Part 1: The Importance of Spatial Reasoning and Geometry in Kindergarten
Kindergarten students benefit from robust spatial skills and geometry education
The simple act of documenting the mathematics learning in my kindergarten students’ block and puzzle play has led me to some revelations about my three-, four-, five- and six-year-old students. They are capable of relatively sophisticated spatial skills such as mental rotation, symmetry, perspective-taking, use of scale, navigation skills, and basic map-making, yet I am just beginning to understand the place of spatial reasoning and geometry in their learning.
A 2012 survey of JK to Grade 2 teachers in Ontario by Cathy Bruce, Joan Moss, and John Ross has made it clear that Ontario Kindergarten teachers like me have apparently been spending the least amount of instructional time on geometry and spatial thinking skills compared to the four other strands of mathematics, when they might be one of most important strands.
To learn more about this important subject, I have banded together with ten like-minded colleagues from the Kawartha Pine Ridge, Hastings-Prince Edward, and Trillium-Lakelands district school boards. We formed the Kindergarten Educator Spatial Reasoning Collaborative Inquiry and have been awarded a $25,000 Teacher Learning and Leadership Program (TLLP) grant for 2017-18 from the Ontario Ministry of Education and the active participation of the Ontario Teachers’ Federation to study the topic.
One of the first things I discovered was the U.S. National Research Council’s (NRC) 398-page research report on the topic. Published in 2009, Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity concludes that learning geometry and spatial skills is so important for children aged 3 to 6 years that it should receive a high priority in early childhood and kindergarten classrooms, yet it often receives the least. Improving spatial skills develops their general mathematics skills.
Spatial and geometric reasoning is not a single ability, skill, or process. It incorporates numerous concepts, processes, as well as tools to represent and communicate ideas about space and spatial relationships. In the 2015 book, Spatial Reasoning in the Early Years, edited by Brent Davis, researchers Yukari Okamoto, Donna Kotsopoulos, Lynn McGarvey, and David Hallowell explain that these skills include perspective taking, visualizing, locating, orienting, dimension shifting, path-finding, sliding, rotating, reflecting, diagramming, modelling, symmetrizing, composing, decomposing, scaling, map-making, and designing.
In their 2014 article, “Finding the missing piece: blocks, puzzles, and shapes fuel school readiness,” researchers Brian N. Verdine, Roberta Michnick Golinkoff, Kathryn Hirsh-Pasek, and Nora S. Newcombe point out that spatial skills are essential for life. They are necessary for example to efficiently pack the trunk of a car, cut equal sized slices of pieces, use a mall map, correctly install a baby car seat from a diagram, and for jobs such as air traffic controller and engineer. Geometric and spatial skills are also seen as essential job skills for 21st century STEAM jobs – Science, Technology, Engineering, the Arts & Architecture, and Mathematics. [see one founder’s definition (https://steamedu.com/about-us/)]
Spatial skills play a critical role in the development of mathematics and reading
Numerous studies also indicate that spatial skills are far more important to academic achievement than many elementary educators previously thought. Research suggests that spatial thinking skills and geometric reasoning play a critical role in the development of problem-solving skills, mathematical learning, and reading comprehension, that more sophisticated spatial reasoning improves the quality of children’s early mathematics performance, and that children’s mathematics learning during kindergarten is “a key predictor of their subsequent success in both reading and mathematics.”
Another important discovery is that children’s early and kindergarten mathematics skill is “the best predictor of later school achievement, regardless of gender or socio-economic status.”
In their important and informative 2011 article, “Early Childhood Teacher Education: The Case of Geometry,” published in The Journal of Mathematics Teacher Education, early mathematics researchers Douglas H. Clements and Julie Sarama cite research which suggests that geometric and spatial thinking support the development of numerical abilities such as understanding quantity, numeral identification, counting, place value, and arithmetic concepts and skills.
The researchers consider geometric and spatial reasoning to be “a gateway skill to the teaching of higher-order mathematics thinking skills.” In their article, they illustrate that geometric knowledge is highly related to problem-solving ability, mathematical reasoning, proportional reasoning, judgmental application of knowledge, concepts and properties, and managing data and processing skills. Geometry education may also contribute to growth in other cognitive abilities including IQ.
How much spatial and geometric content should there be in kindergarten math?
Experienced teachers have always realized that not all curriculum standards (expectations) are created equal, but it appears that based on the foregoing information, geometry and spatial reasoning deserve a great deal more attention in pre-school and kindergarten math programs. What do the experts say? The NRC Committee on Early Childhood Math’s publication recommends on page 21 that mathematics experience and instruction for pre-school and kindergarten children aged three to six years of age concentrate on number sense & operations and geometry & spatial skills.
Whether it’s direct instruction or intentionally designed play-based inquiry learning math experiences and activities, the NRC report urges educators to make number sense and operations the primary focus of kindergarten math programs and geometry and spatial skill development the second goal, but for less focused class time. Everything else should receive only a small investment of time.
As an educator who has always been faithful to the curriculum, revising my kindergarten math program to reflect the NRC position was difficult for me to accept. The change would mean ignoring, downplaying, or integrating three strands of my provincial kindergarten mathematics curriculum: measurement, patterning and algebra, and probability and data management.
I needed to be certain that this was the right move. I delved into my own curriculum, The Ontario Kindergarten Program (2016). On page 75, it also advises educators to focus on two core strands of mathematics: number sense & numeration and spatial thinking skills & geometric reasoning. I found additional support in three other documents published by four noteworthy organizations:
- page 15 of Thinking Mathematically, published in 2010 by The Elementary Teachers’ Federation of Ontario, part of its Thinking It Through: Teaching and Learning in the Kindergarten Classroom,
- pages 1-2 of the National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics’ (NCTM) 2010 co-publications, Focus in Prekindergarten and Focus in Kindergarten by Karen C. Fuson, Douglas H. Clements and Sybilla Beckmann, and,
- page 6 of the 2014 Ontario Ministry of Education monograph, Paying Attention to Spatial Reasoning.
Key points for kindergarten educators improving their mathematics programs
In the face of such evidence, I resolved to improve my mathematics practice, bearing in mind that student achievement is closely linked to:
- profound knowledge of your students’ abilities,
- the typical learning paths of the age group you are teaching,
- effective instructional strategies for the age group,
- carefully selected classroom resources, and,
- deep subject knowledge.
Anyone embarking on this journey should become familiar with the spatial reasoning and geometry learning paths and developmental milestones of kindergarten students. One comprehensive source is the 2014 resource book, Learning and Teaching Early Math: The Learning Trajectories Approach by Douglas H. Clements and Julie Sarama.
It`s also important to have a variety of tried and tested spatial and geometric learning activities. The Clements and Sarama resource book has a wide variety of them, but the typeface is small and the information crowded. I found it easier to start with the suggestions in Taking Shape: Activities to Develop Geometric and Spatial Thinking Grades K-2, a 2016 publication by Joan Moss, Catherine D. Bruce, Bev Caswell, Tara Flynn, and Zachary Hawes. It contains 32 well designed activities which are well explained and easy for busy classroom teachers to implement. However you might want to start with the free lesson packages from M4YC (mathematics for young children) or some of the ideas from EduGAINS – Mathematics.
Since young children learn best by working with concrete objects and through stories, educators should stock their classrooms with picture books that model building and design, spatial vocabulary, spatial gestures, and spatial and geometric concepts, as well as puzzles, wooden unit blocks, attribute blocks, shape sorters, pentominoes, pattern blocks, pattern puzzles, paper folding tasks such as origami and airplane making, and a wide variety of other building toys such as Duplo®, Magna-Tiles®, Lincoln Logs®, Mobilo®, K’nex®, and TinkerToy® products.
My collaborative inquiry group has compiled a wide variety of spatial and geometry education materials and links to important web resources and posted them on a knowledge building wiki. We invite you to visit it, use the resources, or add to them at https://kindergartenspatialreasoning.wikispaces.com/. We tweet about our work at https://twitter.com/EYSR_2017
Edward Schroeter, B.J., B.Ed., OCT, is a Reading Specialist, Kindergarten Specialist, with Special Education, French as a Second Language, and Library Resource Teacher qualifications. He was trained and worked as a newspaper reporter and documentary video producer. He is a four-time national award-winning playwright and former producer of historical theatre for young audiences. During the past 25 years that he has taught Grade 4, Core French, French Immersion SK, Grade 3 French Immersion, and Kindergarten for the Kawartha Pine Ridge District School Board and one of its predecessor boards, the Peterborough County Board of Education.
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Cathy Bruce, Joan Moss, and John Ross. (2012). Survey of JK to Grade 2 teachers in Ontario Canada: Report to the Literacy and Numeracy Secretariat of the Ministry of Education. Toronto, ON. http://www.pme38.com/wp-content/uploads/2014/05/RF-Sinclair-et-al.pdf
Douglas H. Clements & Julie Sarama. (2011). Early childhood teacher education: The case of geometry. Journal of Mathematics Teacher Education, 14(2), 133–148. Retrieved from https://www.researchgate.net/publication/226486133_Early_childhood_teacher_education_The_case_of_geometry
Douglas H. Clements and Julie Sarama. (2014). Learning and teaching early math: the learning trajectories approach. Second edition. New York: Routledge. https://www.routledge.com/Learning-and-Teaching-Early-Math-The-Learning-Trajectories-Approach-2nd/Clements-Sarama/p/book/9780415828505
Elementary Teachers’ Federation of Ontario. (2010). Thinking Mathematically. Thinking It Through: Teaching and Learning in the Kindergarten Classroom. Toronto, ON. Retrieved from http://www.etfo.ca/SupportingMembers/Resources/OrderForms/ETFO%20Professional%20Resource%20Catalogue.pdf
Farmer, Brian N. Verdine, Kelsey R. Lucca, T. Davies, R. Dempsey, Kathryn Hirsh-Pasek, Roberta Michnick Golinkoff, et al. (2013). Putting the pieces together: Spatial skills at age 3 predict to spatial and math performance at age 5. Seattle, WA: Presented at the Society for Research in Child Development.
Tara Flynn and Zachary Hawes. (2014). Paying attention to spatial reasoning. Ontario Ministry of Education. Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/LNSPayingAttention.pdf
Karen Fuson, Douglas H. Clements, and Sybilla Beckmann. (2010). Focus in kindergarten: teaching with curriculum focal points. National Council of Teachers of Mathematics (NCTM) and the National Association for the Education of Young Children (NAEYC). http://www.nctm.org/store/Products/Focus-in-Kindergarten–Teaching-with-Curriculum-Focal-Points/
Karen Fuson, Douglas H. Clements, and Sybilla Beckmann. (2010). Focus in pre-k: teaching with curriculum focal points. National Council of Teachers of Mathematics (NCTM) and the National Association for the Education of Young Children (NAEYC). http://www.nctm.org/store/Products/Focus-in-Pre-K–Teaching-with-Curriculum-Focal-Points/
Kelly S. Mix and Yi-Ling Cheng. The relation between space and math: developmental and educational implications. Advances in child development and behavior, vol. 42. Janette Benson, editor. Burlington: Academic Press; 2012. pp. 197–243. http://www.sciencenewsline.com/news/2013062518090027.html
Joan Moss, Catherine D. Bruce, Bev Caswell, Tara Flynn, and Zachary Hawes. (2016). Taking shape: activities to develop geometric and Spatial Thinking Grades K-2. Toronto: Pearson. http://www.pearsoncanadaschool.com/index.cfm?locator=PS2uLd
Jane F. Schielack, Chair, Sybilla Beckman, Randall I. Charles, Douglas H. Clements, Paula B. Duckett, Francis (Skip) Fennell, Sharon L. Lewandowski, Emma Treviño, and Rose Mary Zbiek. (2006). Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, pp 23-25. Reston, VA: The National Council of Teachers of Mathematics. https://www2.bc.edu/solomon-friedberg/mt190/nctm-focal-points.pdf
National Research Council of the National Academy of Sciences. (2009). Mathematics learning in early childhood: paths toward excellence and equity. Committee on Early Childhood Mathematics, Christopher T. Cross, Taniesha A. Woods, and Heidi Schweingruber, Editors. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press. DOI: 10.17226/11019. Retrieved from https://www.nap.edu/catalog/12519/mathematics-learning-in-early-childhood-paths-toward-excellence-and-equity
Nora S. Newcombe and Andrea Frick. (2010). Early education for spatial intelligence: why, what, and how. Mind, brain, and education. The International Mind, Brain, and Education Society. Wiley-Blackwell Publishing. Volume 4, Number 3, pp. 102-111. http://spatiallearning.org/publications_pdfs/4nora.pdf
Yukari Okamoto, Donna Kotsopoulos, Lynn McGarvey, David Hallowell. (2015). The development of spatial reasoning. Spatial reasoning in the early years: principles, assertions, and speculations. Toronto: Routledge Press, pp. 15-28
Ontario Ministry of Education. (2003). Early Math Strategy — The Report of the Expert Panel on Early Math in Ontario. Toronto: Author. Retrieved from http://www.edu.gov.on.ca/eng/document/reports/math/math.pdf
Ontario Ministry of Education. (2016). Growing success — the kindergarten addendum. Toronto: Author. Retrieved from http://www.edu.gov.on.ca/eng/policyfunding/growingSuccessAddendum.pdf
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Brian N. Verdine, Roberta Michnick Golinkoff, Kathryn Hirsh-Pasek, and Nora S. Newcombe. (2014). Finding the missing piece: blocks, puzzles, and shapes fuel school readiness. Trends in Neuroscience and Education, 3, pp. 7-13. Retrieved from https://www.academia.edu/19508707/Finding_the_missing_piece_Blocks_puzzles_and_shapes_fuel_school_readiness .
Brian N. Verdine, Roberta M. Golinkoff, Kathryn Hirsh-Pasek, Nora S. Newcombe, Andrew T. Filipowicz, and Alicia Chang. Deconstructing Building Blocks: Preschoolers’ Spatial Assembly Performance Relates to Early Mathematics Skills. (2013). Child Development. 2014 May; 85(3): 1062–1076. Published online 2013 Sep 23. doi: 10.1111/cdev.12165. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3962809/pdf/nihms515592.pdf