Ontario Bansho

Math Congress

&

Gallery Walk

The High Yield Strategies for Teaching and

Learning Mathematics

2009

Teaching Mathematics through Problem Solving

Un’rii quite recenTly, understanding The Thinking and learning Thai The mind makes possibie has remained an elusive quest, in par’r because of a lack of powerful research Tools. In facT, many of us learned mai'hema’rics when "file was known aboui' learning or aboui how The brain works. We now know maThema’rics ins'rruc’rion can be developmem‘ally appropria’re and accessibie for Today's learners. MaThemaTics insTr'ucTion has To sTarT from conTexTs ThaT children can relaTe To — so ThaT They can “see Themselves“ in The conTexT of The quesTion. MosT people learned maTh procedures firsT and Then solved word problems reEaTed To The operaTions ofTer pracTising The skills TaughT To Them by The Teacher. The idea of Teaching Through problem solving Turns This process on iTs head.

By sTarTing wiTh a problem in a conTexT (e.g., siTuaTional, inquiry-based) ThaT children can relaTe To, we acTivaTe Their prior knowledge and lived experiences and TaciliTaTe Their access To solving maThemaTicai problems. This acTivaTion connecTs children To The problem; when They can make sense of The deTails, They can engage in problem solving. Classroom insTrucTion needs To provoke sTudenTs To furTher develop Their informal maThemaTical knowledge by represenTing Their maThemaTical Thinking in differenT ways and by adapTing Their undersTandings afTer lis’rening To o’rhers. As They examine The work of oTher sTudenTs and consider The Teacher's commenTs and quesTions, They begin To: recognize paTTerns; idenTify similariTies and differences beTween and among The soluTions; and appreciaTe more formal meThods of represenTing Their Thinking. Through rich maThemaTical discourse and argument sTudenTs (and The Teacher) come To see The maThemaTicai concepTs expressed from many poinTs of view.

The consolida’rion Tha’r follows from such dynamic discourse makes The molhemo'rical represen’raiions explicii' and lets sTudenTs see many ospec’rs and proper'ries of ma’rh concepts, resull'ing in s’rudenTs' deeper understanding. Emphasize when s’ruden’rs share Their work, all benefit. Teachers must show appreciafion for a varie’ry of diverse solul'ions and sira’regies ra’rher' ’rhan only evaluai'ing accuracy and efficiency. The "accuracy" of The answer To a problem is no? The single goal of learning ma’rhemal'ics. The lna’rhemai'ical Thinking is wha’r needs To endure afTer The sTudenTs leave your classroom. And, remember, if Takes a long Time and loTs of experience To build precision in’ro maThema’rics representation.

The connections sTudenTs make To Their own prior knowledge during The solving of problems is whaT informs The Teacher abouT The sTudenTs' undersTanding. Each of us has differenT prior knowledge - each of us has our own lived experiences and we cannoT anTicipaTe ThaT any Two people share eXacTiy The same prior knowledge. As for as knowing ma’rhematical procedures or olgori’rhms is concerned, a less efficient or less sophis’ricci’red me’rhod or s’rro’regy Thai- 0 sl'udenl‘ owns is more valuable +hon o more efficien’r one Tho'l' belongs Someone else. Unders’randing mus’r reside wi’rh The user. Teaching connoi be obou’r Zeroing in on predetermined conclusions. I’r can'l be obou’r The replicolion and perpefuo’rion of a single possible solu’rion given by one student in a Tex’r, or' by a parent nor' can if be aboul' one solu’rion resides in a Teacher's head. Ra'rher', mo’rhema’rics ins’rruefion mus’r be more about bringing The ideas of all s’rudenTs Togei'her, making +he mathematical Thinking explici'r and facili'l’a‘l'ing the discussion That results in an expansion of knowledge, language, and no'rafion.

Compiled from:

Faciiitator’s Handbook: A Guide to Effective instruction in Mathematics, Kindergarten to Grade 6. Teaching and Learning Through Probiem Soiving. The Literacy and Numeracy Secretariat Professional Learning Series, 200?. Ontario, Canada Fosnot, CT. 81 Dolk, M. (2001) Young Mathematicians at Work: Constructing number sense, addition, and subtraction. Portsmouth, NH:Heinemann

iBansho — A Collective Thinkpad

In order To make public The mo’rhemo’rical Thinking s’ruden’rs use To solve o problem, we need a way of organizing The work so everybody can see 1'he range of sTuden’r Thinking. This allows sTudenTs To see Their' own "thinking in The con’rexT of The similar' Thinking of o’rhers in The class. The ma’rching and gcompor'ing process pr'omo’res learning as sTuden’rs Try To unders’rand o‘rhel" solu‘rions and learn from one ano’rher'. Japanese educo’rors use a process They calf bans/10 To organize s‘ruden’r work and To jeod a conversation Tha’r offers everybody a chance To learn more abou’r The mo’rh used in developing Isolufions To a problem.

A bansho process is employed To sorT and classify parTicipanTs' soluTions. The bans/10 process uses a visual display of all sTudenTs' soluTions, organized from leosT To mosT maThemaTicolly rich. This is a process of assessmen’r for learning and allows sTudenTs and Teachers To view The full range of maThemaTical Thinking Their classmaTes and sTudenTs used To solve The problem. STudenTs have The oppor’runi'ly To see and To hear many approaches solving 'lhe problem and lhey are able To consider sTr'a’regies connec’r wiTh The nex’r s’rep in Their' concep’rual unders’randing of The malhema’rics. is NOT aboul assessmen’r of learning, so There should be no alfernpl To classify solu’rions as level 1, level 2, level 3, or' level 4.

'éegin by having represenfafives from each Table/group pos’r Their char’r paper solu’rions on The wali. Facili’raTe The group in organizing The bansho by using promp'rs, such as:

- Which solufions show s’ruden’rs represenfing The ma’rhemcn‘ics using concr'eTe mafer'iais or

pic’rur'es?

'7 Which solutions show siuden’rs working operafions o’rher' Than division?

- Which soluTions show sTudem‘s using an algori’rhm (effec’rively or oTher'wise) Tha’r would have you Think They have a deep unders’randing of The solu’rion? Can They say how and why *the algoriThm

I works?

- How couid we organize The solu’rions ’ro represenT a continuum of gr'ow’rh of ma’rhema‘rical ideas?

lJse The group's Thinking and responses To organize bans/10 by pos’ring The chor'Ts from lef’r To Eigh’r, grouping solu‘rions Tho’r show The same moThema’rics. La’rer, dur'ing discussion, The mo‘rhema’rics Quill be named and subsei‘s labelled To describe specific procedures (e.g., coun’ring, army model, rl‘epea’red

This presenTaTion sTraTegy is NOT meanT for formaTive (evaluaTive) assessment IT should noT be used To represenT 4 levels of performance of ochievemenT on expecToTions. IT shows The range of maThemaTical Thinking and knowledge in The class or group. The display, creaTed by The class, becomes a very powerful Tool To help idenTify The range of undersTandings among The sTudenTs and offers an opporTuniTy To idenTify sTar'Ting poinTs for insTrucTion for The Teacher.

Compiled from: ifaciiitator’s Handbook: A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6. Teaching and Learning Ihrough Probiem Soiving. The Literacy and Numeracy Secretariat Professional Learning Series, 2907. Ontario, Canada

Fo'snot, C.T. Dolk, M. (2001} Young Mathematicians at Work: Constructing number sense, addition, and subtraction. Rortsmouth, NH:Heinemann

The s’ruden’rs examine and discuss The soluTions, comparing Their own 1'0 Those of o’rhers. The process heips problem solvers: 'I - organize Their Thinking; - discover new ideas: and - see connecTionS between por’rs of 1'he lesson, concep’rs, soiu’rions, noi'o'h'ons, and language.

:For' example, if The goal was To show as many polygons as possible wiTh an area of 4 square uniTs, The Teacher would creaTe an arrangemenT of sTudenT work along The following lines:

- posT, on The lefT of your display wall, squares wiTh areas of 4 square units and bases parallel The

bo’r’rom of ’rhe page

7 pos’r, To The righT of Tha’r, r'ec’rangles with bases parallei To The bo’rTom of The page - solu’rions The?

; show similar mcrrhemcn‘ical Thinking are arranged in ver’ricai rows ThciT Together look like a concr'e‘l'e

bar graph

nex‘r, show irregularly shaped figures composed of squares and rec’rangles

- pos’r, in The fourfh column, paralielogr'ams wi’rh areas of 4 square uni’rs

- nex’r, pos’r Triangles wi’rh areas of 4 square uni‘rs

4 Then, on The of The display, show o’rher polygons wi’rh areas of 4 square uni’rs

Your bansho mighT look like This:

Squares Hectangles Polygans Parallelng rams i'liangles Polyg ans

{cumpused of (cmmpasecl. of rectangles; tn 2 variety-I Iii shapes] make irfegula'r tu maika irregular actaggans, hexagnns, etc. hmcagnns, etc.

Compiled from:

Facilitator’s Handbook: A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6. Teaching and Learning Through Probiern Solving. The Literacy and Numeracy Secretariat Professional Learning Series, 2007. Ontario, Canada Fosnot, C.T. Dolk, M. (2001) Young Mathematicians at Work: Constructing number sense, addition, and subtraction. Portsmouth, NH:Heinemann

Whof's imporrcm’r To do and To know obouf in Bansho?

The problem:

0 Find or creole a problem for which muh‘iple solutions and/or s’rro’regies ore

possible. 0 Resources To go To find sirofegies and problems:

O

Small — Making Mo’rh Meaningful, Big Ideas from Dr. Small, PRIME Background and S’rra’regies

Fosno’r - Coniexis for Learning Moihemoiics and Young Moihemaiicions work: Addition and Subirclc’rion; MuIiiplicoiion 0nd Division; Frac’rions, Decimals, and Percenis

Guides Effec’rive lns’rruc’rion in Mo’rhemufics

Van de — S’rudenT Centred Mo’rhemofics

Tex’r books

0 We should do the math first, trying to find as many ditlerent solution strategies as possible. We might ask peers to solve the problem it they get stuck and can't find any more strategies. (Even so, students often come up with solutions we will not have thought of; we really need to listen carefully as the strategies are presented by the students.)

O

Doing The mo’rh helps us begin To have insigh’r in’ro The deveiopmem‘ul sequence 01‘ The concept wi’rh which we are working

The pre—plonning:

0 There are 2 greof resources To prepare us for The importon’r prep work we do

in uncovering 1he developmental sequence:

0 Use The expec’ro’rions of The previous grade and ihe expec’ro’rions of ihe

curremL grade and iu’rure grades To form some idea of what we won’r To look for in The onno’ro’rion & highligh’rs summary and The developmental sequence we’re looking for. Highligh’r words on The lesson plan

0 Then move To the Guide 10 Effective Ins’rruc’rion To see wha’r is wrih‘en There

and highligh’r The sequence vocobuicry There 1‘00.

0 After uncovering The developmen’rol sequence, devise The ucfivu’ring prior

knowiedge port of the lesson.

0 The planning part of The |esson can be recorded on The 3 Pod Lesson Plan,

o’n‘uched.

Fleming, 2009

In The classroom: Preparing The mo’reriols:

The solu’rions are recorded in marker of landscape-oriented 1 1x17 or 8 V2 x 14 paper or 1A shee’r of grid chan‘ paper (when appropria’re to The quesfion). This landscape orien’rafion makes if less space-consuming when consirucfing The bansho.

Put The bansho on a large piece of craft or drawing paper (cu’r from a roli — 3 m X? m) so Thai H can:

0 go up over Top of The blackboard or whi’reboard To leave a clear,

unclufiered working space;

0 be roiled up for ins’rrucfion of o’rher con’ren’r;

o H can be used in The hall or stuff room To promo’re converso’rion among

s’ruH, students, or poren’rs.

Working on i1:

Allow sTuclenTs To begin The problem for obouT 4 minuTes. lnTerrupT To ask: "WhoT is The imporTonT informcn‘ion we need in order To solve This problem?” Record This informoTion as H comes from The sTudenTs and keep iT on The Tor lelT side of The bonsho ThroughouT The enTire lesson.

The teaming does n_ot happen as the students saive the probiem. Therefore it is not necessary that students finish 01‘ have the correct answer. it is necessary that they have answers that are sufficiently completed so that a solution strategy emerges. Early finishers can begin to find a solution pathway on a separate piece of paper.

Preparing for’rhe banshoz

The teacher takes the solutions and looks at them during (1 break or overnight to decide what solution strategies have emerged and the order in which they should be presented. This involves looking at the mathematics present in the solutions,- in other words, we must tocus on what the students $11 know, not on whoT They don'T know. As Teachers consider The sTudenTs’ work, They should look for ways in which The sTudenTs’ work is connecTed To eoch oTher concepTuolly. (e.g.: in mulTiplicoTion, The sTuclenTs will be showing repeoTed quonTiTies across all soluTions.)

Once The Teacher hos grouped The s’ruden’r work and decided on The s’rro’regies or clus’rers of s’ruden’r work They woni presen’r, They should choose The individual pieces of s’ruden’r work from each siru’regy or cluster To serve as pieces Thai represem‘ sirdiegy well. (These mighiL be called anchors.) There may be as few as l sirci’regy or as mcmy us 7 or more.

Fleming, 2009

Conducting The bonsho:

work, adding details where necessary, to make connecting the ideas obvious. As the students are speaking, we need to be asking questions and commenting to draw out the similarities/connecting ideas. These connecting ideas wiii spring from knowing about the deveiopmentai sequence and how the concepT develops in The curriculum and from our undersTonding of The sequence we have read obouT in The Guide To El'lecTive lnsTrucTion on The Topic. Only The sTudenTs who euThored The anchor papers presenT soluTions To The class. This coordinuTion of The discussion is where The real learning occurs.

Anchor papers are affixed horizonToily near, buT noT 01‘ The boTTom, beginning on The leTT side, immedioTely To The righT 0T The informoTion The sTudenTs needed To solve The problem previously recorded UT The Tar IeTT. The ceTegory labels will go beneo’rh anchor papers. Eventually, the siuden’rs’ work will be added To ’rhe bunsho upward from The anchor, as in 0 concre’re graph or o pic’rogroph.

Label each category in ma’rhema’rical language Thai is accessible la the studen’rs — use ma’rhema’rical language as much as possible. This is usually clone as The bansho uniolds.

AT The end of The bonsho, The Teacher highlighTs The learning wiTh respecT To The big idea / key concepT for The lesson and records This as c: sToTemenT direcTly on The bonsho.

Oncejhe anchors have been presen’red by their au’rhors, have The s’ruclen’rs consider where Their work belongs with respect The caiegories on The bansho.

0 Teachers of primary s’ruden’rs often like to have all s’ruclen’rs pos’r Their

work so ThaT each sTuclenT Teels valued

0 Teachers of iunior and inTermecliaTe sTudenTs may or may noT wanT To Take The Time To posT all soluTions, leaving iusT The anchors on The bansho, depending on The level of analysis 0T The enTire bansho ThaT is desired. Discussion might occur about a strategy that the students understood (but didn’t use) that they may like to try next time. They could also choose an expert on that strategy to help them it they get stuck. (Posting the ba nsho somewhere in the room while the next, related problem is being alone can assist as an anchor-chart while students are solving the problem.)

Hove siudem‘s work on one related problem for addi’rionoi practice

Fleming, 2009

Assessmen’r for learning:

Planning for nexir s’reps:

' Plan 0 reIQ’red problem for ono’rher bonsho lesson based on areas of

I Use the student tracking sheet (attached) to record which strategies students

are using and the ways in which students are thinking about the mathematics. Transfer this information to an individual student tracking sheet of your own design so that you have accurate information about student achievement at

The end of each term for repor’ring purposes.

Graphic of the overall loyou’r of u bonsho:

we need To solve ihe problem:

S’ruden’r work A Highlighi/Summury

: Connection 10 big idea

Student work Student work Student work

Student work Student work Student work Student work

Count by ones Count by 25 Count by 3s or 45 Repeated addition

Fleming, 2009

A Math Congress

The Term “congress” comes from The work of Cofhy FosnoT from ClTy College of New York, who has been working successfully wi’rh public school s’ruden’rs and Teachers for obouT 15 years To develop molhemo'rics Thinking and learning. The congress allows The sharing of SelecTed responses, analysis of The ma’rhemafics used in The solu’rions and conjectures, oppori'uni’ry To defend 1*heir' Thinking and promp’rs all s’r'uden’rs To {earn from one another. The purpose is To debr'ief The STmTegies used by sTuden’rs, unear'Th mul‘riple r'epr-esen’ra’rions of ma’rhemaficai Thinking, and assist in The developmen’r of deep understanding of concepts.

By having sTudenTs defend and explain Their Thinking, Teachers give Their sTuclenTs opporTuniTies To see and hear differen’r perspecTives. This gives sTudenTs a chance To examine The maTh very closely — iT is like “laying The concepTs and The miSconcepTions ouT on The Table". STudenTs have a chance To learn from The discourse and clarify Their Thinking abouT The maTh idea. The connecTion To prior maTh knowledge and The developmenT of new knowledge is made expliciT. Enduring learning resulTs.

A Typical congress involves a small number of samples chosen by The Teacher“ and explained by "the s’rudem“. During group work, The Teacher' consider-s wha’r:

- four To six samples To use for a ma’rh congress? - concep’r or property would The samples represcn'r? - facili’rafes The discussion in The classroom?

The ma’rh congress can be sfruc’rur'ed in many ways. If The focus is on a big idea or To illumina’re inafhema’rical modeling, The sfr'uc’rure will logically bring our fhe connecTions befween The different solu’rions and sfr'a’regies. To refine The sfrafegies, scaffold The soluTions and discussion from less efficienf To more efficienf solu’rions. A congress is NOT abou’r assessmen’r of learning, so fhere should be no a’r’rempf To classify solu’rions as level 1, level 2, level 3, or' level 4.

Teacher considerations:

o Wha’r holds up as a proof, as a convincing argumenT?

0 Wha’r coun’rs as a superb idea or" an efficien’r sfr'afegy?

~ How wiil ideas be symbolized?

- Wha’r is mathematical language?

- does it mean To Talk abouT

- Wha’r Toois coun’r as mm‘hemQHcQI Tools?

' WhaT makes a good mafhema’rical question? 3" - serves as a conjec’rure? Compiled from: Facilitator’s Handbook: A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6. Teaching and Learning Through Probiem Soiving. The Literacy and Numeracy Secretariat Professional Learning Series, 2007. Ontario, Canada Fosnot, C.T. Dolk, M. (2001) Young Mathematicians at Work: Constructing number sense, addition, and subtraction. Portsmouth, NHzHeinemann

Whca’r's imporfcm’r To do and To know obou’r in Congress?

Teaching and learning in a communi’ry of learners / establishing The ciimcn‘e:

0 A congress is underpinned by beiief The? knowledge emerges in 0

community of oc’rivi’ry, discourse, and reflection.

0 Students put forth their mathematical ideas to the community and [ustity and

detend their thinking.

0 Teachers encourage siuden’rs To explore, no’rice pofierns, develop efficien’r

strategies, and generalize ideos.

0 Learning happens through ongoing inves’riga’rions deveiopecl wiihin con’rex’rs

and si’ruafions Thai enable s’rudenis to engage in ma’rhemafics.

Q The moth congress provides (:1 forum in which students communicate their

ideas, solutions, problems, proofs, and coniectures.

The landscape of learning:

0 There are resources 10 prepare us {or The imporfcm’r prep work we do in

uncovering 0 porliwlor mu’rherncficul landscape:

'I Use The expecfq’rions of ’rhe previous grade and ’rhe expec’rofions of The

curren’r grade and fu’rure grades To form some idea of who’r we To look for in onno’rofion & highligh’rs summary and The developmem‘ol sequence we're looking for. HighlighlL words on The lesson pion.

0 Check The Guides To Effecfive Insfrucfion, Minisiry of Educoa‘ion

0 Check resources accessible 10 you iho’r identify research based sequences:

Fosno’r, Small, Van de Wolle.

Developing ’rhe con’rex’r:

0 ConTex’rs for invesfigo’rions are crofied suppor’r The developmen’r of big

ideas, sfro’regies, and models.

0 Good conlex’rs are sifua’rions — ei’rher realis’ric or fictional — siuden’rs can imagine, enable Them To realize and rellec’r on whai They are doing, and will po’reniially have an on maihema’rical development

0 Good con’rex’rs allow sfuclen’rs 10 make sense of s’rra’regies, explore and

genera’re pafierns, generalize, and develop 1The abili’ry To use marhemalics solve problems in 1heir world.