Promoting adults’ knowledge for engaging young children with numerical activities.

Esther Levenson; Ruthi Barkai; Pessia Tsamir; Dina Tirosh

doi:10.14712/18047106.2122

Vol. 13 No. 1 (2022): sciED Research Reports

Promoting adults’ knowledge for engaging young children with numerical activities.

Esther Levenson, ¹ Ruthi Barkai, ¹ Pessia Tsamir, ¹ Dina Tirosh, ¹

Tel Aviv University

DOI: https://doi.org/10.14712/18047106.2122

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How to Cite

Levenson, E., Barkai, R., Tsamir, P., & Tirosh, D. (2022). Promoting adults’ knowledge for engaging young children with numerical activities. Scientia in Educatione, 13(1), 26-38. https://doi.org/10.14712/18047106.2122

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Abstract

Considering that young children spend a great deal of time outside of school, there is a need to increase the expertise of adults when it comes to engaging young children with numerical activities. This study presents the Cognitive Affective Mathematics Adult Education framework used to design such a course for interested adults. Focusing on promoting young children’s competency for comparing sets, we illustrate how the framework was used to design the course and how the framework was used to analyze the impact of the course on the relevant expertise of the participating adults. The findings indicated that the adults had gained knowledge in the various strategies one can use to compare sets, had become aware of the importance of evaluating those strategies, and had acquired the ability to take the children’s ages and abilities into account when tailoring activities that engage them with set comparisons.

Keywords Adults’ numerical knowledge; set comparison; young children.

References

Anderson, A., & Anderson, J. (2018). Math-in-context: The types of math preschoolers ‘do’ at home. In I. Elia, J. Mulligan, A. Anderson, A. Baccaglini-Frank, & C. Benz (Eds.), Contemporary research and perspectives on early childhood mathematics education (pp. 183–202). Springer, Cham. Search in Google Scholar
Ball, D. L., Thames, M.H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special. Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554 Open DOI Search in Google Scholar
Barkai, R., Levenson, E., Tsamir, P., & Tirosh, D. (2022). Adults’ awareness of number-related ideas raised by young children. Research in Mathematics Education. https://doi.org/10.1080/14794802.2021.2010238 Open DOI Search in Google Scholar
Baroody, A. J. (1984). Children’s difficulties in subtraction: Some causes and questions. Journal for Research in Mathematics Education, 15(3), 203–213. https://doi.org/10.2307/748349 Open DOI Search in Google Scholar
Baroody, A. J. (2004). The developmental bases for early childhood number and operations standards. In D.H. Clements, & J. Sarama (Eds.), Engaging Young Children in Mathematics Standards for Early Childhood Mathematics Education (pp. 173–220). Taylor and Francis. Search in Google Scholar
Cannon, J., & Ginsburg, H.P. (2008). “Doing the math”: Maternal beliefs about early mathematics versus language learning. Early Education and Development, 19(2), 238–260. https://doi.org/10.1080/10409280801963913 Open DOI Search in Google Scholar
Clarke, B., Cheeseman, J., & Clarke, D. (2006). The mathematical knowledge and understanding young children bring to school. Mathematics Education Research Journal, 18(1), 78–103. https://doi.org/10.1007/BF03217430 Open DOI Search in Google Scholar
Gelman, R., & Gallistel, C. (1978). The child’s understanding of number. Harvard University Press. Search in Google Scholar
Ginsburg, H.P. (2016). Helping early childhood educators to understand and assess young children’s mathematical minds. ZDM Mathematics Education, 48(7), 941–946. https://link.springer.com/article/10.1007/s11858-016-0807-7 Open DOI Search in Google Scholar
Greenes, C., Ginsburg, H.P., & Balfanz, R. (2004). Big math for little kids. Early Childhood Research Quarterly, 19(1), 159–166. https://doi.org/10.1016/j.ecresq.2004.01.010 Open DOI Search in Google Scholar
Greenfield, D.B., & Scott, M. S. (1986). Young children’s preference for complementary pairs: Evidence against a shift to a taxonomic preference. Developmental Psychology, 22(1), 19–21. https://doi.org/10.1037/0012-1649.22.1.19 Open DOI Search in Google Scholar
Gunderson, E.A., & Levine, S.C. (2011). Some types of parent number talk count more than others: Relations between parents’ input and children’s cardinal-number knowledge. Developmental Science, 14(5), 1021–1032. https://doi.org/10.1111/j.1467-7687.2011.01050.x Open DOI Search in Google Scholar
Israel National Mathematics Preschool Curriculum (INMPC). (2010). Retrieved March 18, 2020 from https://edu.gov.il/minhalpedagogy/preschool/subject/math/Pages/math-curriculum.aspx Search in Google Scholar
Levenson, E., Barkai, R., Tirosh, D., & Tsamir, P. (2021). Studying adults’ beliefs related to engaging children with number concepts. In M. Inprasitha, N. Changsri, & N. Boonsena (Eds.), Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 205–215). PME. https://pme44.kku.ac.th/home/uploads/volumn/pme44 vol3.pdf Search in Google Scholar
Levenson, E. S., Barkai, R., Tirosh, D., & Tsamir, P. (2022). Exploring adults’ awareness of and suggestions for early childhood numerical activities. Educational Studies in Mathematics, 109, 5–21. https://doi.org/10.1007/s10649-021-10063-y Open DOI Search in Google Scholar
Levine, S.C., Suriyakham, L.W., Rowe, M. L., Huttenlocher, J., & Gunderson, E.A. (2010). What counts in the development of young children’s number knowledge? Developmental Psychology, 46(5), 1309–1319. https://doi.org/10.1037/a0019671 Open DOI Search in Google Scholar
Marti, M., Merz, E. C., Repka, K.R., Landers, C., Noble, K. G., & Duch, H. (2018). Parent involvement in the getting ready for school intervention is associated with changes in school readiness skills. Frontiers in Psychology, 9, 759. https://doi.org/10.3389/fpsyg.2018.00759 Open DOI Search in Google Scholar
Mason, J. (2011). Noticing: Roots and branches. In M.G. Sherin, V.R. Jacobs, & R.A. Phillipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 35–50). Routledge. Search in Google Scholar
Missall, K., Hojnoski, R. L., Caskie, G. I., & Repasky, P. (2015). Home numeracy environments of preschoolers: Examining relations among mathematical activities, parent mathematical beliefs, and early mathematical skills. Early Education and Development, 26(3), 356–376. https://doi.org/10.1080/10409289.2015.968243 Open DOI Search in Google Scholar
Reardon, S.F., & Portilla, X.A. (2016). Recent trends in income, racial, and ethnic school readiness gaps at kindergarten entry. Aera Open, 2(3), 1–18. https://doi.org/10.1177/2332858416657343 Open DOI Search in Google Scholar
Sarama, J., & Clements, D. (2009). Early childhood mathematics education research: Learning trajectories for young children. Routledge. https://doi.org/10.4324/9780203883785 Open DOI Search in Google Scholar
Sherin, M. G., & Van Es, E.A. (2009). Effects of video club participation on teachers’ professional vision. Journal of Teacher Education, 60(1), 20–37. https://doi.org/10.1177%2F0022487108328155 Open DOI Search in Google Scholar
Skwarchuk, S. L. (2009). How do parents support preschoolers’ numeracy learning experiences at home? Early Childhood Education Journal, 37(3), 189–197. https://doi.org/10.1007/s10643-009-0340-1 Open DOI Search in Google Scholar
Sonnenschein, S., Metzger, S.R., Dowling, R., Gay, B., & Simons, C. L. (2016). Extending an effective classroom-based math board game intervention to preschoolers’ homes. Journal of Applied Research on Children: Informing Policy for Children at Risk, 7(2), Article 1. Available at https://digitalcommons.library.tmc.edu/childrenatrisk/vol7/iss2/1 Search in Google Scholar
Sonnenschein, S., Stites, M., & Dowling, R. (2021). Learning at home: What preschool children’s parents do and what they want to learn from their children’s teachers. Journal of Early Childhood Research, 19(3), 309–322. https://doi.org/10.1177/1476718X20971321 Open DOI Search in Google Scholar
Starkey, P., & Klein, A. (2000). Fostering parental support for children’s mathematical development: An intervention with Head Start families. Early Education and Development, 11(5), 659–680. https://doi.org/s15566935eed1105 7 Search in Google Scholar
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.2307/1175860 Open DOI Search in Google Scholar
Sullivan, P., & Mousley, J. (2001). Thinking teaching: Seeing mathematics teachers as active decision makers. In Making sense of mathematics teacher education (pp. 147–163). Springer, Dordrecht. Search in Google Scholar
Tirosh, D., Tsamir, P., Levenson, E., & Tabach, M. (2011). From preschool teachers’ professional development to children’s knowledge: comparing sets. Journal of Mathematics Teacher Education, 14(2), 113–131. https://doi.org/10.1007/s10857-011-9172-1 Open DOI Search in Google Scholar
Tirosh, D., Tsamir, P., Levenson, E., & Barkai, R. (2017). Working with parents to promote preschool children’s numeracy: Teachers’ attitudes and beliefs. In S. Phillipson, P. Sullivan, & A. Gervasoni (Eds.), Engaging families as the first mathematics educators of children (pp. 181–198). Springer. https://doi.org/10.1007/978-981-10-2553-2_11 Open DOI Search in Google Scholar
Tirosh, D., Tsamir, P., Levenson, E. S., & Barkai, R. (2020). Setting the table with toddlers: a playful context for engaging in one-to-one correspondence. ZDM Mathematics Education, 52, 717–728. https://doi.org/10.1007/s11858-019-01126-9 Open DOI Search in Google Scholar
Tsamir, P. (1999). The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers. Educational Studies in Mathematics, 38, 209–234. https://doi.org/10.1007/978-94-017-1584-3_10 Open DOI Search in Google Scholar
Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive nonexamples: The case of triangles. Educational Studies in Mathematics, 69(2), 81–95. https://doi.org/10.1007/s10649-008-9133-5 Open DOI Search in Google Scholar
Tsamir, P., Tirosh, D., & Levenson, E. (2010). Exploring the relationship between justification and monitoring among kindergarten children. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (CERME 6) (pp. 1891–1901). Institut National de Recherche Pédagogique and ERME. https://hal.archives-ouvertes.fr/hal-02182374/file/cerme6 proceedings.pdf Search in Google Scholar
Tsamir, P., Tirosh, D., Levenson, E., Tabach, M., & Barkai, R. (2014). Employing the CAMTE framework: Focusing on preschool teachers’ knowledge and self-efficacy related to students’ conceptions. In C. Benz, B. Brandt, U. Kortenkamp, G. Krummheuer, S. Ladel, & R. Vogel (Eds.), Early Mathematics Learning – Selected Papers from the POEM 2012 Conference (pp. 291–306). Springer. https://doi.org/10.1007/s10857-013-9260-5 Open DOI Search in Google Scholar
Tsamir, P., Tirosh, D., Levenson, E., Tabach, M., & Barkai, R. (2015). Analyzing number composition and decomposition activities in kindergarten from a numeracy perspective. ZDM Mathematics Education, 47(4), 639–651. https://doi.org/10.1007/s11858-015-0668-5 Open DOI Search in Google Scholar
Tsamir, P., Tirosh, D., Levenson, E., & Barkai, R. (2018). Engaging young children with mathematical activities involving different representations: Triangles, patterns, and counting objects. Center for Educational Policy Studies Journal, 8(2), 9–30. https://doi.org/10.26529/cepsj.271 Open DOI Search in Google Scholar
van Es, E.A., & Sherin, M. G. (2021). Expanding on prior conceptualizations of teacher noticing. ZDM Mathematics Education, 53(1), 17–27. https://doi.org/10.1007/s11858-020-01211-4 Open DOI Search in Google Scholar
Vandermaas-Peeler, M., Nelson, J., Bumpass, C., & Sassine, B. (2009). Numeracy-related exchanges in joint storybook reading and play. International Journal of Early Years Education, 17(1), 67–84. https://doi.org/10.1080/09669760802699910 Open DOI Search in Google Scholar
Vandermaas-Peeler, M., Ferretti, L., & Loving, S. (2012). Playing the ladybug game: Parent guidance of young children’s numeracy activities. Early Child Development and Care, 182(10), 1289–1307. https://psycnet.apa.org/doi/10.1080/03004430.2011.609617 Open DOI Search in Google Scholar

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[1] Anderson, A., & Anderson, J. (2018). Math-in-context: The types of math preschoolers ‘do’ at home. In I. Elia, J. Mulligan, A. Anderson, A. Baccaglini-Frank, & C. Benz (Eds.), Contemporary research and perspectives on early childhood mathematics education (pp. 183–202). Springer, Cham. Search in Google Scholar

[2] Ball, D. L., Thames, M.H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special. Journal of Teacher Education, 59(5), 389–407. https://doi.org/10.1177/0022487108324554 Open DOI Search in Google Scholar

[3] Barkai, R., Levenson, E., Tsamir, P., & Tirosh, D. (2022). Adults’ awareness of number-related ideas raised by young children. Research in Mathematics Education. https://doi.org/10.1080/14794802.2021.2010238 Open DOI Search in Google Scholar

[4] Baroody, A. J. (1984). Children’s difficulties in subtraction: Some causes and questions. Journal for Research in Mathematics Education, 15(3), 203–213. https://doi.org/10.2307/748349 Open DOI Search in Google Scholar

[5] Baroody, A. J. (2004). The developmental bases for early childhood number and operations standards. In D.H. Clements, & J. Sarama (Eds.), Engaging Young Children in Mathematics Standards for Early Childhood Mathematics Education (pp. 173–220). Taylor and Francis. Search in Google Scholar

[6] Cannon, J., & Ginsburg, H.P. (2008). “Doing the math”: Maternal beliefs about early mathematics versus language learning. Early Education and Development, 19(2), 238–260. https://doi.org/10.1080/10409280801963913 Open DOI Search in Google Scholar

[7] Clarke, B., Cheeseman, J., & Clarke, D. (2006). The mathematical knowledge and understanding young children bring to school. Mathematics Education Research Journal, 18(1), 78–103. https://doi.org/10.1007/BF03217430 Open DOI Search in Google Scholar

[8] Gelman, R., & Gallistel, C. (1978). The child’s understanding of number. Harvard University Press. Search in Google Scholar

[9] Ginsburg, H.P. (2016). Helping early childhood educators to understand and assess young children’s mathematical minds. ZDM Mathematics Education, 48(7), 941–946. https://link.springer.com/article/10.1007/s11858-016-0807-7 Open DOI Search in Google Scholar

[10] Greenes, C., Ginsburg, H.P., & Balfanz, R. (2004). Big math for little kids. Early Childhood Research Quarterly, 19(1), 159–166. https://doi.org/10.1016/j.ecresq.2004.01.010 Open DOI Search in Google Scholar

[11] Greenfield, D.B., & Scott, M. S. (1986). Young children’s preference for complementary pairs: Evidence against a shift to a taxonomic preference. Developmental Psychology, 22(1), 19–21. https://doi.org/10.1037/0012-1649.22.1.19 Open DOI Search in Google Scholar

[12] Gunderson, E.A., & Levine, S.C. (2011). Some types of parent number talk count more than others: Relations between parents’ input and children’s cardinal-number knowledge. Developmental Science, 14(5), 1021–1032. https://doi.org/10.1111/j.1467-7687.2011.01050.x Open DOI Search in Google Scholar

[13] Israel National Mathematics Preschool Curriculum (INMPC). (2010). Retrieved March 18, 2020 from https://edu.gov.il/minhalpedagogy/preschool/subject/math/Pages/math-curriculum.aspx Search in Google Scholar

[14] Levenson, E., Barkai, R., Tirosh, D., & Tsamir, P. (2021). Studying adults’ beliefs related to engaging children with number concepts. In M. Inprasitha, N. Changsri, & N. Boonsena (Eds.), Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 205–215). PME. https://pme44.kku.ac.th/home/uploads/volumn/pme44 vol3.pdf Search in Google Scholar

[15] Levenson, E. S., Barkai, R., Tirosh, D., & Tsamir, P. (2022). Exploring adults’ awareness of and suggestions for early childhood numerical activities. Educational Studies in Mathematics, 109, 5–21. https://doi.org/10.1007/s10649-021-10063-y Open DOI Search in Google Scholar

[16] Levine, S.C., Suriyakham, L.W., Rowe, M. L., Huttenlocher, J., & Gunderson, E.A. (2010). What counts in the development of young children’s number knowledge? Developmental Psychology, 46(5), 1309–1319. https://doi.org/10.1037/a0019671 Open DOI Search in Google Scholar

[17] Marti, M., Merz, E. C., Repka, K.R., Landers, C., Noble, K. G., & Duch, H. (2018). Parent involvement in the getting ready for school intervention is associated with changes in school readiness skills. Frontiers in Psychology, 9, 759. https://doi.org/10.3389/fpsyg.2018.00759 Open DOI Search in Google Scholar

[18] Mason, J. (2011). Noticing: Roots and branches. In M.G. Sherin, V.R. Jacobs, & R.A. Phillipp (Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 35–50). Routledge. Search in Google Scholar

[19] Missall, K., Hojnoski, R. L., Caskie, G. I., & Repasky, P. (2015). Home numeracy environments of preschoolers: Examining relations among mathematical activities, parent mathematical beliefs, and early mathematical skills. Early Education and Development, 26(3), 356–376. https://doi.org/10.1080/10409289.2015.968243 Open DOI Search in Google Scholar

[20] Reardon, S.F., & Portilla, X.A. (2016). Recent trends in income, racial, and ethnic school readiness gaps at kindergarten entry. Aera Open, 2(3), 1–18. https://doi.org/10.1177/2332858416657343 Open DOI Search in Google Scholar

[21] Sarama, J., & Clements, D. (2009). Early childhood mathematics education research: Learning trajectories for young children. Routledge. https://doi.org/10.4324/9780203883785 Open DOI Search in Google Scholar

[22] Sherin, M. G., & Van Es, E.A. (2009). Effects of video club participation on teachers’ professional vision. Journal of Teacher Education, 60(1), 20–37. https://doi.org/10.1177%2F0022487108328155 Open DOI Search in Google Scholar

[23] Skwarchuk, S. L. (2009). How do parents support preschoolers’ numeracy learning experiences at home? Early Childhood Education Journal, 37(3), 189–197. https://doi.org/10.1007/s10643-009-0340-1 Open DOI Search in Google Scholar

[24] Sonnenschein, S., Metzger, S.R., Dowling, R., Gay, B., & Simons, C. L. (2016). Extending an effective classroom-based math board game intervention to preschoolers’ homes. Journal of Applied Research on Children: Informing Policy for Children at Risk, 7(2), Article 1. Available at https://digitalcommons.library.tmc.edu/childrenatrisk/vol7/iss2/1 Search in Google Scholar

[25] Sonnenschein, S., Stites, M., & Dowling, R. (2021). Learning at home: What preschool children’s parents do and what they want to learn from their children’s teachers. Journal of Early Childhood Research, 19(3), 309–322. https://doi.org/10.1177/1476718X20971321 Open DOI Search in Google Scholar

[26] Starkey, P., & Klein, A. (2000). Fostering parental support for children’s mathematical development: An intervention with Head Start families. Early Education and Development, 11(5), 659–680. https://doi.org/s15566935eed1105 7 Search in Google Scholar

[27] Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. https://doi.org/10.2307/1175860 Open DOI Search in Google Scholar

[28] Sullivan, P., & Mousley, J. (2001). Thinking teaching: Seeing mathematics teachers as active decision makers. In Making sense of mathematics teacher education (pp. 147–163). Springer, Dordrecht. Search in Google Scholar

[29] Tirosh, D., Tsamir, P., Levenson, E., & Tabach, M. (2011). From preschool teachers’ professional development to children’s knowledge: comparing sets. Journal of Mathematics Teacher Education, 14(2), 113–131. https://doi.org/10.1007/s10857-011-9172-1 Open DOI Search in Google Scholar

[30] Tirosh, D., Tsamir, P., Levenson, E., & Barkai, R. (2017). Working with parents to promote preschool children’s numeracy: Teachers’ attitudes and beliefs. In S. Phillipson, P. Sullivan, & A. Gervasoni (Eds.), Engaging families as the first mathematics educators of children (pp. 181–198). Springer. https://doi.org/10.1007/978-981-10-2553-2_11 Open DOI Search in Google Scholar

[31] Tirosh, D., Tsamir, P., Levenson, E. S., & Barkai, R. (2020). Setting the table with toddlers: a playful context for engaging in one-to-one correspondence. ZDM Mathematics Education, 52, 717–728. https://doi.org/10.1007/s11858-019-01126-9 Open DOI Search in Google Scholar

[32] Tsamir, P. (1999). The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers. Educational Studies in Mathematics, 38, 209–234. https://doi.org/10.1007/978-94-017-1584-3_10 Open DOI Search in Google Scholar

[33] Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive nonexamples: The case of triangles. Educational Studies in Mathematics, 69(2), 81–95. https://doi.org/10.1007/s10649-008-9133-5 Open DOI Search in Google Scholar

[34] Tsamir, P., Tirosh, D., & Levenson, E. (2010). Exploring the relationship between justification and monitoring among kindergarten children. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (CERME 6) (pp. 1891–1901). Institut National de Recherche Pédagogique and ERME. https://hal.archives-ouvertes.fr/hal-02182374/file/cerme6 proceedings.pdf Search in Google Scholar

[35] Tsamir, P., Tirosh, D., Levenson, E., Tabach, M., & Barkai, R. (2014). Employing the CAMTE framework: Focusing on preschool teachers’ knowledge and self-efficacy related to students’ conceptions. In C. Benz, B. Brandt, U. Kortenkamp, G. Krummheuer, S. Ladel, & R. Vogel (Eds.), Early Mathematics Learning – Selected Papers from the POEM 2012 Conference (pp. 291–306). Springer. https://doi.org/10.1007/s10857-013-9260-5 Open DOI Search in Google Scholar

[36] Tsamir, P., Tirosh, D., Levenson, E., Tabach, M., & Barkai, R. (2015). Analyzing number composition and decomposition activities in kindergarten from a numeracy perspective. ZDM Mathematics Education, 47(4), 639–651. https://doi.org/10.1007/s11858-015-0668-5 Open DOI Search in Google Scholar

[37] Tsamir, P., Tirosh, D., Levenson, E., & Barkai, R. (2018). Engaging young children with mathematical activities involving different representations: Triangles, patterns, and counting objects. Center for Educational Policy Studies Journal, 8(2), 9–30. https://doi.org/10.26529/cepsj.271 Open DOI Search in Google Scholar

[38] van Es, E.A., & Sherin, M. G. (2021). Expanding on prior conceptualizations of teacher noticing. ZDM Mathematics Education, 53(1), 17–27. https://doi.org/10.1007/s11858-020-01211-4 Open DOI Search in Google Scholar

[39] Vandermaas-Peeler, M., Nelson, J., Bumpass, C., & Sassine, B. (2009). Numeracy-related exchanges in joint storybook reading and play. International Journal of Early Years Education, 17(1), 67–84. https://doi.org/10.1080/09669760802699910 Open DOI Search in Google Scholar

[40] Vandermaas-Peeler, M., Ferretti, L., & Loving, S. (2012). Playing the ladybug game: Parent guidance of young children’s numeracy activities. Early Child Development and Care, 182(10), 1289–1307. https://psycnet.apa.org/doi/10.1080/03004430.2011.609617 Open DOI Search in Google Scholar

Scientia in educatione

Promoting adults’ knowledge for engaging young children with numerical activities.

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