extraburden if you are working with students whoare having a difficult time just in learningprocedures. On the contrary, the structurethat a focus on reasoning brings can providevital support for understanding and continuedlearning.Often students struggle because they findmathematics meaningless. Instruction thatfails to help them find connections throughreasoning and sense making may lead to aseemingly endless cycle of reteaching. However,with purposeful attention and planning,teachers can hold all students in every highschool mathematics classroom accountable forpersonally engaging in reasoning and sense

making, thus leading students to reason forthemselves instead of merely observing andapplying the reasoning of others.Student 1: Oh, yeah, I remember—there’s agreat big square root sign, but Idon’t remember what goes underit.What can you do in your classroomto ensure that reasoning and sensemaking are paramount?Student 3: I know! It’s x1 plus x2, all over 2,isn’t it?Student 4: No, that’s the midpoint formula.You can make reasoning and sense making afocus in any mathematics class. A crucial step is todetermine how reasoning and sense making serveas integral components of the material that youteach.Even with topics traditionally presentedthrough procedural approaches, you can teachthe concepts in ways that allow students to reasonabout what they are doing. Although proceduralfluency is important in high school mathematics,it should not be sought in the absence—or at theexpense—of reasoning and sense making.What exactly do reasoning and sensemaking “look like” in the mathematics classroom?The following example illustrates the needto infuse reasoning and sense making into aclassroom experience. The scenario illustrateswhat frequently happens when students areasked to recall a procedure taught withoutunderstanding—in this case, the distanceformula.Teacher:Today’s lesson requires that wecalculate the distance betweenthe center of a circle and a pointon the circle to determine the circle’s radius. Who remembers howto find the distance between twopoints?Student 1: Isn’t there a formula for that?Student 2: I think it’s x1 plus x2 squared, orsomething like that.2The discussion continued in the sameway until the teacher reminded the class ofthe formula. The next year, the same teacherdecided to try a different approach—one withthe potential to engage the students in reasoningabout the distance formula as they solved aproblem. The following scenario shows studentsreasoning about mathematics, connecting whatthey are learning with the knowledge that theyalready have and making sense of the distanceformula:Teacher:Let’s take a look at a situation inwhich we need to find the distancebetween two locations on a map.Suppose that this map [shown atthe top of the next page] shows yourschool; your house, which is located two blocks west and five blocksnorth of school; and your bestfriend’s house, which is locatedeight blocks east and one blocksouth of school. Also suppose thatthe city has a system of evenlyspaced perpendicular and parallel streets. How many blocks wouldyou have to drive to get from yourhouse to your friend’s house?Student 1: Well, we would have to drive tenblocks to the east and six blocksto the south, so I guess it would besixteen blocks, right?The National Council of Teachers of Mathematics

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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