Investigating young students’ multiplicative thinking : The 12 little ducks problem

Journal article

Cheeseman, Jill, Downton, Ann, Roche, Anne and Ferguson, Sarah. (2020). Investigating young students’ multiplicative thinking : The 12 little ducks problem. The Journal of Mathematical Behavior. 60, p. 100817. https://doi.org/10.1016/j.jmathb.2020.100817
AuthorsCheeseman, Jill, Downton, Ann, Roche, Anne and Ferguson, Sarah
Abstract

Children’s multiplicative thinking as the visualization of equal group structures and the enumeration the composite units was the subject of this study. The results were obtained from a small sample of Australian children (n = 18) in their first year of school (mean age 5 years 6 months) who participated in a lesson taught by their classroom teacher. The 12 Little Ducks problem stimulated children to visualize and to draw different ways of making equal groups. Fifteen children (83 %) could identify and create equal groups; eight of these children (44 %) could also quantify the number of groups they formed. These findings show that some young children understand early multiplicative ideas and can visualize equal group situations and communicate about these through their drawings and talk. The study emphasises the value of encouraging mathematical visualization from an early age; using open thought-provoking problems to reveal children’s thinking; and promoting drawing as a form of mathematical communication.

Keywordsvisualization; multiplicative reasoning; children’s drawings; early mathematics; problem solving
Year2020
JournalThe Journal of Mathematical Behavior
Journal citation60, p. 100817
PublisherElsevier Inc.
ISSN0732-3123
Digital Object Identifier (DOI)https://doi.org/10.1016/j.jmathb.2020.100817
Scopus EID2-s2.0-85091259637
Research or scholarlyResearch
Page range1-16
Publisher's version
License
All rights reserved
File Access Level
Controlled
Output statusPublished
Publication dates
Online23 Sep 2020
Publication process dates
Accepted01 Sep 2020
Deposited07 Jul 2021
Permalink -

https://acuresearchbank.acu.edu.au/item/8w53v/investigating-young-students-multiplicative-thinking-the-12-little-ducks-problem

Log in to edit

Restricted files

Publisher's version

  • 182
    total views
  • 0
    total downloads
  • 4
    views this month
  • 0
    downloads this month
These values are for the period from 19th October 2020, when this repository was created.

Export as

Related outputs

Investigating teacher noticing and learning in Australia, China, and Germany : A tale of three teachers
Damrau, Milena, Barton, Daniel, Huget, Judith, Chan, Man Ching Esther, Roche, Anne, Wang, Chongyang, Clarke, Doug M., Cao, Yiming, Liu, Bing, Zhang, Shu and Peter-Koop, Andrea. (2022). Investigating teacher noticing and learning in Australia, China, and Germany : A tale of three teachers. ZDM Mathematics Education. 54(2), pp. 257-271. https://doi.org/10.1007/s11858-022-01361-7
Aspirations for mathematics learning : The voice of primary mathematics middle leaders
Roche, Anne, Russo, James, Kalogeropoulos, Penelope and Vale, Colleen. (2020). Aspirations for mathematics learning : The voice of primary mathematics middle leaders. Mathematics Education Research Journal. pp. 1-25. https://doi.org/10.1007/s13394-020-00360-9
Using contextualized tasks to engage students in meaningful and worthwhile mathematics learning
Clarke, Doug and Roche, Anne. (2018). Using contextualized tasks to engage students in meaningful and worthwhile mathematics learning. The Journal of Mathematical Behavior. 51, pp. 95 - 108. https://doi.org/10.1016/j.jmathb.2017.11.006
Learning from lessons: Studying the structure and construction of mathematics teacher knowledge in Australia, China and Germany
Chan, Man Ching Esther, Clarke, David J., Clarke, Doug, Roche, Anne, Cao, Yiming and Peter-Koop, Andrea. (2018). Learning from lessons: Studying the structure and construction of mathematics teacher knowledge in Australia, China and Germany. Mathematics Education Research Journal. 30(1), pp. 89 - 102. https://doi.org/10.1007/s13394-017-0214-6
Learning from lessons: Teachers' insights and intended actions arising from their learning about student thinking
Roche, Anne, Clarke, Douglas McLean, Clarke, David and Chan, Ching Esther. (2016). Learning from lessons: Teachers' insights and intended actions arising from their learning about student thinking. In B. White, M. Chinnappan and S. Trenholm (Ed.). 39th Annual Conference of the Mathematics Education Research Group of Australasia. Australia: Mathematics Education Research Group of Australasia. pp. 560 - 567
Supporting teachers in structuring mathematics lessons involving challenging tasks
Sullivan, Peter, Askew, Mike, Cheeseman, Jill, Clarke, Doug, Mornane, Angela, Roche, Anne and Walker, Nadia. (2015). Supporting teachers in structuring mathematics lessons involving challenging tasks. Journal of Mathematics Teacher Education (online version). 18, pp. 123 - 140. https://doi.org/10.1007/s10857-014-9279-2
Describing the nature and effect of teacher interactions with students during seat work on challenging tasks
Roche, Anne and Clarke, Douglas. (2015). Describing the nature and effect of teacher interactions with students during seat work on challenging tasks. Annual Conference of the Mathematics Education Research Group of Australasia (MERGA). Australia: Mathematics Education Research Group of Australasia. pp. 532 - 539
Students' willingness to engage with mathematical challenges :Implications for classroom pedagogies
Sullivan, Peter, Clarke, Doug, Cheeseman, Jill, Mornane, Angela, Roche, Anne, Sawatzki, Carly and Walker, Nadia. (2014). Students' willingness to engage with mathematical challenges :Implications for classroom pedagogies. In J. Anderson, M. Cavanagh and A. Prescott (Ed.). Curriculum in focus: Research guided practice (Proceedings of the 37th annual conference of the Mathematics Education Research Group of Australasia). Australia: Mathematics Education Research Group of Australasia. pp. 597 - 604
Primary teachers' written unit plans in mathematics and their perceptions of essential elements of these
Roche, Anne, Clarke, Doug M., Clarke, David J. and Sullivan, Peter. (2014). Primary teachers' written unit plans in mathematics and their perceptions of essential elements of these. Mathematics Education Research Journal. 26(4), pp. 853 - 870. https://doi.org/10.1007/s13394-014-0130-y
Teaching strategies for building student persistence on challenging tasks : Insights emerging from two approaches to teacher professional learning
Clarke, Doug, Cheeseman, Jill, Roche, Anne and van der Schans, Stephanie. (2014). Teaching strategies for building student persistence on challenging tasks : Insights emerging from two approaches to teacher professional learning. Mathematics Teacher Education and Development Journal. 16(2), pp. 46 - 70.
Teachers' views of the challenging elements of a task
Cheeseman, Jill, Clarke, Doug, Roche, Anne and Wilson, Karen. (2013). Teachers' views of the challenging elements of a task. In V Steinle, L Ball and C Bardini (Ed.). Mathematics Education: Yesterday, Today and Tomorrow (MERGA36). Australia: Mathematics Education Research Group of Australasia. pp. 154 - 161
Teachers' decisions about mathematics tasks when planning
Sullivan, Peter, Clarke, David, Clarke, Doug and Roche, Anne. (2013). Teachers' decisions about mathematics tasks when planning. In V. Steinle, L. Ball and C. Bardini (Ed.). Mathematics Education: Yesterday, Today and Tomorrow. Melbourne, Australia: Mathematics Education Research Group of Australasia. pp. 626 - 633
Teachers and observers learning from demonstration lessons in mathematics education
Clarke, Doug, Roche, Anne and Knight, Rose. (2013). Teachers and observers learning from demonstration lessons in mathematics education. In A. M. Lindmeier and A. Heinze (Ed.). Mathematics learning across the life span. Germany: International Group for the Psychology of Mathematics Education. pp. 44 - 44
Primary teachers' representations of division: Assessing mathematical knowledge that has pedagogical potential
Roche, Anne and Clarke, Douglas M.. (2013). Primary teachers' representations of division: Assessing mathematical knowledge that has pedagogical potential. Mathematics Education Research Journal. 25(2), pp. 257 - 278. https://doi.org/10.1007/s13394-012-0060-5
Demonstration lessons in mathematics education: Teachers' observation foci and intended changes in practice
Clarke, Doug, Roche, Anne, Wilkie, Karina, Wright, Vince, Brown, Jill, Downton, Ann, Horne, Marjorie, Knight, Rose, McDonough, Andrea, Sexton, Matthew and Worrall, Chris. (2013). Demonstration lessons in mathematics education: Teachers' observation foci and intended changes in practice. Mathematics Education Research Journal. 25(2), pp. 207 - 230. https://doi.org/10.1007/s13394-012-0058-z
Supporting teachers in choosing and using challenging mathematics tasks
Sullivan, Peter, Clarke, Doug M., Michels, Deborah, Mornane, Angela and Roche, Anne. (2012). Supporting teachers in choosing and using challenging mathematics tasks. In J. Dindyal, L. P. Cheng and S. F. Ng (Ed.). Mathematics education: Expanding horizons. Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia. Australia: Mathematics Education Research Group of Australasia. pp. 688 - 695
One-to-one student interviews provide powerful insights and clear focus for the teaching of fractions in the middle years
Clarke, Doug, Roche, Anne and Mitchell, Annie. (2011). One-to-one student interviews provide powerful insights and clear focus for the teaching of fractions in the middle years. In In Way, Jenni and Bobis, Janette (Ed.). Fractions : Teaching for understanding pp. 23-32 Australian Association of Mathematics Teachers.
Some lessons learned from the experience of assessing teacher pedagogical content knowledge in mathematics
Roche, Anne and Clarke, Doug. (2011). Some lessons learned from the experience of assessing teacher pedagogical content knowledge in mathematics. In J. Clark, B. Kissane and J. Mousley (Ed.). Mathematics: Traditions and [new] practices: Proceedings of the 2011 AAMT-MERGA conference. Australia: AAMT and MERGA. pp. 658 - 666
Building teachers' expertise in understanding, assessing and developing children's mathematical thinking : The power of task-based, one-to-one assessment interviews
Clarke, Doug, Clarke, Barbara and Roche, Anne. (2011). Building teachers' expertise in understanding, assessing and developing children's mathematical thinking : The power of task-based, one-to-one assessment interviews. ZDM Mathematics Education. 43(6-7), pp. 901-913. https://doi.org/10.1007/s11858-011-0345-2
Teachers' extent of the use of particular task types in mathematics and choices behind that use
Clarke, Doug M. and Roche, Anne. (2010). Teachers' extent of the use of particular task types in mathematics and choices behind that use. In L. Sparrow, B. Kissane and C. Hurst (Ed.). 33rd Annual Conference of the Mathematics Education Research Group of Australasia. John Curtin College of the Arts, Fremantle, Western Australia, Australia 03 - 07 Jul 2010 Australia: Mathematics Education Research Group of Australasia. pp. 153 - 160
The power of a single game to address a range of important ideas in fraction learning
Clarke, Doug and Roche, Anne. (2010). The power of a single game to address a range of important ideas in fraction learning. Australian Primary Mathematics Classroom. 15(3), pp. 18 - 24.
Decimats: Helping students to make sense of decimal place value
Roche, Anne. (2010). Decimats: Helping students to make sense of decimal place value. Australian Primary Mathematics Classroom. 15(2), pp. 4 - 12.
Students' fraction comparison strategies as a window into robust understanding and possible pointers for instruction
Clarke, Doug M. and Roche, Anne. (2009). Students' fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics. 72(1), pp. 127 - 138. https://doi.org/10.1007/s10649-009-9198-9
Opportunities and challenges for teachers and students provided by tasks built around ''real'' contexts
Clarke, Douglas McLean and Roche, Anne. (2009). Opportunities and challenges for teachers and students provided by tasks built around ''real'' contexts. In R. Hunter, B. Bicknell and T. Burgess (Ed.). Crossing Divides: Proceedings of the 32nd Annual Conference of the Mathematics Education Research Group of Australasia (MERGA). Palmerston North, New Zealand: Mathematics Education Research Group of Australasia. pp. 722 - 725
Making Sense of Partitive and Quotitive Division :A snapshot of Teachers' Pedagogical Content Knowledge
Clarke, Douglas McLean and Roche, Anne. (2009). Making Sense of Partitive and Quotitive Division :A snapshot of Teachers' Pedagogical Content Knowledge. In Roberta Hunter, Brenda Bicknell and Tim Burgess (Ed.). Crossing Divides: Proceedings of the 32nd Annual Conference of the Mathematics Education Research Group of Australasia. Palmerston Nth, New Zealand: Mathematics Education Research Group of Australasia. pp. 467 - 474
Using mathematical tasks built around "real" contexts: Opportunites and challenges for teachers and students: The need for greater challenge and relevance for students in the middle years
Clarke, Doug and Roche, Anne. (2009). Using mathematical tasks built around "real" contexts: Opportunites and challenges for teachers and students: The need for greater challenge and relevance for students in the middle years. Australian Primary Mathematics Classroom. 14(2), pp. 24 - 31.
Ten practical tips for making fractions come alive and make sense
Clarke, Doug, Roche, Anne and Mitchell, Annie. (2008). Ten practical tips for making fractions come alive and make sense. Mathematics Teaching in the Middle School. 13(7), pp. 373 - 380.
Eliciting growth in teachers' proportional reasoning :Measuring the impact of a professional development program
Dole, Shelley, Clarke, Douglas McLean, Wright, Tony, Hilton, Geoff and Roche, Anne. (2008). Eliciting growth in teachers' proportional reasoning :Measuring the impact of a professional development program. In Merrilyn Goos, Ray Brown and Katie Makar (Ed.). Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia. Brisbane, Australia: Mathematics Education Research Group of Australasia. pp. 163 - 170
Year Six Fraction Understanding :A Part of the Whole Story
Clarke, Douglas McLean, Mitchell, Annie and Roche, Anne. (2007). Year Six Fraction Understanding :A Part of the Whole Story. In J Watson and K Beswick (Ed.). 30th annual conference of the Mathematics Education Research Group of Australasia. Wrest Point Hotel Casino, Hobart, Tasmania, Australia 02 - 06 Jul 2007 Adelaide,Australia: Mathematics Education Research Group of Australasia. pp. 207 - 216
When successful comparison of decimals doesn't tell the story
Roche, Anne and Clarke, Douglas McLean. (2006). When successful comparison of decimals doesn't tell the story. In J. Novotna, H. Moraova and M. Kratka (Ed.). Proceedings of the 30th Conference of the International Group of Psychology of Mathematics Education. Prague, Czech Republic: Charles University. pp. 425 - 432
Assessing student understanding of fractions using task-based interviews
Clarke, Douglas McLean, Roche, Anne, Mitchell, Annie and Sukenik, Michal. (2006). Assessing student understanding of fractions using task-based interviews. In J. Novotna, H. Moraova and M. Kratka (Ed.). Proceedings of the 30th Conference of the International Group of Psychology of Mathematics Education. Prague, Czech Republic: Charles University. pp. 337 - 344
Longer is larger - or is it?
Roche, Anne. (2005). Longer is larger - or is it? Australian Primary Mathematics Classroom. 10(3), pp. 11 - 16.
When does the successful comparison of decimals reflect conceptual understanding?
Clarke, Douglas McLean and Roche, Anne. (2004). When does the successful comparison of decimals reflect conceptual understanding? In Ian Putt, Rhonda Farragher and Mal McLean (Ed.). Mathematics Education for the Third Millennium: Towards 2010 (Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australia). Townsville,Australia: James Cook University. pp. 486 - 493