Problem Solving in the Early Grades
Young children are problem solvers, and yet early childhood classrooms (overly) focus on number sense. Why not teach number sense through problem solving? For over 3 school years, our research-practice partnership (five researchers and seven classroom teachers) has been examining how young children’s problem-solving practices and mathematical self-efficacy beliefs evolve over time. Rather than cooperating in our efforts, we are collaborating by sharing in our work teaching and researching as well as learning with and from one another and the children.
Related Publications and National Presentations
Deal, E., Mooney, E., Cullen, A. L., Kroesch, A., Bajwa, N. P., Corven, J., & MacDonald, B. (in press). Accessing and assessing components of elementary and middle school students’ mathematical disposition through metaphors. Investingations in Mathematics Learning. https://doi.org/10.1080/19477503.2024.2419291
Kroesch, A., Bajwa, N. P., MacDonald, B. L., Mattoon, C., Hatch, A. M. G.. Cullen, A. L., Mooney, E. S., & Corven, J. (2024). Kindergarteners as sense makers! Mathematics Teacher: Learning and Teaching PK–12, 117(11), 802–812. http://dx.doi.org/10.5951/MTLT.2023.0284
Cullen, A. L., Strawder, S., & Talbot, J. (2024, September). How young children solve problems from kindergarten through grade 2. Presented at the National Council of Teachers of Mathematics Annual Meeting and Exposition, Chicago, IL.
MacDonald, B. L., Cullen, A. L., Kroesch, A. M., Bajwa, N. P., Mooney, E., & Corven, J. (2023). Reimagining the emergent perspective within research-practice learning communities. In T. Lamberg, D. Moss, R. Welder, G. Waddell, & L. Wiest (Eds.), Proceedings of the 45th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 910–914), Reno, NV.
MacDonald, B. L., Cullen, A. L., Kroesch, A. M., Bajwa, N. P., Mooney, E., & Corven, J. (2023, October). Reimagining the emergent perspective within research-practice learning communities. Paper presented at the 45th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Reno, NV.
Odell, J., Langrall, C. W., & Cullen, A. L. (2022). Challenging students to experience mathematics as mathematicians. Mathematics Teacher: Learning and Teaching PK–12, 115(3), 191–201. https://doi.org/10.5951/MTLT.2021.0160
Geometric Measurement
Geometric measurement is a commonly taught topic in school mathematics. Typically, measurement instruction has a strong emphasis on procedures (e.g., how to use a ruler) and computing (e.g., “with area you multiply and perimeter you add”). In my research, I focus on how to support and promote students’ thinking and learning about geometric measurement concepts in the domains of length, area, volume, and angle.
Related Publications and National Presentations
Talbot, J., Cullen, A. L., & Lizano, C. (2023). Fraction as a quantity: Describing students’ reasoning. In T. Lamberg, D. Moss, R. Welder, G. Waddell, & L. Wiest (Eds.), Proceedings of the 45th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 333–341), Reno, NV.
Eames, C., Cullen, A. L., & Talbot, J. (2021, September). Reasoning about area formulas: From rectangles to triangles and back again. Presentation to be presented at the National Council of Teachers of Mathematics Annual Meeting and Exposition. Atlanta, GA. Canceled due to COVID-19.
Cullen, A. L., & Barrett, J. E. (2020). Area measurement: Structuring with nonsquare units. Mathematical Thinking and Learning, 22(2), 85–115. doi:10.1080/10986065.2019.1608619
Cullen, A. L., Lawton, C. A., Patterson, C. S., & Cullen, C. J. (2020). All the way around a circle. Mathematics Teacher: Mathematics Learning and Teaching PreK–12, 113(8), 637–642.
Barrett, J. E., Wickstrom, M. H., Tobias, J. M., Cullen, C. J., Cullen, A. L., & Baek, J. M. (2019). Children’s measurement project: Sharing trajectories with teachers. In P. Sztajn & P. H. Wilson (Eds.), Learning trajectories for teachers: Designing effective professional development for math instruction (pp. 48–74). New York, NY:Teachers College Press.
Clements, D. H., Sarama, J., Van Dine, D. W., Barrett, J. E., Cullen, C. J., Hudyma, A., Dolgin, R., Cullen, A. L., & Eames, C. L. (2018). Evaluation of three interventions teaching area measurement as spatial structuring to young children. Journal of Mathematical Behavior, 50, 23–41. doi:10.1016/j.jmathb.2017.12.004
Cullen, A. L., Cullen, C. J., & O’Hanlon, W. A. (2018). Effects of an intervention on children’s conceptions of angle measurement. International Journal of Research in Education and Science, 4(1), 136–147. doi:10.21890/ijres.382941
Cullen, A. L., Eames, C. L., Cullen, C. J., Barrett, J. E., Sarama, J., Clements, D. H., & Van Dine, D. W. (2018). Effects of three interventions on children’s spatial structuring and coordination of area units. Journal for Research in Mathematics Education, 49(5), 533–574.
Barrett, J. E., Clements, D. H., Sarama, J., Miller, A. L., Cullen, C. J., Van Dine, D. W., . . . Klanderman, D. (2017). Integration of results: A new learning trajectory for area. In J. E. Barrett, D. H. Clements, & J. Sarama (Eds.), Children’s measurement: A longitudinal study of children’s knowledge and learning of length, area, and volume. Journal for Research in Mathematics Education monograph series(Vol. 16, pp. 129–148). Reston, VA: National Council of Teachers of Mathematics.
Barrett, J. E., Cullen, C. J., Miller, A. L., Eames, C. L., Kara, M., & Klanderman, D. (2017). Area in the middle and later elementary grades. In J. E. Barrett, D. H. Clements, & J. Sarama (Eds.), Children’s measurement: A longitudinal study of children’s knowledge and learning of length, area, and volume. Journal for Research in Mathematics Education monograph series (Vol. 16, pp. 105–127). Reston, VA: National Council of Teachers of Mathematics.
Cullen, A. L., Cullen, C. J., & O’Hanlon, W. A. (2017). Elementary students’ reasoning about angle constructions. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 347–354). Indianapolis, IN: Hoosier Association of Mathematics Teacher Educators.
Cullen, C. J., Barrett, J. E., Kara, M., Eames, C. L., Miller, A. L., & Klanderman, D. (2017). Integration of results: A new learning trajectory for volume. In J. E. Barrett, D. H. Clements, & J. Sarama (Eds.), Children’s measurement: A longitudinal study of children’s knowledge and learning of length, area, and volume. Journal for Research in Mathematics Education monograph series (Vol. 16, pp. 181–201). Reston, VA: National Council of Teachers of Mathematics.
Miller, A. L., Cullen, C. J., & O’Hanlon, W. A. (2016, April). Third graders’ development of quantitative reasoning about angle measure. Brief research report presented at the 2016 Research Conference of the National Council of Teachers of Mathematics, San Francisco, CA.
Cullen, C. J., Barrett, J. E., Clements, D. H., Sarama, J., Eames, C. L., Miller, A. L.,& Klanderman, D. (2014, April). Supporting children’s learning of area measurement: A microgenetic study. Paper presented at the 2014 Annual Meeting of the American Educational Research Association, Philadelphia, PA.
Miller, A. L. (2017). Area. In J. E. Barrett, D. H. Clements, & J. Sarama (Eds.), Children’s measurement: A longitudinal study of children’s knowledge and learning of length, area, and volume. Journal for Research in Mathematics Education monograph series (Vol. 16, pp. 71–81). Reston, VA: National Council of Teachers of Mathematics.
Miller, A. L., Eames, C. L., & Cullen, C. J. (2014, April). Thinking deeply about area measurement. Session presented at the 2014 Annual Conference of the National Council of Teachers of Mathematics, New Orleans, LA.
Eames, C. L., Miller, A. L., Kara, M., Cullen, C. J., & Barrett, J. E. (2013). The longitudinal development of unit concepts in area and volume measurement contexts: A case study. In M. Martinez & A. Castro Superfine (Eds.), Proceedings of the 35th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 171–178). Chicago, IL. Retrieved from http://pmena.org/2013/files/PMENA2013_Conference_Proceedings.pdf
Miller, A. L. (2013). Investigating conceptual, procedural, and intuitive aspects of area measurement with non-square area units (Doctoral dissertation). Available from ProQuest Dissertations and Theses database. (Order No. 3572989)
Eames, C. L., Miller, A. L., Kara, M., Van Dine, D., Cullen, C. J., Barrett, J. E., . . . Sarama, J. (2013, June). The longitudinal development of children’s conceptions of spatial measurement. Paper presented at the 43rd Annual Meeting of the Jean Piaget Society, Chicago, IL.
Van Dine, D., Kara, M., Miller, A. L., Eames, C. L., Sarama, J. Clements, D. H., . . . Cullen, C. J. (2013, June). Validating a learning trajectory for volume measurement with Rasch modeling. Poster presented at the 43rd Annual Meeting of the Jean Piaget Society, Chicago, IL.
Barrett, J. E., Cullen, C. J., Miller, A. L., Van Dine, D. W., Eames, C. L., Kara, M., . . . Clements, D. H. (2013, April). Framing and revising a hypothetical learning trajectory for area measurement. Research symposium conducted at the Research Pre-session of the National Council of Teachers of Mathematics, Denver, CO.
Barrett, J. E., Van Dine, D. W., Cullen, C. J., Eames, C. L., Kara, M., & Miller, A. L. (2013, April). Using Rasch modeling to support validation of a developmental progression for area. Paper presented at the 2013 Annual Meeting of the American Educational Research Association, San Francisco, CA.
Miller, A. L. (2013, April). Investigating conceptual, procedural, and intuitive aspects of area measurement with non-square area units. Paper presented at the Society for Research in Child Development Biennial Meeting, Seattle, WA.
Miller, A. L., Eames, C. L., Kara, M., & Cullen, C. J. (2013, January). Hypothetical learning trajectories as works-in-progress tools: Insights into the revising and refining process. Individual session presented at the 17th Annual Conference of the Association of Mathematics Teacher Educators, Orlando, FL.
Kara, M., Miller, A. L., Cullen, C. J., Barrett, J. E., Sarama, J., & Clements, D. H. (2012). A retrospective analysis of students’ thinking about volume measurement across grades 2-5. In L. R. Van Zoest, J. Lo, & J. L. Kratky (Eds.), Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1016–1023). Kalamazoo, MI.
Kara, M., Miller, A. L., Cullen, C. J., Barrett, J. E., Sarama, J., & Clements, D. H. (2012, November). A retrospective analysis of students’ thinking about volume measurement across grades 2-5. Paper presented at the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Kalamazoo, MI.
Miller, A. L., Kara, M., Eames, C. L., Cullen, C. J., & Barrett, J. E. (2012). A comparison of three students’ responses to area invariance tasks across grades 2-5. In L. R. Van Zoest, J. Lo, & J. L. Kratky (Eds.), Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1042). Kalamazoo, MI.
Cullen, C. J., Miller, A. L., & Van Dine, D. W. (2012, April). Reaching the measurement objectives in the CCSSM. Individual session presented at the Annual Meeting of the National Council of Teachers of Mathematics, Philadelphia, PA.
Miller, A. L., Eames, C. L., Kara, M., Barrett, J. E., & Cullen, C. J. (2012, April). Trajectories of three students’ learning of area measurement: Grades 2 – 5. Poster presented at the Research Pre-session of the National Council of Teachers of Mathematics, Philadelphia, PA.
Miller, A. L., Kara, M., & Eames, C. L. (2012, April). Area measurement: What’s the square got to do with it? Individual session presented at the Annual Meeting of the National Council of Teachers of Mathematics, Philadelphia, PA.
Eames, C. L., Kara, M., Cullen, C. J., & Miller, A. L. (2012, February). Implications for the integration of learning trajectories in teacher preparation. Individual session presented at the Annual Meeting of the Association of Mathematics Teacher Educators, Fort Worth, TX.
Barrett, J. E., Cullen, C. J., Sarama, J., Clements, D. H., Klanderman, D., Miller, A. L., & Witkowski Rumsey, C. (2011). Children’s unit concepts in measurement: A teaching experiment spanning grades 2 through 5. ZDM: The International Journal on Mathematics Education, 43(5), 637–650. doi:10.1007/s11858-011-0368-8
Cullen, C. J., Miller, A. L., Barrett, J. E., Clements, D. H., & Sarama, J. (2011). Unit eliciting task structures: Verbal prompts for comparative measures. In B. Ubez (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. II, pp. 249–256). Ankara, Turkey.
Kara, M., Eames, C. L., Miller, A. L., Cullen, C. J., & Barrett, J. E. (2011). Developing an understanding of area measurement concepts with triangular units. In L. R. Wiest & T. Lamberg (Eds.), Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1015–1023). Reno, NV.
Kara, M., Eames, C. L., Miller, A. L., Cullen, C. J., & Barrett, J. E. (2011, October). Developing an understanding of area measurement concepts with triangular units. Paper presented at the 33rd Annual Meeting of the North American Chapter of International Group for the Psychology of Mathematics Education, Reno, NV.
Barrett, J. E., Cullen, C. J., Miller, A. L., & Witkowski-Rumsey, C., & Klanderman, D. (2011, April). Designing integrative learning tasks for unit concepts through length, area, and volume measurement (grades 2-5). Poster presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.
Cullen, C. J., Miller, A. L., & Barrett, J. E. (2011, April). Connecting length, area, and volume tasks through units and comparison. Individual session presented at the Annual Meeting of the National Council of Teachers of Mathematics, Indianapolis, IN.
Cullen, C. J., Miller, A. L., Witkowski-Rumsey, C., Barrett, J. E., & Sarama, J. (2011, April). Area hypothetical learning trajectory: Relating square units to non-rectilinear regions. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.
Miller, A. L., Cullen, C. J., & Tobias, J. M. (2011, January). Using empirical classroom results to inform modifications on a hypothetical learning trajectory for area. Individual session presented at the Annual Meeting of the Association of Mathematics Teacher Educators, Irvine, CA.
Cullen, C. J., Witkowski, C., Miller, A. L., Barrett, J. E., Sarama, J. A., & Clements, D. H. (2010). The key components for measurement tasks. In P. Brosnan, D. B. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. VI, pp. 599–606). Columbus, OH.
Cullen, C. J., Witkowski, C., Miller, A. L., Barrett, J. E., Sarama, J. A., & Clements, D. H. (2010, October). The key components for measurement tasks. Paper presented at the 32nd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH.