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ALEXANDER BEDRIN/THINKSTOCKMANIPULTHE T O ATINGOLST: Well, if you go, how many 2s are in 16—so, 2,4, 6, 8, 10, 12, 14, 16 [writing the numbers whileshe counts by twos]. How many is that? [Theteacher points along the list of numbers whileshe counts aloud.] 1, 2, 3, 4, 5, 6, 7, 8, right?P: Mmm-hmm.T: So, 20 goes into 160, which is just [attaching]a zero. [The teacher points at the appropriatespot on the paper for Penny to write.]P: [writing] Eight.T: Mmm-hmm. Twenty times 8. Yes, ’cause 20times 8 is 160, so this would be an 8, right?P: Mmm-hmm.With the answer of 18 now written, the teacherchecks Penny’s understanding of what theyhave just done with another series of closedquestions.T: So, how many boxes do we need? [WhenPenny does not respond, the teacher points tothe answer of 18.] What does this represent? Doyou know?P: Eighteen.T: Mmm-hmm, but do you know like in thisproblem how we would P: Eighty-one? I mean T: Do you know what this [18] represents? Likethis 20 represents the 20 books that can fit ineach box.P: Mmm-hmm.T: And 360 represents the total number ofbooks. So, 18 represents P: The boxes.T: How many boxes?P: Eighteen.T: There you go. Does that make sense?P: Mmm-hmm.T: ’Cause you just have to divide them into thedifferent boxes.In this example, the teacher began theinteraction with moves that supported Penny’sthinking (e.g., probing her initial strategy and110September 2014 teaching children mathematics Vol. 21, No. 2understanding of the problem) and then helpedher reach a correct answer. However, we sharethis illustration because it also highlights thethree moves that should serve as warning signsbecause they often, and in this case did, lead totaking over the child’s thinking: interrupting thechild’s strategy, manipulating the tools, and asking a series of closed questions.1. Interrupting the child’s strategyWhen a teacher interrupts a child’s strategyto suggest a different direction, the teacher’sthinking becomes privileged because the child’sthinking—which was “in process”—is halted.This interruption may involve talking over achild who is already speaking, or jumping inwhen a child is working silently. In both cases,this warning sign generally accompanies thehazard of breaking the child’s train of thought—the child may struggle to regain momentumin solving the problem or may lose the threadof his or her idea altogether. Additionally, theteacher may introduce a strategy that does notmake sense to the child. In the example above,Penny had a viable strategy and was in theprocess of executing it when her strategy wasinterrupted with a different approach proposedby her teacher. Perhaps the teacher thoughtthat Penny’s strategy of counting up by twentieswould take too long or that she would struggletoo much to find each multiple. Or perhaps theteacher had expected (or hoped) that Pennywould use the standard division algorithm. Inany case, Penny had no opportunity to return toher original strategy and complete it. Furthermore, Penny was making sense of the problemsituation with her original strategy, but thissense making disappeared when the teacherintroduced the algorithmic strategy.In our larger study, we observed that somechildren, like Penny, had viable strategies forsolving their problems, whereas other children’sstrategies and intent were unclear. However,in all cases, their thinking was “in process” inthat they were writing, counting aloud, movingfingers while working silently, and so on. Theteachers’ interruptions sometimes introducedcompletely new strategies (as in Penny’s case)and other times pushed children to engagewith their partial strategies in specific ways thatchanged children’s problem-solving approachesand were inconsistent with their reasoning. Inwww.nctm.org

each case, teachers risked impeding or abortingchildren’s thinking by inserting and privilegingtheir own ideas while halting the children’s inprocess thinking.2. Manipulating the toolsTABL E 1Another warning sign teachers should notice iswhen they visibly take control of the interactionby manipulating the pen, cubes, or other tools.In the example above, Penny had a writtenrecording of her strategy in progress at the top ofthe page when the teacher’s writing of the standard division algorithm shifted Penny’s focus tothe teacher’s strategy. The teacher then retainedcontrol of the pen for much of the interactionwhile she wrote and talked her way throughthis algorithm. In doing so, she changed therepresentation of the problem from Penny’swritten recording of the multiples of twentyand the accompanying tallying of boxes to anapproach that was abstract for Penny and nota good match for her thinking—as evidencedin Penny’s struggles to make sense of both thecalculation and the result.In our larger study, we observed teachers writing things or moving manipulatives,although sometimes they did so without changing the course of conversations so completely.However, taking over tools was inherently riskybecause doing so sent children a message aboutwho owned the thinking. Teachers also riskedaltering problem representations to representations unclear to children—teachers and childrenmay be thinking differently, even when lookingat the same manipulatives or written representations (Ball 1992).3. Asking a series of closed questionsThis third warning sign highlights a situationthat may begin nonhazardously—when theteacher asks a question with a simple and oftenobvious answer. The danger arises when thisquestion is followed by another and another andanother such question. The net effect of a seriesof closed questions is that the problem getsbroken down for the child into tiny steps thatrequire minimal effort and little understandingof the problem situation. Such was the case forPenny after the standard division algorithm wasintroduced because the teacher asked questionsthat required little more than Penny’s agreement(“Mmm-hmm”). Penny did not have to thinkabout the underlying ideas of division, and theproblem-solving endeavor was instead reducedto following directions.In our larger study, we observed teachersgiving directions that were sometimes phrasedas questions and other times as steps to follow. In either case, when the answer was finallyreached, the children had often forgotten theBecome aware of teaching moves and of potentially taking over students’ thinking.Warning signs for taking over children’s thinkingWarning signsQuestions to consider before proceedingPotential alternative moves1. Interruptingthe child’sstrategyDo I understand how the child is thinking and willmy ideas interfere with that thinking? Slow down: Allow the child to finishbefore intervening.Will the child be able to make sense of my ideas?2. Manipulatingthe toolsWill the child still be in control of the problem solving? Encourage the child to talk about hisor her strategy so far.3. Askinga seriesof closedquestionsWill my questions be about the child’s thinking ormy thinking?www.nctm.orgWill my problem representation make sense tothe child?Will the child still have an opportunity to engagewith substantive mathematics, or will my questionsprevent him or her from doing so? Ask questions to ensure that thechild understands the problemsituation and how the strategyrelates to that situation. Ask whether trying another tool orstrategy would help.Vol. 21, No. 2 teaching children mathematics September 2014111

CLOSEDSTUDIOARZ/THINKSTOCKoriginal goal and wererarely able to relate thesolution to the problem situation. Wesaw this confusionwith Penny whenshe guessed,“Eighty-one?”i n re s p o n s eto a questionabout howmany boxeswere needed. This apparent stab in the dark wasa signal that the teacher’s sequence of closedquestions did not help Penny make sense of theteacher’s algorithmic strategy or relate it to theoriginal problem.Heeding the warning signsThe warning signs exemplified in Penny’sinteraction arose often in our study, sometimesin isolation and sometimes as a set. So, whatcan teachers do? When possible, we encourageteachers to heed the warning signs by choosing alternative moves that are more likely topreserve children’s thinking. The questions intable 1 are designed to help teachers consideralternative moves. We do not suggest that thesealternative moves are foolproof—unfortunately,no moves are. Engaging with children’s thinkingis a constant negotiation, fraught with trial anderror, as teachers work to find ways to elicit andrespect children’s thinking while nudging thatthinking toward reasoning that is more sophisticated. However, in analyzing our data, we werestruck with how often the three warning signswere unproductive in achieving this goal, thusprompting us to consider alternative moves.For example, how might the interactionhave been different if Penny had not beeninterrupted and had been able to complete herinitial strategy? The teacher could have probedPenny’s completed strategy, validating and eliciting her ways of thinking about the problem.If the teacher still wondered about efficiency,she might have asked if Penny could think ofanother way of solving the problem, perhapsin a way that was more efficient. This approachwould have built on Penny’s ways of thinkingabout the problem while still preserving thegoal of efficiency. Alternatively, if the teacher didchoose to suggest the division algorithm, she112September 2014 teaching children mathematics Vol. 21, No. 2could have left Penny in control of the pen andposed some open-ended questions to explorePenny’s understanding of the algorithm and itsconnection to the problem situation. Anotheroption would have been to ask Penny to consider efficiency while she was still solving theproblem with her original strategy. After Pennyhad completed 160 books (8 boxes) by counting by 20s, the teacher could have asked her toreflect on what she had done so far and if thatwork could help her proceed more quickly. (Thisquestion might prompt Penny to recognize thatdoubling 160 books [and 8 boxes] would be closeto the needed 360 books, but she would alsohave the option of continuing with her originalstrategy.) Although there is no perfect move inany situation, these types of alternative movesmight have increased the likelihood that theteacher would have supported and extendedPenny’s thinking without taking over that thinking. (See Jacobs and Ambrose [2008–2009] formore on alternative moves.)Are these moves ever productive?Our data convinced us that the warning signswere generally unproductive moves, but wewondered if these same moves could ever beproductive. After all, teaching moves need tobe considered in context because the samemove can be productive in one situation butunproductive in another. We found that thethree warning signs were occasionally usedproductively but, to us, they almost seemed likedifferent moves because, although they lookedsimilar on the surface, they were coupled withthe preservation of children’s thinking.For example, teachers sometimes productively interrupted a child going far off track orengaging in an extremely inefficient strategyby discussing with the child how he or she wasthinking. This move was not, as we saw withPenny, used to immediately suggest a different direction but instead deepened the child’s(and teacher’s) understanding of how the childwas thinking about the problem. Similarly,teachers sometimes productively manipulatedthe tools to help organize the workspace byremoving “extra” cubes after ensuring that theywere considered “extra” by the child (versus,for example, removing cubes to ensure that thecorrect quantities were represented). This moveprovided some organizational scaffolding whilewww.nctm.org

preserving the child’s way of thinking about theproblem. Finally, teachers sometimes productively asked a series of closed questions to checkon their understanding of a child’s strategy. Thismove kept the focus on the student’s thinkingby putting the child in position to confirm ordeny what he or she had already done, said,or thought. Thus, we are not suggesting thatthe three warning signs can never be used productively. However, our data overwhelminglyshowed that these moves typically led to takingover children’s thinking and thus should be usedwith caution.Good intentionsAll four authors have had the experience of solving a problem for a child without gaining anyidea what the child does or does not understand.We always begin these interactions with goodintentions, but other pressures (e.g., shortnessof time) or goals (e.g., desire to see the child usea more sophisticated strategy) often derail ourefforts. Our data also showed that taking over achild’s thinking was not linked to any particulartone or interaction style. In other words, in anygiven situation, any of us can be tempted to takeover a child’s thinking.In summary, avoiding the impulse to take overa child’s thinking in one-on-one conversations(either inside or outside the classroom) is challenging. We also recognize that the task becomeseven more challenging in social situations likesmall-group work or whole-class discussions.Nonetheless, in all these instructional situations,the same goals exist: eliciting, supporting, andextending children’s thinking. Further, the movesidentified as warning signs are likely to thwartefforts to achieve these goals because childrenget transported to the answer without actuallyengaging in problem solving. In identifying thewarning signs, our hope is that teachers will bemore likely to pause and consider alternativemoves to avoid the dangers of taking over children’s thinking. As a first step, we invite readersto go online (see the More4U box to the right) topractice recognizing these warning signs in aninteraction with a first grader.R EFER E N CE SBall, Deborah Loewenberg. 1992. “Magical Hopes:Manipulatives and the Reform of Math Education.”American Educator 16 (2): 16–18, 46–47.www.nctm.orgCarpenter, Thomas P., Elizabeth Fennema,Megan Loef Franke, Linda Levi, and SusanB. Empson. 1999. Children’s Mathematics:Cognitively Guided Instruction. Portsmouth,NH: Heinemann.Common Core State Standards Initiative (CCSSI).2010. Common Core State Standards forMathematics. Washington, DC: NationalGovernors Association Center for BestPractices and the Council of Chief State SchoolOfficers. th Standards.pdfJacobs, Victoria R., and Rebecca C. Ambrose.2008–2009. “Making the Most of StoryProblems.” Teaching Children Mathematics 15(December/January): 260–66.This research was supported in part by agrant from the National Science Foundation(ESI0455785). The opinions expressed in thisarticle do not necessarily reflect the position,policy, or endorsement of the supporting agency.Victoria R. Jacobs,vrjacobs@uncg.edu,is a mathematicseducator at theUniversity of NorthCarolina at Greensboro. Heather A.Martin, hmartin@ncbb.net, andRebecca C. Ambrose,rcambrose@ucdavis.edu, are mathematics educatorsat the University of California–Davis. Randolph A.Philipp, rphilipp@mail.sdsu.edu, is a mathematicseducator at San Diego State University in California.They all collaborate with teachers to explore children’smathematical thinking and how that thinking caninform instruction.Download one of the freeapps for your smartphoneto scan this code, or go towww.nctm.org/tcm067 toaccess an appendix.Vol. 21, No. 2 teaching children mathematics September 2014113

All Means ALLSave the Date!Equity:Washington DC April 22–25, 2009NCTM 2009 AnnualMeeting & ExpositionVisit www.nctm.org/washingtondc for the most up-to-date information.The NCTM 2009 AnnualMeeting and Exposition inWashington D.C. will bethe mathematics teachingevent of the year. This is oneprofessional developmentopportunity you can’t afford tomiss. Conference attendees will: Learn from more than 800presentations in all areas ofmathematics Network with other educatorsfrom around the world Explore the NCTM ExhibitHall and experience the latestproducts and services Develop your mathematicsresource library with booksand products from theNCTM Bookstore Enjoy Washington D.C.and all the nation’scapitol has to offer

Making the Most ofHonoring students’ solutionapproaches helps teachers capitalizeon the power of story problems.No more elusive train scenarios!By Victoria R. Jacobs and Rebecca C. AmbroseVictoria R. Jacobs, vjacobs@mail.sdsu.edu, is a mathematics educator at San Diego StateUniversity in California. Rebecca C. Ambrose, rcambrose@ucdavis.edu, is a mathematics educator at the University of California–Davis. They collaborate with teachers to explore children’smathematical thinking and how that thinking can inform instruction.260Story problems are an important component ofthe mathematics curriculum, yet many adultsshudder to remember their own experienceswith them, often recalling the elusive train problems from high school algebra. In contrast, researchshows that story problems can be powerful tools forengaging young children in mathematics, and manystudents enjoy making sense of these situations(NCTM 2000; NRC 2001). Honoring children’sstory problem approaches is of critical importance sothat they construct strategies that make sense to themrather than parrot strategies they do not understand.To explore how teachers can capitalize onthe power of story problems, we chose to studyTeaching Children Mathematics / December 2008/January 2009Copyright 2008 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.Photograph by Rebecca C. Ambrose; all rights reservedStory Problems

teacher-student conversations in problem-solvinginterviews in which a K–3 teacher worked one-onone with a child. The skills needed for productiveinterviewing are the same as those needed in theclassroom: Teachers must observe, listen, question,design follow-up tasks, and so on. We focused ourinvestigation on interviews because interviews isolate these important teacher-student conversationsfrom other aspects of classroom life.Supporting and ExtendingMathematical ThinkingAfter analyzing videotaped problem-solving interviews conducted by 65 teachers interviewing 231children solving 1,018 story problems, we identifiedeight categories of teacher moves (i.e., intentionalactions) that, when timed properly, were productive in advancing mathematical conversations. Weseparately considered (a) the supporting moves thata teacher used before a student arrived at a correctanswer and (b) the extending moves that a teacherused after the child gave a correct answer. We wantto be clear that the eight categories of teacher moveswe present are not intended to be a checklist thata teacher executes on every problem. Instead, weconsider these moves to be a toolbox from which ateacher can draw, after considering the specific

The teacher wants to pack 360 books in boxes. If 20 books can fit in each box, how many boxes does she need to pack all the books? Penny pauses after initially hearing the prob-lem, and the teacher supports her by discussing the problem situation, highlighting what she is trying to find: Teacher [T]: So, she has 360