CMEF Early Geometry
This is page for the CMEF working group Geometry Pre-K to Grade 3: What Children can do, and how Teachers can support them.
We are working towards the Canadian Mathematics Education Forum in Vancouver, May 2009. The main page for the forum is http://www.cms.math.ca/Events/CMEF2009/
As an initial task, we are working to generate a 'landscape' of geometry/visual reasoning landmarks for early childhood, with some sequencing among the pieces.
While there is a fairly rich landscape for early numeracy, we have not found an landscape for early geometry. There is an consensus among the many people we have talked with, that such a landscape would assist the development of this field.
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Many strands of research confirm that young children live in 3-D, and can master a range of geometry far richer than the current curriculum offers. Some research (e.g. Clements) shows that delaying the introduction actually develops misconceptions which make the material harder, later.
Related work over several decades in Quebec (Denes, Gaulin) and Ontario (del Grande) seems to have been lost in the fog of ongoing curriculum revisions which target 'numeracy' and 'pre-algebra'.
The developments within the NCTM Standards to not seem to address these concerns and appear to work within a box which does not recognize the capacities and the signiifance of these abilities. Scattered research indicates that spatial abilities actually decline during schooling (particularly in the 9-12 grades) and that difficulties faced by students who have become weak in these abilities.
Failure to develop these abilities in students who are weak has significant consequences later, and failure to engage these abilities in students for whom these are strengths has profound, and discriminatory impact on their futures.
On the other hand, the current curriculum, and the current preparation of pre-K to Grade 3 teachers does not support this richer work. A recent study by one of the participants showed that this general description applies to current Ontario classrooms, both teachers and students.
A second recent report (CIIM report at http://www.edu.gov.on.ca/eng/studentsuccess/lms/) confirms that even the limited curriculum is very uneven in its implementation, including the use of visual and hands-on materials which would support these abilities.
We start from the work of those who have carefully observed young children (e.g Froebel, Montessori, Piaget, Clements), and work in a currennt SSHRC project on the concerns, abilities, and learning trajectories of in-service elementary teachers. We will also draw on resources developed in the work of Denes, Gaulin, del Grande, and others.
From this, we start with a range of critques, questions, tentative suggestions and possible materials and activities which could modify classroom experiences, the curriculum, and the assessment as well as the necessary professional development of pre-service and inservice teachers for pre-K to Grade 3 geometry and spatial reasoning.
The goals include: - a more complete review of the research literature in this area, including research on gesture and kinesthetic reasoning / memory in children pre-K to 3, - a collection of resources in accessible forms (e.g a wiki) drawn from a variety of sources, including current practices in alternative programs (e.g. Waldorf, Montessori, Froebel), - some reports on experiences in classrooms, - some reports on work with pre-service and in-service teachers. - where appropriate, study the use of technology (e.g. computer games, GSP, Cabri Junior, … ) to support these activities, in conjunction with manipulatives and body activities.
We continue to seek contacts with other projects to generate ‘out-of-the-box’ proposals with supporting evidence and resources.
We would be delighted to have contacts with those in other regions working on related issues and resources.
Doug Clements, formerly of University of Buffalo and his collaborators have done some in-depth work:
Here are some other links to his work
- Building Blocks Curriculum : http://www.ubbuildingblocks.org
- Conference on Early Math Standards : http://www.gse.buffalo.edu/org/conference/
- Professional Development in Preschool Mathematics: http://www.gse.buffalo.edu/org/profdev/
This page is under construction
A focus on Symmetry
At the CMEF in Vancouver, we have been exploring this topic bringing different backgrounds and goals. After some exploration, including the pages on 'Big Ideas in Geometry", we agreed to focus on the theme of 'symmetry'.
We discussed why geometry had continued to be in decline in North American curriculum revisions, and what factors would contribute to making it 'essential' to writers of curriculum. One factor we identified was its connections to areas of science, art, design and math. The future of Geometry is in the realization it is necessary for a number of areas outside of mathematics, and these connections are part of the
We outlined a sequence of tasks covering different stages of the trajectory from early childhood through to work beyond school in life and in intellectual areas including mathematics, design, sciences, arts, ... .
(a) early children's capacity to perceive, and clear enjoyment of play with symmetry and develop symmetry; (b) role of affect and imagination in learning symmetry; (c) development through pre-school programs and early elementary school; (d) rich in meaning and value of structure and reasoning with composition / decomposition creation of symmetric objects)(trajectories beyond classification); (e) embodied gestural ways of knowing symmetry; (f) "Why do now?" modeling activities during the school years? (g) "Why symmetry is important" - studies university and in key areas beyond mathematics;
We will work on these through: - an annotated bibliography (to appear on this wiki); - coauthor an advocacy article for this topic; - initial presentations: CMESG 2009 Ad Hoc (June 2009), possible AERA proposal, MAA January 2010 meeting session on Visualization in Mathematics.
Walter Whiteley May 2009
The evidence I am collecting points to the lack of solid research about what topics are appropriate, what materials, activities, adult interactions best support geometry learning for young children ... .
Most importantly, they suggest no consensus on 'learning trajectories' or 'learning landscapes' - what landmarks and pieces are around, and how they connect/sequence, at least in small chunks. I have not found that the current curriculum reflects much more than 'received wisdom' inherited from generation to generation and curriculum to curriculum. Too often it actually wanders with no real goal, and very limited learning from age 5 to age 10.
So the rich work of Froebel (the man who invented Kindergarten in the 1850's) seems isolated in its emphasis on 3-D, and symmetry/decomposition prior to 2-D (with more symmetry and play).
The work of Clements (above) does give some key pointers about the need for a good range of examples/non-examples for whatever students are learning (in his case primarily 2-D concepts). It also reinforces concerns that waiting to do some key ideas actually makes it harder for children, as they develop misconceptions and seriously incomplete conceptions of objects such as triangles, rectangles etc. - both in shape and in orientation.
Already, there are some lessons which publishers could take to heart, about showing a good variation within the 'sample space' of triangles (not just the iconic, almost equilateral, or perhaps almost isosceles) triangles as illustrations and exercises - as well as a good selection of 'non-examples' so the boundaries of the concept are clarified. It would be good to contribute some analyses of current texts which do, and do not, respect this finding.
Walter Whiteley May 8, 2008
A few of us had a supper meeting at the CMESG meeting in Sherbrooke. We now have (informally) about 10 members, including two at the University of Sherbrooke, two in Alberta, and about six in southern Ontario.
Go to http://www.founders.net/ to download the Early Years Study 2. This is a rich collection of background material and references and action plans, focusing on 0-5, but with an emphasis showing the great importance of ages 0-3. Note that, like many other studies, there is a conflation of mathematics with numeracy. There is an interest in broader connections to spatial reasoning, pattern recognition, etc. which are also developing (or developing deficits) in those years, but not yet a broad literature or general recommendations.
See also the more general pages on geometric / spatial reasoning: http://wiki.math.yorku.ca/index.php/Spatial_Reasoning_Links
A site at the Ontario Ministry related to early childhood implementation is:
There are some interesting position papers from the National Kindergarten Alliance (US) at
The following papers are downloadable: Childhood a Time for Play; and: Visual Learning is More Than Meets the Eye
We might think about a submission related to early childhood geometry!
If you are interested in joining - contact Walter Whiteley at firstname.lastname@example.org
Members of the group (April 2009) who will be at the CMEF include: Lynn McGarvey Susan Gerofsky Pam Hagan Geri Lorway