Investigating functional thinking in the elementary classroom: Foundations of early algebra reasoning

Journal article

Warren, Elizabeth, Cooper, Tom and Lamb, Janeen. (2006). Investigating functional thinking in the elementary classroom: Foundations of early algebra reasoning. The Journal of Mathematical Behavior. 25(3), pp. 208 - 223. https://doi.org/10.1016/j.jmathb.2006.09.006
AuthorsWarren, Elizabeth, Cooper, Tom and Lamb, Janeen
Year2006
JournalThe Journal of Mathematical Behavior
Journal citation25 (3), pp. 208 - 223
PublisherElsevier
ISSN0732-3123
Digital Object Identifier (DOI)https://doi.org/10.1016/j.jmathb.2006.09.006
Scopus EID2-s2.0-33751318004
Page range208 - 223
Research GroupSchool of Education
Place of publicationUnited States of America
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Warren, Elizabeth, Young, Janelle and De Vries, Eva. (2008). The impact of early numeracy engagement on four-year-old indigenous students. Australasian Journal of Early Childhood. 33(4), pp. 2 - 8.
Bridging the educational gap: Indigenous and non-indigenous beliefs, attitudes, and practices in a remote Australian school
Warren, Elizabeth, Cooper, Tom and Baturo, Annette. (2008). Bridging the educational gap: Indigenous and non-indigenous beliefs, attitudes, and practices in a remote Australian school. Educational Practice and Theory. 30(1), pp. 41 - 56.
The role of oral language in assisting young Australian Indigenous students' numeracy outcomes
Warren, Elizabeth Anne, Young, Janelle Patricia and de Vries, Eva. (2008). The role of oral language in assisting young Australian Indigenous students' numeracy outcomes. Changing Climates: Education for Sustainable Futures. Brisbane, Australia: Australian Association for Research in Education.
Generalising mathematical structure in Years 3-4 :A case study of Equivalence of Expression
Cooper, Tom and Warren, Elizabeth Anne. (2008). Generalising mathematical structure in Years 3-4 :A case study of Equivalence of Expression. In Olimpia Figueras, Jose Luis Cortina and Silvia Alatorre (Ed.). International Group for the Psychology of Mathematics Education: Proceedings of the Joint Meeting of PME 32 and PME-NA XXX. Morelia, Mexico: Cinvestav-UMSNH. pp. 369 - 376
Patterns that support early algebraic thinking in the elementary school
Warren, Elizabeth Anne and Cooper, Tom. (2008). Patterns that support early algebraic thinking in the elementary school. In Algebra and Algebraic Thinking in School Mathematics: Seventieth Yearbook pp. 113 - 126 National Council of Teachers of Mathematics.
Research empowering the researched :Reflections on supporting Indigenous teacher Aides to tutor mathematics
Underwood, Petrina, Warren, Elizabeth Anne, Cooper, Tom, Matthews, Chris and Baturo, Annette. (2008). Research empowering the researched :Reflections on supporting Indigenous teacher Aides to tutor mathematics. In Olimpia Figueras, Jose Luis Cortina and Silvia Alatorre (Ed.). Proceedings of 32nd Annual Conference of the International Group for the Psychology of Mathematics Education. The Joint Meeting of PME 32 and PME-NA XXX.. Morelia, Mexico: Cinvestav-UMSNH. pp. 137 - 144
Oral language, representations and mathematical understanding: Indigenous Australian students
Warren, Elizabeth and Young, Janelle. (2008). Oral language, representations and mathematical understanding: Indigenous Australian students. The Australian Journal of Indigenous Education. 37(1), pp. 130 - 137. https://doi.org/10.1017/S1326011100016173
The effect of different representations on Years 3 to 5 students’ ability to generalise
Cooper, Tom J. and Warren, Elizabeth A.. (2007). The effect of different representations on Years 3 to 5 students’ ability to generalise. ZDM. 40(1), pp. 23 - 37. https://doi.org/10.1007/s11858-007-0066-8
The use of a new mathematics text book scheme :support or impediment
Heirdsfield, Ann, Dole, Shelley and Warren, Elizabeth Anne. (2007). The use of a new mathematics text book scheme :support or impediment. AARE Focus 2007. Australia: Australian Association for Research in Education.
Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking
Warren, Elizabeth A. and Cooper, Thomas J.. (2007). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics. 67(2), pp. 171 - 185. https://doi.org/10.1007/s10649-007-9092-2
Current issues of vocational education and technical training in Australia
Warren, Elizabeth. (2007). Current issues of vocational education and technical training in Australia. Educational Practice and Theory. 29(2), pp. 81 - 92.
Repeating patterns and multiplicative thinking: Analysis of classroom interactions with 9 year old students that support the transition from known to the novel
Warren, Elizabeth and Cooper, Tom. (2007). Repeating patterns and multiplicative thinking: Analysis of classroom interactions with 9 year old students that support the transition from known to the novel. Journal of Classroom Interaction. 41(2), pp. 7 - 17.
Functional thinking in the year 1 classroom
Warren, Elizabeth and Benson, Samantha. (2007). Functional thinking in the year 1 classroom. Australian Primary Mathematics Classroom. 12(3), pp. 14 - 17.
Australian Indigenous Students :The role of oral language and representations in the negotiation of mathematical understanding
Vries, Eva de, Young, Janelle Patricia and Warren, Elizabeth Anne. (2007). Australian Indigenous Students :The role of oral language and representations in the negotiation of mathematical understanding. In Jane Watson and Kim 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. 775 - 784
Combining teaching experiments and professional learning: Conflicts between research and teacher outcomes
Lamb, Janeen, Cooper, Tom and Warren, Elizabeth. (2007). Combining teaching experiments and professional learning: Conflicts between research and teacher outcomes. Mathematics Education Research Journal. 19(3), pp. 73 - 92. https://doi.org/10.1007/BF03217463
Exploring an understanding of equals as quantitative sameness with 5 year old students
Warren, Elizabeth Anne. (2007). Exploring an understanding of equals as quantitative sameness with 5 year old students. In Jeong-Ho Woo, Hee-Chan Lew and Kyo-Sik Park (Ed.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, PME31, Seoul, Korea, July 8-13, 2007. Haifa, Israel: International Group for the Psychology of Mathematics Education. pp. 1 - 8
Teacher actions :Enhancing the learning of mental computation in Year 2
Heirdsfield, Anne and Lamb, Janeen Therese. (2006). Teacher actions :Enhancing the learning of mental computation in Year 2. In Jarmila Novotna, Hana Moraova and Magdalena Kratka (Ed.). Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague, Czech Republic: Atelier Guimaec, sro. pp. 281 - 288
Using repeating patterns to explore functional thinking
Warren, Elizabeth and Cooper, Tom. (2006). Using repeating patterns to explore functional thinking. Australian Primary Mathematics Classroom. 11(1), pp. 9 - 14.
Teacher Professional Development in Patterns and Algebra :Being Sensitive to a Teacher's Zone of Proximal Development
Lamb, Janeen Therese, Warren, Elizabeth Anne and Cooper, Tom. (2006). Teacher Professional Development in Patterns and Algebra :Being Sensitive to a Teacher's Zone of Proximal Development. In Peter Grootenboew, Robyn Zvenbrergen and Mohan Chinnappan (Ed.). Identities; Cultures and Learning Spaces: Proceedings of the 29th Annual Conference of the Mathematics Education Research Group of Australasia. Adelaide, Australia: Mathematics Education Research Group of Australasia. pp. 543 - 550
Teacher actions that assist young students write generalizations in words and in symbols
Warren, Elizabeth Anne. (2006). Teacher actions that assist young students write generalizations in words and in symbols. In Jarmila Novotna, Hana Moraova and Magdelena Kratka (Ed.). Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague, Czech Republic: Charles University. pp. 377 - 384
Aim high - Beat Yourself :Effective mathematics teaching in a remote indigenous community
Cooper, Tom, Baturo, Annette, Warren, Elizabeth Anne and Grant, Edlyn. (2006). Aim high - Beat Yourself :Effective mathematics teaching in a remote indigenous community. In Jarmila Novotna, Hana Moraova and Magdelena Kratka (Ed.). Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education. Prague, Czech Republic: Charles University. pp. 339 - 376
Supporting learning in early algebra :A model of professional learning
Warren, Elizabeth Anne. (2006). Supporting learning in early algebra :A model of professional learning. In Peter Grootenboew, Robyn Zvenbrergen and Mohan Chinnappan (Ed.). Proceedings of the 29th annual conference of the Mathematics Education Research Group of Australasia. Adelaide, Australia: Mathematics Education Research Group of Australasia. pp. 535 - 542
Year 2 inaccurate but flexible mental computers: Teacher actions supporting growth
Heirdsfield, Ann and Lamb, Janeen. (2006). Year 2 inaccurate but flexible mental computers: Teacher actions supporting growth. In P. L. Jeffery (Ed.). AARE 2006 International Education Research Conference Engaging Pedagogies. Melbourne, Australia: Australian Association for Research in Education. pp. 1 - 8
Comparative mathematical language in the elementary school: A longitudinal study
Warren, Elizabeth A.. (2006). Comparative mathematical language in the elementary school: A longitudinal study. Educational Studies in Mathematics. 62(2), pp. 169 - 189. https://doi.org/10.1007/s10649-006-4627-5
Patterns supporting the development of early algebraic thinking
Warren, Elizabeth Anne. (2005). Patterns supporting the development of early algebraic thinking. In P. Clarkson, A. Downton and M. Horne (Ed.). Proceedings of the 28th conference of the mathematics education research group of Australasia.. Melbourne, Australia: Deakin University Press. pp. 759 - 766
Young children's ability to use the balance strategy to solve for unknowns
Warren, Elizabeth Anne and Cooper, Tom J.. (2005). Young children's ability to use the balance strategy to solve for unknowns. Mathematics Education Research Journal. 17(1), pp. 58 - 72. https://doi.org/10.1007/BF03217409
Lifelong learning and the reform of the teaching practicum in Australia
Chapman, Judith, Toomey, Ron, Gaff, Janet, McGilp, Jacqueline, Walsh, Maureen, Warren, Elizabeth and Willia, Irene. (2005). Lifelong learning and the reform of the teaching practicum in Australia. Asia-Pacific Journal of Teacher Education. 33(1), pp. 23 - 34.
Introducing functional thinking in year 2: A case study of early algebraic thinking
Warren, Elizabeth and Cooper, Tom. (2005). Introducing functional thinking in year 2: A case study of early algebraic thinking. Contemporary Issues in Early Childhood. 6(2), pp. 150 - 162.
Young children's ability to generalize the pattern rule for growing patterns
Warren, Elizabeth Anne. (2005). Young children's ability to generalize the pattern rule for growing patterns. In H. Chick and J. Vincent (Ed.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Melbourne, Australia: University of Melbourne. pp. 305 - 312
Queensland Indigenous students in rural and remote classrooms. Pedagogical practices in numeracy learning
Cooper, Tom, Baturo, Annette and Warren, Elizabeth Anne. (2005). Queensland Indigenous students in rural and remote classrooms. Pedagogical practices in numeracy learning. In J. Zajda and P. Ninnes (Ed.). Proceedings of the 23rd annual conference of the Australian and New Zealand comparative and international education society.. Armidale, Australia: Centre for Research on Education in Context, University of New England. pp. 154 - 165
Mental Computation :The benefits of informed teacher instruction
Heirdsfield, Ann and Lamb, Janeen. (2005). Mental Computation :The benefits of informed teacher instruction. In P. Clarkson, A. Downton and D. Gronn (Ed.). Building connections: Research; theory and practice (Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia). Sydney, Australia: Mathematics Education Research Group of Australasia. pp. 419 - 426
Indigenous and Non Indigenous teaching relationships in the three mathematics classrooms in remote Queensland
Baturo, Annette, Cooper, Tom and Warren, Elizabeth Anne. (2005). Indigenous and Non Indigenous teaching relationships in the three mathematics classrooms in remote Queensland. In Helen Chick and Jill Vincent (Ed.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education. Melbourne, Australia: Melbourne University Press. pp. 262 - 272
Generalising arithmetic :Supporting the process in the early years
Warren, Elizabeth Anne. (2004). Generalising arithmetic :Supporting the process in the early years. In Marit Johnsen Hoines and Anne Berit Fuglestad (Ed.). Proceedings of the 28th Conference of the International group for the psychology of mathematics education. Bergen,Norway: Bergen University College. pp. 425 - 432
Lifelong learning and the assessment and evaluation practices in some Australian faculties of education
Toomey, Ron, Chapman, Judith, Gaff, Janet, McGilp, Jacqueline, Walsh, Maureen, Warren, Elizabeth and Willia, Irene. (2004). Lifelong learning and the assessment and evaluation practices in some Australian faculties of education. Professional Development in Education. 30(2), pp. 225 - 243.
Indigenous students and mathematics: Teachers' perceptions of the role of teacher aides
Warren, Elizabeth, Cooper, Tom and Baturo, Annette. (2004). Indigenous students and mathematics: Teachers' perceptions of the role of teacher aides. The Australian Journal of Indigenous Education. 33, pp. 37 - 46.
Learning and teaching algebra
Warren, Elizabeth Anne and Pierce, Robyn. (2004). Learning and teaching algebra. In In B. Perry, G. Anthony and C. Diezmann (Ed.). Research in Mathematics Education in Australasia pp. 291 - 312 Post Pressed.
The impact of developing teacher conceptual knowledge of division
Booker, George and Lamb, Janeen Therese. (2004). The impact of developing teacher conceptual knowledge of division. In M. J. Hoines and A. B. Fuglestad (Ed.). Proceedings of the 28th Conference of the International group for the psychology of mathematics education. Bergen, Norway: Bergen University College. pp. 177 - 184
A review of research in learning and teaching algebra
Warren, Elizabeth Anne and Pierce, Robyn. (2004). A review of research in learning and teaching algebra. In In B. Perry, G. Anthony and C. Diezmann (Ed.). Research in Mathematics Education in Australasia pp. 291 - 321 Post Pressed.
Young white teachers' perceptions of mathematics learning of aboriginal and non-aboriginal students in remote communities
Doig, Shani, Baturo, Annette, Warren, Elizabeth Anne and Cooper, Tom. (2004). Young white teachers' perceptions of mathematics learning of aboriginal and non-aboriginal students in remote communities. In Marit Johnsen Hoines and Anne Berit Fuglestad (Ed.). Proceedings of the 28th Conference of the International group for the psychology of mathematics education. Bergen,Norway: Bergen University College. pp. 239 - 246
The role of arithmetic structure in the transition from arithmetic to algebra
Warren, Elizabeth. (2003). The role of arithmetic structure in the transition from arithmetic to algebra. Mathematics Education Research Journal. 15(2), pp. 122 - 137.
Open-ended realistic division problems, generalization and early algebra
Coope, Tom and Warren, Elizabeth A.. (2003). Open-ended realistic division problems, generalization and early algebra. Proceedings of the 27th conference of the international group for the psychology of mathematics education. Honolulu, Hawaii: International Group for the Psychology of Mathemati.... pp. 245 - 253
Parent partnerships within Catholic school communities: Values underpins success
Warren, Elizabeth, Young, Janelle and Hanifin, Pam. (2003). Parent partnerships within Catholic school communities: Values underpins success. Journal of Empirical Theology. 4(1), pp. 63 - 80.
Young children's understanding of equals: A longitudinal study
Warren, Elizabeth Anne. (2003). Young children's understanding of equals: A longitudinal study. In Neil Pateman, Barbara Dougherty and Joseph Zilliox (Ed.). Navigating between theory & practice PME27. Honolulu, Hawaii USA: CRDG College of Education - University of Hawaii. pp. 379 - 386
Introducing equivalence and inequivalence in year 2
Warren, Elizabeth and Cooper, Tom. (2003). Introducing equivalence and inequivalence in year 2. Australian Primary Mathematics Classroom. 8(1), pp. 9 - 14.
Language arithmetic & young children's interpretations
Warren, Elizabeth. (2003). Language arithmetic & young children's interpretations. Focus on Learning Problems in Mathematics. 25(4), pp. 22 - 35.
The impact of teachers' understanding of division on their students' knowledge of division
Lamb, Janeen Therese. (2003). The impact of teachers' understanding of division on their students' knowledge of division. In In B.A. Knight and W. Scott (Ed.). Learning Difficulties: Multiple Perspectives pp. 147 - 163 Pearson Education.
Arithmetic pathways towards algebraic thinking: Exploring arithmetic compensation in year 3
Warren, Elizabeth and Cooper, Tom. (2003). Arithmetic pathways towards algebraic thinking: Exploring arithmetic compensation in year 3. Australian Primary Mathematics Classroom. 8(4), pp. 10 - 16.
Number combinations and arithmetic structure: Implications for early algebra
Warren, Elizabeth Anne. (2003). Number combinations and arithmetic structure: Implications for early algebra. In L. Bragg, C. Campbell and G. Herbert (Ed.). Mathematics Education Research: Innovation; Networkings; Opportunity: Proceedings of the 26th Annual Conference of the Mathematics Education Research Group of Australia. Melbourne, Australia: Mathematics Education Research Group of Australasia. pp. 712 - 719
Collaboration and sharing as crucial elements of professional development
Warren, Elizabeth Anne, Nisbet, Stephen and Cooper, Tom. (2003). Collaboration and sharing as crucial elements of professional development. In In A. Peter-Koop, V. Santos-Wagner and C. Breen et al. (Ed.). Collaboration in Teacher Education: Examples from the Context of Mathematics Education pp. 23 - 40 Kluwer Academic Publishers.
Lifelong learning and teacher education
Chapman, Judith, Toomey, Ron, McGilp, Jacqueline, Walsh, Maureen, Warren, Elizabeth Anne and Gaff, Janet. (2003). Lifelong learning and teacher education Melbourne, Australia: Australian Teacher Education Association.
Learning for life
Walsh, Maureen Patricia, Chapman, Judith Dorothy, McGilp, Jacqueline, Toomey, Ron and Warren, Elizabeth A.. (2003). Learning for life. In In bFrank Crowther (Ed.). Teachers as Leaders in a Knowledge Society pp. 69 - 80 Australian College of Educators.
Literacy and numeracy in the home: Parents supporting children's learning
Young, Janelle Patricia and Warren, Elizabeth Anne. (2002). Literacy and numeracy in the home: Parents supporting children's learning. In Bill Cope and Mary Kalantzis (Ed.). Learning for the Future: Proceedings of the Learning Conference 2001. Melbourne, Australia: Common Ground Publishing. pp. 369 - 376
Arithmetic and quasi-variables: A year 2 lesson to introduce algebra in the early years
Warren, Elizabeth Anne. (2002). Arithmetic and quasi-variables: A year 2 lesson to introduce algebra in the early years. In A. Cockburn and E. Nardi (Ed.). International Group for the Psychology of Mathematics Education: Proceedings of the 26th Annual Conference. Norwich, England: University of East Anglia. pp. 369 - 376
Children's understanding of 'turn arounds'
Warren, Elizabeth. (2002). Children's understanding of 'turn arounds'. Australian Primary Mathematics Classroom. 7(1), pp. 9 - 13.
Parent and school partnerships in supporting literacy and numeracy
Warren, Elizabeth and Young, Janelle. (2002). Parent and school partnerships in supporting literacy and numeracy. Asia-Pacific Journal of Teacher Education. 30(3), pp. 217 - 228.