Wa ika t o Te Wh a re Wa n a n ga o Wa ika t o. Email: [email protected]. Jo Lelieveld and John Brooker teach at. Knighton Normal School, Hamilton.
M AT H E M AT I C S
Using multiplication and division contexts with young children to develop part–whole thinking BR ENDA BICK NEL L , JENN Y YOUNG - L OV ER IDGE, JO L EL IE V EL D, and JOHN BROOK ER
K E Y P OIN T S t
'JWFBOEZFBSPMEDIJMESFOBSFDBQBCMFPGTPMWJOHNVMUJQMJDBUJPOBOE EJWJTJPOQSPCMFNT
t
8PSEQSPCMFNTBOESFBMXPSMEDPOUFYUTDBOTVQQPSUDIJMESFOUPEFWFMPQ part–whole thinking.
t
*UJTJNQPSUBOUUPQSPWJEFZPVOHDIJMESFOXJUICPUIQBSUJUJWF TIBSJOH BOERVPUJUJWF NFBTVSFNFOU EJWJTJPOQSPCMFNTUPTVQQPSUUIFJSQBSUo whole thinking.
t
4FRVFODFEMFTTPOTUSVDUVSFTBOEXIPMFDMBTTEJTDVTTJPOTIFMQZPVOH DIJMESFOUPNBLFLFZNBUIFNBUJDBMDPOOFDUJPOT OBNFMZCFUXFFO – – o o
multiplication and addition multiplication and division EJWJTJPOBOESFQFBUFETVCUSBDUJPO JODMVEJOHSFNBJOEFST
HSPVQTPGUFOBOEQMBDFWBMVFVOEFSTUBOEJOH
http://dx.doi.org/10.18296/set.0018
set ,
ôò
M AT H E M AT I C S
This article reports on a New Zealand study with 5- and 6-year-old children that focused on the use of multiplication and division word problems to build part–whole thinking. Using a design research methodology, researchers and two teachers developed word problems using familiar contexts and materials with groups of two, five, and ten. Children were assessed before and after a series of targeted lessons and showed improvement in basic number knowledge, including subitising, counting, recall of number facts, place value, and addition, subtraction, multiplication, and division problem-solving. The researchers and teachers found that using problems JOWPMWJOHNVMUJQMJDBUJPOBOEEJWJTJPO OVNCFSTCFZPOE BOENVMUJQMF SFQSFTFOUBUJPOT IFMQFEFYUFOEDIJMESFOTNBUIFNBUJDBMSFBTPOJOHCFZPOE commonly held expectations.
Introduction A key concept underpinning mathematics VOEFSTUBOEJOHJTUIFJEFBPGUIFVOJU TFFGPSFYBNQMF -BNPO ðøøó 4PQIJBO ñïïö *OJUJBMMZDIJMESFOXPSL with units of one as they learn to count by ones, a fundamental process for the early development of number. This one-to-one relationship is reinforced as children learn to count the number of objects in a set. Related to this idea is the notion of iteration—the repetition of the unit. As children construct number sequences of increasing length and consistency, and begin to use units greater than one, they develop understanding of “increasingly more abstract unit types, such as iterable units, composite units, and JUFSBCMFDPNQPTJUFVOJUTw -BOHSBMM .PPOFZ /JTCFU +POFT ñïï÷ Qððó *UFSBCMFVOJUTBSFUIPTFUIBU are repeated and composite units are units larger than one. Understanding and using iteration of the units is important when multiplying and dividing, measuring, BOEXPSLJOHXJUIGSBDUJPOT .VMMJHBO ñïðð Children need to have two complementary ways PGUIJOLJOHBCPVUBOVNCFSBTTFRVFODF PSEJOBMJUZ
BOEBTDPNQPTFEPGQBSUTXJUIJOBXIPMF DBSEJOBMJUZ :BDLFM ñïïð ɨFZDBOVTFQSPCMFNTPMWJOH strategies based on sequence, such as starting from one of the numbers and adding on, or taking away the other, either in ones or in chunks. This has sometimes CFFODBMMFEUIFi+VNQwTUSBUFHZ ɨPNQTPO ñïðï Another way to solve problems is to use the so-called “Split” strategy, where the numbers are partitioned into groups such as tens and ones, and the problem
set ,
solved by working separately with each kind of group ɨPNQTPO ñïðï "DDPSEJOHUP:BDLFM ñïïð JUJT important that both counting-based and group-based approaches are used so that children develop a deep and connected understanding of number. In using counting-based strategies, children can think about moving forwards or backwards along the number line, whereas group-based approaches support strategy development in partitioning numbers, especially when working with multi-digit numbers that require placevalue understanding. Solving multiplication and division problems JOWPMWFTBEJĊFSFOUUZQFPGUIJOLJOHGSPNBEEJUJPO BOETVCUSBDUJPO 1BSL/VOFT ñïïð "EEJUJPO and subtraction problems are based on one-to-one relationships, whereas multiplication and division originate from an understanding of many-to-one. Current thinking supports the notion that young DIJMESFOTNBUIFNBUJDTMFBSOJOHDPVMECFTUSFOHUIFOFE in the long term by an earlier focus on providing FYQFSJFODFTXJUIHSPVQTPUIFSUIBOPOF 4PQIJBO ñïïö *UNJHIUTFFNFBTJFSUPTUBSUXJUIXIBUJT QFSDFJWFEUPCFTJNQMFS JF BEEJUJPOBOETVCUSBDUJPO
but Sophian warns that this narrow focus may just make it more difficult for students to understand the ideas of multiplication, division, and fractions when they later meet them. We also have to be wary about grounding the concept of multiplication simply on repeated addition as this could become just another MFBSOFEDBMDVMBUJPOQSPDFEVSF 1BSL/VOFT ñïïð 1BSLBOE/VOFT ñïïð TVHHFTUBOBQQSPBDIXIFSF the focus is placed on the schema of one-to-many
M AT H E M AT I C S
correspondence. For example, instead of seeing three NPOLFZTFBDIXJUImWFCBOBOBTBTô ô ô SFQFBUFE BEEJUJPO
BDPOTUBOUDPSSFTQPOEFODFJTFTUBCMJTIFE CFUXFFOFBDIPGUIFNPOLFZT ò BOEUIFOVNCFSPG CBOBOBT ô UIBUJT ôCBOBOBTQFSNPOLFZ TPðïCBOBOBT GPSñNPOLFZT BOEðôCBOBOBTGPSòNPOLFZT
0VSTUVEZ In this article we report on the first phase of a project funded by the Teaching and Learning Research Initiative 5-3* GVOE #JDLOFMM:PVOH-PWFSJEHF ñïðô *UXBT a collaborative project developed in partnership with a culturally diverse primary school. The initial seed for the study was sown by the associate principal who noticed the absence of multiplication in junior classes, possibly the result of its omission in the Numeracy Development 1SPKFDUT"TTFTTNFOU /VN1" 'PSN"UIBUJNQMJDJUMZ gives a message that multiplication and division is not appropriate in the junior school. This omission has since been addressed in the Junior Assessment of Mathematics [JAM] which provides tasks in multiplication and EJWJTJPO .JOJTUSZPG&EVDBUJPO ñïðó Two researchers worked together with two teachers +PBOE+PIO BOEUIFJSDMBTTFTPGôBOEõZFBSPMETUP investigate how young children solve word problems using multiplication and division contexts. In our experience, most junior-class teachers in New Zealand focus on addition and subtraction problems, so children of this age do not usually solve problems of this type. We investigated the use of multiplication and division problems situated in meaningful contexts, to explore how young children solve such problems. We wanted to NFBTVSFBOZDIBOHFTJOUIFDIJMESFOTOVNCFSLOPXMFEHF including number-word sequences, subitising, basic facts, and early place-value understanding. Additionally, we wanted to provide opportunities for students to work together as a whole class of mixed ability as suggested by #PBMFS ñïï÷ BOEGPSUFBDIFSTUPFODPVSBHFBMMMFBSOFSTUP FOHBHFJOSFBTPOJOHEJTDPVSTF )VOUFS ñïðï With these ideas in mind, we worked collaboratively to decide on suitable contexts, materials, and word problems for children to solve over two -week periods UISFFDMBTTFTQFSXFFL ɨFòóDIJMESFOJO+PBOE +PIOTDMBTTFTXIPQBSUJDJQBUFEJOUIFTUVEZXFSFEJWFSTF learners from a range of cultural backgrounds which included Māori, Pasifika, African, and Asian students. Approximately one third of the students were English language learners. It was important to take this diversity into account when planning the word problems. Each child was interviewed individually using purposefullydesigned assessment tasks before and after the two focused teaching periods in order to ascertain gains
in mathematical understanding. The teachers were interviewed at the start and end of the project.
The lesson structure A common lesson structure was followed in each class with the teachers leading the whole-class discussion, and the researchers working alongside the teachers during the problem introduction and follow-up independent time. As part of lesson preparation and our practice we considered the five practices of anticipating, monitoring, selecting, sequencing, and connecting as presented in Smith and 4UFJOT ñïðð GSBNFXPSLGPSPSDIFTUSBUJOHNBUIFNBUJDBMMZ productive discussions. To start the lesson, for to minutes, the teachers usually began with a number knowledge warm-up activity. For example, Jo would roll a large dice and each child would make a pattern on their paper plates with counters showing the number that was “one more than” the quantity displayed on the dice. The objective of the number knowledge warm-up activities was to get children focused on mathematics, and to be XPSLJOHJOUIFNBUIFNBUJDTSFHJTUFS -BHFS ñïïõ 8F wanted children to learn to make connections between the “number just after”, “the number one more than” and “the number plus one”, so their knowledge of sequence became connected to concepts of quantity. *OUIFOFYUQIBTFPGUIFMFTTPO UIFEBZTQSPCMFNXBT JOUSPEVDFEJOBNPEFMMJOHCPPL BMBSHFTDSBQCPPL *G OFXNBUFSJBM FH TPDLT 6OJmYDVCFT XBTJOUSPEVDFE there was a brief time of exploratory play before using it for instructional purposes. The children and their teacher read the word problem together and particular XPSETXFSFIJHIMJHIUFE FH FBDI QBJS BMUPHFUIFS iHSPVQTPGw GVMM BOEUIFUFBDIFSEJTDVTTFEUIFTFXPSET with the children to ensure that they understood the QSPCMFN$IJMESFOTOBNFT QBSUJDVMBSUPFBDIDMBTT XFSF occasionally incorporated in the problems along with two DIBSBDUFST %PHBOE3BCCJU XIPIBECFFOQBSUPGUIF initial diagnostic interview. The children then collectively modelled a solution to the problem. The teacher selected children to model the problem, to talk about what they were doing, and have others revoice, or describe their BDUJPOT"TUIFDIJMESFODPNQMFUFEEJĊFSFOUQBSUTPGUIF QSPCMFNBOEPĊFSFEXBZTPGSFDPSEJOHUIFJSUIJOLJOH “like a mathematician”, the teacher recorded it in the modelling book. This large book provided a summary PGUIFDMBTTFĊPSUT TFF'JHVSFð
BOETPNFUJNFTBDIJMET name was put next to particular pictures or equations to acknowledge individual contributions. Jo explained: *UTBWFSZJNQPSUBOUQBSUUPPVSMFBSOJOH JTPVSNBUIT CPPL*UTBHSFBUSFDPSEPGPVSMFBSOJOHBOEJUTBMTPMPWFMZ for the parents to come in and have a little look around and see what it is and to share it.
set ,
M AT H E M AT I C S
'*(63&456%&/5"/%5&"$)&33&$03%*/('03 "2605*5*7&%*7*4*0/ #:580 130#-&.
&BDIDIJMEIBEBQSPKFDUCPPLXJUIUIFEBZTQSPCMFN pasted in, ready for the follow-up independent work. This problem, with the same wording as the class problem, had EJĊFSFOUOVNCFST XJUIBDIPJDFPGUXPOVNCFSTGPSFBDI QSPCMFN TFF'JHVSFñ *OMBUFSQSPCMFNTXFQSPWJEFEB TQBDF VTJOH GPSDIJMESFOUPJOTFSUUIFJSPXOOVNCFS The children called this option the “mystery number”, and they mostly opted for numbers that provided an appropriate level of challenge. Materials were always available for the children to use to support their thinking. ɨFZXPSLFEJOEFQFOEFOUMZBUHSPVQUBCMFTPOUIFEBZT problem and were encouraged to show their thinking in their project books using pictures and then equations. All children were encouraged to complete at least one, if not two examples.
5IFQSPCMFNT We started with groups of two, and used familiar objects such as socks, sneakers, gumboots, and jandals. Materials GPSUIFTFQSPCMFNTDPOTJTUFEPGQBJSTPGWFSZTNBMM CBCZ socks, collections of their own footwear, and pictures of gumboots and other groups of two. We began with multiplication problems: t ɨFóDIJMESFOFBDIHFUñTPDLTGSPNUIFCBH )PXNBOZTPDLTEPUIFDIJMESFOIBWFBMUPHFUIFS
Later we included a few partitive TIBSJOH EJWJTJPO problems where objects were shared equally to find how many in each group: t 3BCCJUIBTðïCJTDVJUTBOEXBOUTUPTIBSFUIFN evenly with Dog. How many biscuits does each of them HFU
set ,
'*(63&456%&/53&$03%*/('03"2605*5*7& %*7*4*0/ #:5&/ 130#-&.
After introducing partitive problems, all the division problems on subsequent days were of the quotitive NFBTVSFNFOU UZQFXIFSFHSPVQTPGPCKFDUTBSF constructed, and the purpose was to find out how many HSPVQTPGBQBSUJDVMBSTJ[F FH UXP DPVMECFNBEF t 8FIBWFõHVNCPPUT)PXNBOZQBJSTEPXFIBWF
Later we deliberately used odd numbers so that, in making the groups of two, there was one remainder. The idea of division with remainder was to be particularly important later for division into groups of ten. The use of odd numbers also set the scene for discussion and generalisation about odd and even numbers: t ɨFSFBSFøTPDLTJOUIFCBTLFU)PXNBOZQBJST DBOXFNBLF
The next series of problems focused on groups of ve. For this context, we began by using fingers and gloves but continued with images showing groups of five, such as DBOEMFTPOBDBLFBOEQFUBMTPOBnPXFS t *IBWFUXPHMPWFTPO)PXNBOZmOHFSTBSFJOTJEFNZ HMPWFTBMUPHFUIFS t .BSJOBIBTóDBLFTɨFSFBSFôDBOEMFTPOFBDI DBLF)PXNBOZDBOEMFTBSFUIFSFBMUPHFUIFS
We then again deliberately focused on quotitive or measurement division problems, as this meant that the DIJMESFOIBEUPNBLFHSPVQTPGBQBSUJDVMBSTJ[F JF mWF Instead of sharing objects evenly and finding how many JOFBDIHSPVQ QBSUJUJWF
UIFTFQSPCMFNTJOWPMWFEmOEJOH how many groups of a particular size could be made from a given quantity. In the last phase of the study, we presented quotitive division into groups of ten and used the context of eggs
M AT H E M AT I C S
in cartons. We provided the children with egg cartons that held ten eggs and either plastic eggs or Unifix cubes to model the problems. The context was varied to include chocolates in trays of ten. We then progressed to measurement division problems and filling egg cartons using multiples of ten and then other two-digit numbers. t ɨFSFBSFóïFHHT&BDIDBSUPOIPMETðïFHHT )PXNBOZGVMMDBSUPOTBSFUIFSF
After presenting problems where the number was a multiple of ten, numbers were used that were not multiples of ten. In both cases, the number of full cartons corresponded to the “tens” digit, while the objects that XFSFMFGU UIFSFNBJOEFS DPSSFTQPOEFEUPUIFiPOFTw digit. Some children chose to use large numbers that were multiples of ten and wrote the equation followed by “ remainder” or “ r”. t ɨFSFBSFñóFHHT&BDIDBSUPOIPMETðïFHHT How many full cartons are there.
The children, when working as a class, recorded on pieces PGQBQFS UIFOVNCFSPGGVMMDBSUPOT BOEQMBDFEUIJT CFMPXUIFDBSUPOT BOEUIFOVNCFSPGMFGUPWFST UIFPOFT Recording division as repeated subtraction helped the children better understand the way quotitive division is EJĊFSFOUGSPNQBSUJUJWFEJWJTJPOɨFDIJMESFOXFSFBMTP encouraged to check division problems using the inverse operation, multiplication.
Making mathematical connections During the lessons, we explicitly focused on helping students make three key mathematical connections. The first was helping students make connections between addition and multiplication, then multiplication and division, and between division and repeated subtraction. The second connection was the recognition of the EJĊFSFODFCFUXFFOPEEBOEFWFOOVNCFSTUIBUBSPTFGSPN making pairs. As children made pairs of socks from an odd number of socks, some initially described the one sock left over as “a loner” or “leftover” and were then introduced UPUIFXPSEiSFNBJOEFSwBOETZNCPMAS#ZMPPLJOHBU QBUUFSOTPOUIF)VOESFET#PBSEBOEIJHIMJHIUJOHUIF numbers from which they could make pairs without leftovers and those they could not, the students came to SFDPHOJTFUIFEJĊFSFODFCFUXFFOPEEBOEFWFOOVNCFST The final connection was between the groups of ten BOEQMBDFWBMVFVOEFSTUBOEJOH#ZQMBOOJOHBTFSJFTPG quotitive division problems that involved making groups of ten, explicit links were made with the number of HSPVQT UFOT BOESFNBJOEFS POFT
BOEQMBDFWBMVF TFF 'JHVSFñ )PXFWFS UIJTDPOOFDUJPOXBTTUJMMJOTFDVSFGPS most of these young children. For a few students, it led to important growth in two-digit number understanding.
Children’s understanding The findings from this study, derived from pre- and post-intervention task-based interviews, showed that children improved their number knowledge, and their understanding of addition, subtraction, multiplication, BOEEJWJTJPO$IJMESFOTLOPXMFEHFPGOVNCFSTFRVFODF JODMVEJOHDPVOUJOHCZPOFT UXPT mWFT BOEUFOT improved substantially. Working with multiplication and division problems led to increased recall of doubles, halving, and early place-value knowledge. Although money was not used as a context in the lessons, the number of children who could recognise the number PGçðïOPUFTOFFEFEUPQBZGPSBOç÷ïJUFNJODSFBTFECZ óöQFSDFOU3FTQPOTFTUPUIJTBTTFTTNFOUUBTLSFnFDUBO early understanding of place value. There was noteworthy increase in the number of children who could solve BEEJUJPOBOETVCUSBDUJPOQSPCMFNTTVDIBT÷ ôBOEðóø There was improvement in multiplication and division as evidenced by the number of children who successfully TPMWFEóHSPVQTPGô GSPNóöǷUPøðǷ
òHSPVQTPGðï GSPNòñǷUP÷ôǷ
ðïTPDLTJOUPQBJST GSPNñóǷUPöóǷ
BOE÷QJFDFTPGDIPDPMBUFTIBSFEFRVBMMZBNPOHóDIJMESFO GSPNóöǷUP÷÷Ƿ
Challenges for the children and teachers The teachers believed that there were several challenges for the children, beginning with the reading of the word problem. These children were beginning readers and also many of them were English language learners so careful consideration had to be given to the use of language. The teacher and children read the problem aloud together and the teacher questioned the children to ensure they understood what was being asked in the problem. Another challenge for the children was moving between solving multiplication and division problems from one day to another. Jo explained that: “they tended to apply the same pattern and strategy of solving multiplication problems to division problems”. This confusion between the two operations was addressed in the class work by providing considerable time for children to model the action of the word problem. Teachers responded by also TDBĊPMEJOHUIFQSPDFTTUISPVHIRVFTUJPOJOH UIFVTFPG prompts, monitoring students as they explored the task, BOEDBSFGVMMZTFMFDUJOHBOETFRVFODJOHTUVEFOUTTPMVUJPO strategies. To further help children they guided them in supporting their representations with recorded equations. Writing the problems was a more time consuming and challenging task than we had all originally anticipated. We usually began with a context chosen by the teacher, but as Jo explained:
set , ôö
M AT H E M AT I C S
*UIJOLBMPUPGJUTMBOHVBHFCBTFE UIFIVHFSPMFPG MBOHVBHF JTOUJU *UTBCPVUHFUUJOHUIFMBOHVBHFSJHIUBOE making the problems that we agonised over, contextual. #FDBVTF JGUIFZEPOUVOEFSTUBOEUIFDPOUFYU XFSF XBTUJOHPVSUJNF*UTHPUUPCFBCPVUUIFN UIFZSFmWF UIFZSFTJY JUTBMMBCPVUUIFNIPXNBOZDBLFT DBOEMFT TIPFT TMJQQFST JUTBCPVUUIFN4PJUTEFmOJUFMZUIBU QMVT the equipment role, that is so important, but also exposing them to being grown-up mathematicians who would use a multiplication sign.
Considerable time was spent, once a context had been decided on, in using minimal language, but at the same time ensuring that the operation, the problem type, and UIFRVFTUJPOXFSFFYQMJDJU.BLJOHTFOTFPGUIFEJĊFSFOU types of division problems prompted considerable discussion when writing the word problems. The initial response was to write partitive division problems, as the sharing process seemed to occur more naturally in UIFDIJMESFOTSFBMXPSMEɨFUFBDIFSTBMTPTFFNFEUP CFNPSFGBNJMJBSXJUIQBSUJUJWF TIBSJOH EJWJTJPOUIBO XJUIRVPUJUJWF NFBTVSFNFOU EJWJTJPO)PXFWFS PODF the problems involved making groups of twos, problems making pairs as a naturally-occurring group supported quotitive or measurement division problems. This problem structure was then applied to writing subsequent RVPUJUJWFEJWJTJPOQSPCMFNTXJUIEJĊFSFOUDPOUFYUTBOE groups. The collaborative and iterative nature of the project meant that our problem writing skills improved and by the end of the study we had a comprehensive collection of well-sequenced problems that we have subsequently shared with many teachers. The teachers saw working as a whole class, not as a huge challenge, but as a change in practice. They reported enjoying the practice and felt that it was beneficial for their learners to support their mathematical understanding and use of multiple representations. In this situation, all children were exposed to a range of mathematical ideas, including various solution strategies. I like the fact [that] we had one problem and we all solved POFQSPCMFN BOEUIFOXFTFOUUIFNPĊUPEPBQSPCMFNUIBU XBTQJUDIFEBUUIFJSMFWFM*UIJOLUIBUTQBSBNPVOU +P
The biggest change in my practice has been the idea of us working together as a class… I have noticed that with the whole-class activities, that I think the children who are very competent mathematicians have brought along the PUIFSDIJMESFO UIFZWFCFFOIFMQJOHUIFN TP*UIJOLUIBUT WFSZJNQPSUBOU +PIO
Implications We have sufficient evidence that the use of multiplication and division contexts and whole-class discussion can FOIBODFTUVEFOUTQBSUoXIPMFUIJOLJOH*OTUFBEPG
ô÷ set ,
approaching the teaching of multiplication as the NFNPSJTBUJPOPGOVNCFSGBDUT FH 4NJUI4NJUI ñïïõ
UIFTUVEFOUTJOUIJTTUVEZMFBSOFEBCPVU multiplication as being about making equal-sized groups. While this is only one way of viewing multiplication, JUJTUIFFBTJFTUQSPCMFNTUSVDUVSF .VMMJHBO .JUDIFMNPSF ðøøö )PXFWFS UIFDPOUFYUPGUIFXPSE problem determines how children will interpret and solve the problem. Working with the “groups of” idea JONVMUJQMJDBUJPOBOEEJWJTJPOTVQQPSUFEDIJMESFOT performance in addition and subtraction. Teachers could use quotitive division problems, constructing groups of ten with remainder, to help young children understand place value in a meaningful way. This contrasts with a more typical approach using “bundling tens” activities without a problem-solving context.
Acknowledgements This project was made possible by funding from the Teaching and Learning Research Initiative [TLRI] administered through the New Zealand Council for Educational Research and the interest and support of the teachers and children involved in the project.
References #JDLOFMM #:PVOH-PWFSJEHF + ñïðô
6TJOHNVMUJQMJDBUJPO BOEEJWJTJPODPOUFYUTUPFOIBODFZPVOHDIJMESFOTQBSUo whole thinking in mathematics. Teaching and Learning Research Initiative. Available at IUUQXXXUMSJPSH O[UMSJSFTFBSDISFTFBSDIDPNQMFUFTDIPPMTFDUPSVTJOH multiplication-and-division-contexts-enhance #PBMFS + ñïï÷ 1SPNPUJOHASFMBUJPOBMFRVJUZBOEIJHI mathematics achievement through an innovative mixedability approach. British Educational Research Journal, 34 ñ
ðõöoðøóIUUQEYEPJPSHðïðï÷ïïðóððøñïöïðôòñðóô )VOUFS 3 ñïðï $IBOHJOHSPMFTBOEJEFOUJUJFTJOUIF construction of a community of mathematical inquiry. Journal of Mathematics Teacher Education, 13 ô
òøöoóïø IUUQEYEPJPSHðïðïïöTðï÷ôöïðïøðôñY -BHFS $" ñïïõ 5ZQFTPGNBUIFNBUJDTMBOHVBHFSFBEJOH interactions that unnecessarily hinder algebra learning and assessment. Reading Psychology, 27 ñoò
ðõôoñïóIUUQ EYEPJPSHðïðï÷ïïñöïñöðïõïïõóñóöô -BNPO 4 ðøøó 3BUJPBOEQSPQPSUJPO$PHOJUJWFGPVOEBUJPOT in unitizing and norming. In G. Harel and J. Confrey &ET
e development of multiplicative reasoning in the learning of mathematics QQ÷øðñï /FX:PSL /:4UBUF 6OJWFSTJUZPG/FX:PSL1SFTT Langrall, C. W., Mooney, E. S., Nisbet, S., & Jones, G. " ñïï÷ &MFNFOUBSZTUVEFOUTBDDFTTUPQPXFSGVM NBUIFNBUJDBMJEFBT*O-%&OHMJTI &E
Handbook of international research in mathematics education ñOEFE QQ ðïøoðòô /FX:PSL /:5BZMPS'SBODJT
M AT H E M AT I C S
.JOJTUSZPG&EVDBUJPO ñïðó Junior assessment of mathematics (JAM): Teachers’ guide"WBJMBCMFGSPNIUUQO[NBUITDPO[ junior-assessment-mathematics .VMMJHBO + ñïðð 5PXBSETVOEFSTUBOEJOHUIFPSJHJOTPG DIJMESFOTEJċDVMUJFTJONBUIFNBUJDTMFBSOJOHAustralian Journal of Learning Disabilities, 16 ðøoòøIUUQEYEPJPSHð ïðï÷ïðøóïóðô÷ñïððôõòóöõ .VMMJHBO +5 .JUDIFMNPSF .$ ðøøö :PVOHDIJMESFOT intuitive models of multiplication and division. Journal for Research in Mathematics Education, 28 ò
òïøoòòïIUUQ EYEPJPSHðïñòïööóøö÷ò 1BSL +) /VOFT 5 ñïïð ɨFEFWFMPQNFOUPGUIFDPODFQU of multiplication. Cognitive Development, 16 öõòoööòIUUQ EYEPJPSHðïðïðõ4ï÷÷ôñïðó ïð ïïïô÷ñ 4NJUI 4;4NJUI .& ñïïõ "TTFTTJOHFMFNFOUBSZ understanding of multiplication concepts. School Science & Mathematics, 106 ðóïoðóøIUUQEYEPJ PSHðïððððKðøóø÷ôøóñïïõUCð÷ðöðY 4NJUI .4 4UFJO ., ñïðð 5 Practices for orchestrating productive mathematics discussions. Reston, VA: NCTM. 4PQIJBO $ ñïïö e origins of mathematical knowledge in childhood/FX:PSL /:&SMCBVN
ɨPNQTPO * ñïðï (FUUJOHZPVSIFBEBSPVOENFOUBM DBMDVMBUJPO*O*ɨPNQTPO &E
Issues in teaching numeracy in primary schools (ñOEFE QQðõðoðöò Maidenhead, UK: McGraw Hill. :BDLFM & ñïïð 1FSTQFDUJWFTPOBSJUINFUJDGSPNDMBTTSPPN based research in the United States of America. In J. "OHIJMFSJ &E
Principles and practices in arithmetic teaching: Innovative approaches for the primary classroom QQðôoòð #VDLJOHIBN 6,0QFO6OJWFSTJUZ1SFTT
Br e n d a Bic k n e ll a n d Je n n y Yo u n g Lo v e r id g e le ct u re a t Th e Un iv e r s it y of Wa ika t o Te Wh a re Wa n a n ga o Wa ika t o. Em a il: bickn e ll@w a ika t o.a c.n z Jo Le lie v e ld a n d Jo h n Br o o k e r t e a ch a t Kn ig h t on Nor m a l Sch o ol, Ha m ilt on .
set , ôø
