8 - 2 - lecture- establishing routines in mathematics classrooms (39-34)

8/13/2019 8 - 2 - Lecture- Establishing Routines in Mathematics Classrooms (39-34)

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Establishing Routines in MathematicsClassrooms.Hello, I'm Victoria Bill.What's the vision for the future?It's safe to say that discussions areessential.The role of the teacher's changing, we'reno longer going to be dispensers ofknowledge.We're no longer going to be arbiters ofthe mathematical correctness.what is our role then?Our role is to be engineers of learningenvironments.This is founded in many research studiesStigler and Stevensonas well as Paul Cobb the NCTM Standards.The new common core standards adopted by,many states in the United States,actually, 48 states in the United Stateshave adopted the new common corestandards.And they've gone as far as to identify

a set of eight common core mathematicalpractices that they've identified.That kind of outline, what's the cla,mathematicsclassroom have to look like in the future?And this is founded and based on many,manyyears, 30 years of research in the area ofmathematics.So if mathematical discourse, classroomtalk is essential, where does mathematicaltalk live out in these eight mathematicalpractices identified?

Making sense of problems and persevering,students haveto kind of grapple with a problem, figureoutwhat to do first, what to do second andthis is easiest when done in small groupwork.They have to construct a viable argumentand critique the reasoning of others.So, they have to not only be good atputting a logical sequenceof ideas together, but they have toalso listen to others and critique others'

reasoning.This, of course, is easy if they'remodeling with mathematics, if they'reable to write equations or toshow representations and thenrepresentations of what.Number seven talks about look for and makeuse ofstructures of mathematics, this is whatmathematicians really care about.


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What is the meaning of the mathematics,and how do these ideas fit together?How do, what patterns do they see, and howdo they use these patterns?And this is made easier by number eight,looking for and expressing regularity andrepeated reasoning.If something happens over and over again,mathematicians are able to say,this is a structure that lives, that is atruth of mathematics.All of these practices rely ontalk and sense making.But something that's critical in Ball andLambert, Lambert talk aboutthis is that this orchestrating of wholeclass discussions is really difficult.because what's key is that we build thesediscussions off of students ideas, webecome verygood listeners, of what students do anddon'tunderstand and build the ideas off ofstudent ideas.

So, that we kind of build, build and linkto that prior knowledge, but also advancetheir thinking throughout the whole classdiscussion, so if this is, is difficultas it seems to be, this kind of gettingclassroom talk going.Building the talk off of students'thinking and also making itrecognizable that students know that youare building off of their thinking.Then what might help to getthis process going in classrooms?And one of the things we know for sure is,

it's establishment of classroom routines.So, let's just define routines for aminute.They're repetitive, recognizable patternsof interdependent actions accordingto Feldman and Pentland, they involvemultiple actors.So what's the benefit of routines?Help the learners in the classroom,because they provide stability,continuity over time.Actors, it allows the individuals who areinto

these routines, to know exactly what'scoming next.And it allows them to relax, and know thatthere willbe time for the things that they careabout in a lesson.The other reason that routines areimportant, though, is thatthey are good for propagating new forms ofpractice in schools.


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this routine, so if studentsknow regularly that they will have privatetime to think, it's amean of giving all students time to thinkand reason andit signals to students that they areaccountable for their own thinking.It permits the teacher to monitorindividual students abilityto enter the problem to monitor anddetermine which representationsstudents are using if they havemisconceptions, if they have errors.exactly what they understand about themathematical idea before theyhave, before they work with other studentsin small groups.Next, after the private time, is anopportunityfor students to work in small groups, thisis when the teacher really begins tocirculateand begins to assess an advanced studentlearning.

So let's watch the segment from theExplore Phase and thinkabout how do students benefit from theExplore Phase of the lesson?As well as how does the teacher benefitfrom havingstudents explore and work together priorto the group discussion.>>[CROSSTALK]x because that's constant, that's alwaysgoing to stay the same.x[CROSSTALK]

>> Okay.>> So, now you have to find the numberthat's before X to get to that.You already found what you have to add togetthat, what do you have to multiply to getto that?>> We figured this.>> Not a lot better.>> But I can't, I can't do equations[INAUDIBLE]faster.So, yeah, we figured it out on first

period.>> Okay, so, you're, okay so you'regoing to start with thetable, and see if you can find an equationfor the table?>> Yeah.>> Okay.>> But we don't know how, we don't know.>> I don't even know how to start tofind the equation then.


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>> We know what the d is, we don't knowif it should bex plus 3, or x plus an odd, cause it's xplus an odd.>> Right.>> Cause these are all odds onthe bottom, and they're always going to beodds.And it's plus 2.Betweenthem.There's two different.>> Okay.>> A difference of 2 between them.>> Okay.>> We already know this, the top rowsare x, 1, 2, 3, 4.Right, that's our x.>> Right.>> For pattern the growth is.I don't know what the growth is, that'swhat.>>[LAUGH]

That[UNKNOWN].>> If the growth isn't odd, isn't, it'slike odd numbers.>> Okay.So, you have the table, you have thesenumbers written out, right?>> We have the equations.>>[UNKNOWN]well no kidding oh yeah, if I gave you theequation life would be great.What do you have?

>> This.>> Which is a what?>> s.>> Okay.It has what?>> Lots.>> Okay, how many?>> 26>> In number five?>> Yeah.>> Okay, that's 26.There's no other way you can come up withthat number 26 and just count it.

>> Youcan go by like,[CROSSTALK]>> So x plus[UNKNOWN]He's leaving us.>> broke it down real easy, real simple.So obviously it looks real simple so, doyou guys know what he's doing?>> Yeah.


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[CROSSTALK]>> Alright, well hold up.You did this, so tell me what he did.[CROSSTALK]You don't know?>> I was working by myself.>> Oh, okay.[CROSSTALK]>> Whatever the pattern number is.>> Okay.>> Not even looking at, not even lookingat this, just whatever the pattern numberis.>> Right.>> You take it.And you times it by 2, because there's 2,there's obviouslytwo rows and each, the top, top row and abottom row.Both have the nu, this number in the 2.And then.>> Okay.>> Times that by 2, that'll give you thetop

and the bottom and then the middle, is asquare, so.>> Right.>> Minu, you do 2, you do the I'mgoing to call this x, x minus 1 squared,that'll giveyou the middle, and then you just add themtogether.>> I understand them, I just can'texplain them.>> Did you understand that,[UNKNOWN]?>> Yes.

>> And then I got another one, I gotanother one that I don't if you take itand go this way, rectangle, If you take xplus 1, and then do x minus 1.And that'll give you, that'll give youthis dimension right here.>> Okay.Just, go back to the first one,[INAUDIBLE].>> Alright.>> Look at what he's doing.Tell me what he is doing?>> I mean, what do you mean?

Like.>> Well, he came up with.When he was explaining what the top row,bottom row,and the centre, do you know what he wastalking about?>> Yes.>> What?>> I don't know how to say it, but Ihelped him do that, too like, it's not


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just all him.>> Okay.So tell me, show me, I mean, do you haveit inyour head, is it on paper?>> It's in my head.He said that, okay in the middle, there's1and 1 square in the middle, and thenthere's2 on the top subtract 1 and square it toget the number of boxes in the middle.>> Okay.>> So,x minus 1, so 3 minus 1 is 2 and then yousquare 2 to get 4, in the middle and thenyou multiply the whatever sequence your ontimes 2, cause there's a top and a bottom.>> Okay and that's how you came up withthe equation.So, okay, can you take his equations, 2 xplus xminus x squared, and could you paint itinto a picture?

[CROSSTALK]Can you put it to these pictures?Like let's, let's, let's pull out, let'ssay, number 4, okay.If we take this, how does this picturerighthere relate to 2x plus x minus 1 squared?[CROSSTALK][CROSSTALK]>> Yeah.>> You see what I'm saying there?>> That's[UNKNOWN]

>> That's the easy way to break it down.>> Yeah.>> You go from.>> Hold on.>> You just take these two[UNKNOWN]>> Ready?>> Okay, go ahead.>> Here's X.That's X.The box, so important.>> Correct.>> That's our X.

Plus one on each side, okay?>> Okay?And you have the length and the width plusone.>> With the flow?>> Okay.Square that, to be[UNKNOWN],right?>> Right.


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>> Minus 2 x, because these are the 2 xthat we added, because it's 10?>> All right.>> So, because, 2 x, 2, 5.>> Yes.>> 10.That's how many we need to minus, and itworks for all of them same here, like>> So you took tiles that weren't evenon your paneand you put them in and then you took themback out.>> To make your the full[UNKNOWN].>> Very interesting, so now what I'masking you this, which one is right?>> Both!>> Both!>> Both of them.>> How can they both be right, they'retwo totally different equations.>> Because[CROSSTALK]they came out with the same number.

>> You have a lot, you have a lot of.>> Okay, I'm back.Alright.So, when I left, I asked the group to comeup and explain how wecame up with 2 x plus x minus 1 squared,and where that relates to the picture.So are you ready to tell me?>> Yeah, well, actually I think that weneed a different equation[INAUDIBLE][CROSSTALK]>> Yeah, but that doesn't explain the

picture.>> What I asked when I left was, 2x plusx minus 1 squared.Fit the pattern, correct?>> It fit the pattern, yes.>> Okay.What I wanted to know when I left was Howdoes it relate to the tiles?>>>> Where is 2x in these tiles?>> 2.>> Where is the x minus 1 squared inthese tiles?

>> Well x is that number right there>> Okay.>> And 2, you just multiply 2 bythat number.>> Why?Why?>> Because>> The tile?Because it gets bigger, it doubles.>> Well we have a whole bunch of square


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roots, so it's not the answer.[CROSSTALK]>> But yeah, it doubles every time.>> What doubles?>> That, the tiles.Like for one it doubles, and then for twoit doubles, three it doubles.[CROSSTALK]Oh, oh, yeah, that, those two.>> The x is the top part, and thebottom.>> What's the matter?>> I don't, I mean, I understand, but Ican't really explain it.Like, those two.>> If you want, see, my thoughts alwayswere ifyou really truly understood, thenexplaining would be the easy part.>> Well I do understand.[CROSSTALK]>> Like, 2x, because just take out thosetwo,and use that.

And then x minus 1 is, like, 4 minus 1.Yeah.>> Which is squared, and there's nineright there.>> Then you square it, and that's howmuch is in middle.>> Okay, so, take this sheet of paperright now, take number four.Separate, I want you to actuallymanipulate those black tiles on hereshow me the 2 X, show me the X minus 1squared.>> Alright.

>> Okay.>> Then you add one.>> You listening?>> I am.>> Alright you had two in the bottom twoin the top.>> I get it.>> Youhave that one in the middle, that's anextra.>> Yes.>> So you add that, that's plus 1.So 2 and 2 is 4.

>> Right.>> That's 2 squared is 4.You add this extra one in the middle,that's five.>> Okay, what is he telling me?Where x squared is.>> Are you asking me?Or are you just.>> Yeah, I'm asking you.>> I get it!


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It's like you said, you start out withone, andthen you cancel that over, and you'd be atone.>> Okay.>> And then you just keep going, untildo you wantme to keep going?>> No!What I want you to do, is I want you totake these, these blacktiles that are sitting right here, andshow me, I want you to show me.[CROSSTALK]>> Where do you see 2 squared?>>[CROSSTALK]>> And then where's the plus 1 at?>> Right.>> Where's the 2 squared?>> Right here.>> What's that.>> And right here.These are 2.

>> Okay.>> Then, the ones in the middle.>> Okay, so for number two, for task,for pattern.>> Mm, hmm.>> Top row and the bottom row.>> Mm, hmm.>> You're putting those two together.>> And that's.>> Making a square and adding one to it.[CROSSTALK].I did the same thing in pattern numberthree.

>> Yeah.>> I took the top row and the bottomrow.And I put them together.Is that 3 times 3?[CROSSTALK].>> Is there one left over?So what I'm telling you is how do youmanipulate these tiles for your x squaredplus 1?>> I mean, cause, look.>> If it obviously works.[INAUDIBLE].

1,2,3.1,2,3.1,2,3, we can just did it like that.>> Where?>> And there the one left over.>> It's cool.>>[LAUGH].>> You can do that?>>


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[LAUGH].Oh, God.If I can do it, then[INAUDIBLE]I can do it.>>[INAUDIBLE].>> I am not.I am not[INAUDIBLE].>> Okay.You have.>> Four.Four.>> How many tiles do you have?>> Four.About one.>> How many tiles do you have in patternthree?How many tiles do you have in patternthree?>> Ten.>> Okay.I'm giving you ten individual tiles on

this piece of paper.>> huh.>>[CROSSTALK]They are not touching.[CROSSTALK]I want you to take those ten tiles and Iwant you toshow me how you put them together to get xsquared plus 1.That's what I want you to show me.>> Well as you can see students in thisclassroom were really engaged in the

problem solving, the making sense that wesawas one of the mathematical practicestandard earlier.they were working together, solving theproblem.They could share their solutionpaths with each other and their reasoning,as well.They were also held accountable by theteacher to critique thereasoning of others and over time the,they'll do that independently.

So what was the teacher doing, as thestudents were working in small groups.he was monitoring, he was assessingstudents problem solving process.You heard the teacher pressing thestudentsto respond to questions related to eachother'sproblem solving process and theirunderstanding of the mathematics.


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So the teacher was assessing studentunderstanding.not only was the teacher assessing studentunderstanding, but the teachers wasworking hard toadvance the students' understanding of theproblemsolving process, their understanding ofthe mathematical ideas.And the teacherleaves them with an idea, so that theyhave to grapple and work in small groups,andthen he returns, and this may occur three,four times within one Explore Phase of thelesson.The Explore Phase, could last anywherefrom 15 to 20minutes, sometimes longer depending on thesize of the task.So, this routine if done regularly andstudentsknow to expect it, it gives them time tomake

sense of the task, they know that they'regoing tohave to listen and talk with their peers,so thereforethey'll get others' ideas, they're notleft out there, working independently.And therefore when the whole class shareoccurs, they won't have to be embarrassedif they don't understand something becausetheyhave this opportunity to work with small,in small groups.They have a chance to represent the

problem with models, theyalso get this private one on one time withthe teacher,as he walks around and works with each ofthe smallgroups important, though, in this processis the use of multiple representation.You saw on Jeff Zigler pressing thestudents to talk about the squarepattern task, he was asking them to pointspecifically to the picture of the squarepattern task.He was asking them to talk about their

thinking related to the pattern.He was holding them accountable fortalkingabout each others thinking related to thepattern.And then he eventually would be pressingthem to write theformal representation, the algorithms, andto come up with the formal,the equation to represent the nth figure


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in the pattern.So, this movement between theserepresentation actually serves as a meansof scaffolding the, the talk that canhappen in a small group.So, it's not just the routines but it'salso the use of othertools that help us to kind of sustain thetalk in these small groups.And then finally, for the routine, thefinal part of the routine ofstructures and routines for a classroom isthe Share, Discuss and Analyze Phase.this is, it lasts anywhere from 20 minutesit could last up to40 minutes and once again, it's dependentupon the size of the task.But this is a time that where theideas come together, and students have to,have tolisten attentively to each other.It gives students an opportunity to speakpublicly, to share their reasoning.It also gives them an opportunity to hear

various,levels of sophistication of mathematicalreasoning from their peers.Those that are clear, those that areunclear and therefore there isa need for students to have to critic thereasoning of others.Okay so, so let'swatch a segment from the Share, Discuss,and Phase of a 6th grade class room.the students are determining how far apartthey are.from after walking in two different

directions after having a picnic together.One walks to the right and one walks tothe left a certain number of steps.And they have to determine how far total,what's their totaldistance and they have to determine thetotal distance apart.think about how do students benefit fromhaving this share discussed.They've worked privately and now they'reengaged in awhole group, Share, Discuss and AnalyzePhase of this lesson.

And then what is the teacher's role, andhow doesthe teacher benefit from having studentshave this Share, Discuss Phase?So Igot to subtract 17 miles, becauseDimoniquewent 17 miles, and then Jenellie went 14miles, soI subtracted, and we're three miles apart,


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that'smy opinion.>> The thing I was wondering was, howdid you get negative fourteen?>> It's just the subtraction sign.>> Oh, sorry.>> Darius.>> Well if she would subtract it shewould havegotten negative three and that is morethan three miles apart.I mean not negative, I mean three.>> So explain to me what you're sayingagain.Tell me again?>> Like, I used, I used addition becausea neg,because I would get three miles if I wouldsubtract it.So I did addition because I knew they'remore than three miles apart.>> Okay, can you come show us on theboard what you mean?Say that again?

>> I don't understand how they werethreemiles apart if they went in oppositedirections.>> Okay, so Brandon, can you come andshow us on Janelle's number line maybewhat you, what you're thinking, how is it.>> I don't really.>> Come give it a try.>> Can youexplain to me how you gotthey were three milesapart if they went in

opposite directions?>> Well.From the start, They had togo like, farther away from each other.>> Okay.>> So, I guess I had toI think I changemy opinion.>>[NOISE]because, speak up so we can hear you andtalk really loud Janelle.>> Christopher what do you think?

>> I disagree with all of them.[LAUGH]>> Alright!>> Because there are two sides in thenumber line.>> What do you mean?>> There are two sides in the numberline with positive and negative.And since they go opposite direction, Isay they both


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start at zero, and go, one goes negative,one goes positive.>> Okay Shilah, can you repeat whatChristopher just said?>> He said there's.>> Speak up!>> He said that there's a negative and apositive side.And you start at zero.And they have to subtract it.>> Okay.What was that zero?>> The picnic.>> The picnic.Okay.So Jonelle.Let's go back to you for right now.And explain to us what you have got here.>> Okay.seehow right here, there is like three milesapart to that town.I thought they were three miles apart.and then I got the 31 from adding them

also.>> Sowhat is the 31?>> From 17 and 14.>> Seventeen and fourteen?So how did you, why did you add thosetogether?To get 31.>> But what does 31 tell you?>> So, that would be like my stop in mynumber line.>> Your stopping point?So, do both you and Diamonique end

up at 31?>> No.>> No?Can you.Tell me what you're thinking?[NOISE]okay.>> on the left side of her number line,she should have one at the first.>> Go show us.>> She should have one right here, and17 at the end.>> How come,

[UNKNOWN]?>> Because just like Chris said, if youhave negative on those, youwill start with negative 1, negative 2,negative 3, and they weren't there.>> Okay, so if they're both starting at0, like Christopher said, and you'regoing in opposite directions, what wouldbe the first one this direction from 0?Tell me again.


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>> So you went, so you would have oneright here andnegative one right here.>> Okay, so Janelle do you agree withhim, or what do you think?>> I think I'm agreeing with Karon andChristopherbecause they started at zero and they bothwent.Opposite directions.>> Okay and Janelle canyou tell me why you're putting negativesigns in front of all the numbers.>> Because they went like a negativeamount of distance from[INAUDIBLE].>> Okay.So, Dominique.Can you explain to me what Janelle isdoing now?>> Well, she just changed her decisionof just going positive.>> Okay.>> So she's making her, other side of

her number line negative.Very good.Anthony, tell me what you're thinkingright now.>> Well, what I was thinking was, like,like, if it, they're going the oppositedirections,then, like you're supposed to start at 0and do like, to the numberuntil where they're going, so like,you'll,you'll stop at the number where they stop.Which means like then you'll add up, you

just count how many numbers they areapart.>> Okay, so why don't you come on uphere and show us what you're saying.I want you to show me, use Janelle'snumber line.>> Write Dominic's right there andJanelle is right here, then,like you like, count the numbersto, like where they are like,[INAUDIBLE]1, 2, 3, 4, 5, 6.[SOUND].

[INAUDIBLE].31.The'd be like 31[INAUDIBLE]apart.>> Okay, so,[INAUDIBLE].Can you tell us what Anthony just saidagain?>> That's what I was also going to say


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but, like, he said, it would be, for thenumbers that he has, it would take 31slashes to get to the opposite side.>> Okay.Okay, thank you.Let's give all of these people a round ofapplause, a round of applause.Okay, here's a few equations that I thinkwe've seen explanations of allof these up here on the board with Kenly'sand with Janelly's anda little bit with Darius' too.>> Oh.I changed my mind.>> Okay.So what.[LAUGH]>> The, I think it's the last one.>> Okay.Point to it and tell me, tell me why youchanged your mind.>> Well.Because I changed my mind because, if youdo both two negatives, equals a positive.

>> Okay.>> 14, plus 17 plus 14 equals 31.>> Christopher, can you tell me whatyou'e thinking?>> I agree with her.Because if you're starting at the negativeside thenyou'll have to switch because a negativemeans reverse.>> Mm, hm.>> To go the opposite direction.And since, 14 is a negative.the, the subtraction sign is telling it to

flip to turn into a positive.So it's 17 plus 14.>> Okay.Can you come show us on the number linewhat you mean?>> Like if you start at negative 14,the negative sign is telling you toswitch, so the negative 14 turnsinto a positive 14 plus a positive 17and that will equal a positive 31.>> Iwas about to raise my hand.>> Okay.

>> but, to say that it was D, because,like twonegatives, like, two negatives do equal apositive, and it would make it 31, butthen I changed my mind again and put itback to C, because,in real world matters, you wouldn't reallysay that you're going negative 14.>> Okay, so if we, let's label themlike this.


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Do you still mean B and C?>> Yeah.>> Which equation relates to that?>> C.>> C?>> C.No!>> Kenley, what do you think?>> Because, because you said because yousaidshe was at zero, that means so, it's D.>> Why?>> Because you said it came to zero?>> Okay,so why does that mean it's D?>> Because before zero, it's negativenumbers, and after zero, it's positivenumbers.>> Okay, Amani?I think.>>[COUGH].>> I think negative 14, it should not benegative.

It should just have a positive.When Dominique went positive, she went, noshe went positive, she went negative.>> Okay, so which one tells you that?Which equation on the board deals withthat?>> D.>> D?Okay,[UNKNOWN],what do you think?>> I agree with D but that's not theequation that I used.

>> Okay, but why do you agree with D?>> Because Janelle, Janelleshe went 14 miles in the oppositedirection, Dominique just went 17 miles.>> Okay, so we're going to go ahead andwrap this up.We know that it's not A, and we know thatit's not B.The other way that we can do this iswe can use absolute value.How many of you remember absolute value?Good, so if we wanted to add the twotogether So we

wanted to add the two together, and wecould add the 17 milesthat Janelle walked and from negative 14miles, I'm sorry, the 14 milesthat Janelle walked, and the 17 miles thatDaileen walked.So what's the absolute value of negative14?14.>> Shila?


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>> 14.>> 14 and what's the absolute value of17?Destiny.>> 17.>> 17.So, what would 14 plus 17 be?>> 31.>> 31.Right.>> Well, as you could see from thevideo,the students were very engaged in sharingtheir solution path.They not only shared their solution paths,but itwas clear they were critiquing thereasoning of their peersand anxious to go to the board and showanother way of representing the distancebetween the two students.They added on to each others' ideas onestudent even offered a counter example,he, he could take the student who showed a

representation on the boardand suggest an alternative way ofrepresenting the distance, on the numberline.the teacher was monitoring and assessingstudents' problemsolving process, and what they understoodabout the mathematics.She didn't intervene when the studentincorrectly represented the distanceon the number line of the positive andnegative distance.Instead, she left called on students

strategicallyand let a student do the critiquing ofthe, her peers' reasoning the studentswerepressed though and challenged to thinkabout thinking.So, the stu, the teacher was asking foranother way of representing the distance.The teacher was challenging the studentstothink about the two different solutionpaths andwhich was correct, which was incorrect.

Kay, and it was clear that many, manystudents in the classroomwere engaged in the discussion about thedistance between the two students.How do students benefit?Well, they learned to listen attentivelyto each other.They were learning to speak publiclybecause iftheir ideas weren't cleared there peers


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were quick tojump on the ideas and to ask questions orto challenge what they were saying.it provided them an opportunity to learnhow to critique each other as well, okay.They had to take a position, do they agreeor disagree?There were two different solution pathspresented,one was correct and one was incorrect butthey had to critique which was correct andwhy, and therefore, they had to sharetheir reasoning.It allowed students, though, to hearseveraldifferent students share their reasoning,some ofthe reasoning shared was incomplete, notincorrect,necessarily, but maybe not as complete asothers.So then it required the students add, addon to each others' ideas so that theycan build a complete logical argument for

whyone solution was accurate and one was notaccurate.So what are the benefits for the teacher?This is where the teacher really doesthe engineering of the math classroomenvironment.It allows the teacher to asses studentsthinking and reasoning, but it, this iswhen the teacher builds that classroomcommunity,where the teacher says who can add on.Who understands what she's saying, who can

say it inanother way, who can come up to therepresentation andpoint to it, and this is where the teacheris sending the message of, this is howmathematicians work.So she's kind of shaping how students workas mathematicians.This is also the time where the teacheradvances student's mathematical ideas ortheir problem solvingstrategies, so although she's not talkingmuch she's

very strategic about who she's calling ontocome to the front of the room.She was strategic in having two differentsolution paths shared, okay?But just having those shared isn't enoughit's the sequence, the order in which sheasked them to share, share, so shehas to weave together the multiplesolution paths.


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So that they're presented in a whole, sothatstudents begin to understand theconnectivity between, among the ideasthat were shared.So that they begin to build thatconceptual understanding that we'restriving for in mathematics.I've just shared on routine and that washow to establish the structures androutines for talk.How does the teacher go about getting thetalk established though?Asking students to refer to manipulativeand equations.Did you notice how many times the studentscame to the board, and they pointed to therepresentation and how that helped makethe ideas clear.The teacher might also list and postcriteria for qualitywork, this would have students understand,if they've been clear enough,or what else they could add to their,

their representations, totheir diagram or their equation or couldthey add labels andso forth.So, what counts as quality work?Okay.She, the teacher might also identify waysstudentscan get started when problem solving andpost anddiscuss these strategies so that theirstudents knowthese are the ways in which a

mathematician works.Math, for example, mathematicians offercounter examples, andthis is accepted and expected in amathematics classroom.teacher might also consider lettingstudents partnershipwith others and have turn and talks, sothat they actually have little mini dressrehearsalsbefore they have to publicly share theirideas.So I hope this was beneficial today and

that you'llgo back to your classrooms and try some oftheseideas and, or, if you work with teachers,that you'llengage teachers in discussions about theimportance of engaging their studentsin talk.And establishing routines like thisbecause they allow you to


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8 - 2 - lecture- establishing routines in mathematics classrooms (39-34)

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