Early childhood educators are becoming increasingly aware of the fundamental importance and efficacy of putting developmentally appropriate learning paths at the centre of mathematics education for children from birth to 8 years of age.
In the last four years, research has revealed that children’s preschool mathematics knowledge predicts math achievement into secondary school. To the surprise of many, it also predicts reading achievement even better than early reading skills. Doing more mathematics actually increases oral language abilities such as vocabulary, inference, grammatical complexity and independence.
Four-year-old Ava Smiley, an Ontario kindergarten student, practises sorting incongruent rectangles—rectangles of different heights and widths. The activity, from the developmental level on the Shape Learning Trajectory, proved to be a little advanced for her. She needs to practise skills from the previous developmental step, Constructor of Shapes from Parts—Looks Like.
Wilma Armstrong is a kindergarten teacher at Millbrook-South Cavan Public School in Ontario who began using mathematics learning trajectories for assessment and programming in September 2017.
“Math learning trajectories have been useful in programming and evaluation in our Kindergarten program. I refer to the site LearningTrajectories.org (LT2). Each area of each math strand is broken down into small tasks. I can look to see what is developmentally appropriate for a child at a certain age and then find out if they are demonstrating that particular learning and possible next steps.”
Mathematics learning trajectories are one of several “learning paths,” a general term for a number of approaches that include a developmental continuum. In addition to mathematics learning trajectories, there are growth points, learning progressions, developmental continua, developmental maps, diagnostic maps, landscapes, pathways and/or cognitively guided instruction (CGI). Some of the mathematics educators who have developed these learning paths include Douglas Clements and Julie Sarama, Catherine Twomey Fosnot, Lynn McGarvey, John A. Van de Walle, Alex Lawson, and Sue Willis (First Steps Mathematics).
The learning paths are all slightly different, but as Mrs. Armstrong’s classroom experience attests, they are more detailed, clear, precise, developmentally appropriate, and helpful for assessing student learning and for designing instruction than provincial mathematics curricula. Learning paths do not replace mathematics curriculum but, instead, support it.
They are particularly important because a considerable body of evidence suggests that early childhood and primary grade math educators need a great deal more knowledge to be able to help children develop mathematical competencies—much more than a bare curriculum can provide. In their 2016 article, “Professional development in early mathematics: effects of an intervention based on learning trajectories on teachers’ practices,” co-written with C. B. Wolfe and M. E. Spitler, Dr. Julie Sarama and Dr. Douglas Clements report that this is due in part to low levels of both math and pedagogical content knowledge among many early grade educators.
This unfortunate situation seems to be improving as developmentally appropriate learning paths move from research to the classroom and are integrated into best teaching practices. Mrs. Armstrong, for example, adopted the mathematics learning trajectories (LTs) approach by Clements and Sarama in 2017 and is currently using it in her kindergarten classroom because the math trajectories are clear, comprehensive and intuitive. They are also supported by a companion website for educators and parents, Learning and Teaching with Learning Trajectories (LT2) at LearningTrajectories.org. It is free and open access thanks to the Heising-Simons Foundation and the Bill and Melinda Gates Foundation.
Math learning trajectories consist of three parts:
- a learning goal (aka: a target, benchmark, expectation),
- a developmental path along which children develop to reach that goal
- a set of activities matched to each of the levels of thinking in that path that help children develop the next higher level of thinking.
Explain Clements and Sarama: “Curriculum research has revealed sequences of activities that are effective in guiding children through the developmental levels of thinking. These developmental paths are the basis for the learning trajectories.”
The developmental levels in learning trajectories help educators and curriculum developers assess students’ abilities, determine their next steps in learning, and provide effective activities that build math competence. “One of the most powerful techniques for teaching… is the use of formative assessment: knowing where kids are in their thinking and knowing how to get them to the next step.”
According to the two researchers, learning trajectories allow teachers to build the mathematics of children through guided play as it develops naturally. “We know that all the goals and activities are within the developmental capacities of children. We know that each level provides a natural developmental building block to the next level. Finally, we know that the activities provide the mathematical building blocks for school success.”
The tried, tested and true activity suggestions contained in each learning trajectory on the LT2 website and in the quintessential book, Learning and Teaching Early Math: The Learning Trajectories Approach, are an added bonus for busy classroom teachers.
Says Wilma Armstrong: “The LearningTrajectories.org website offers many different types of support for small and large group learning. I have viewed several short videos under the heading “You May See This.” In them, students are asked to demonstrate their learning by completing a variety of tasks. The questions and prompts modelled in this section help me to assess the learning of my students. The site also provides small group and whole group activities for suggestions of follow up activities.”
Learning trajectories have been thoroughly researched as well as tested in classrooms with students. Any remaining skeptics can review the body of research, which is widely available on the Internet. “Through planned teaching of early childhood mathematics and by also encouraging informal, incidental mathematics, teachers can help children learn at an appropriate and deep level,” say Dr. Clements and Dr. Sarama.
Learning trajectories are extremely valuable to teachers and their students another way. Clements and Sarama explain that research shows that when teachers understand how children develop mathematical understanding, they are more effective in questioning, analyzing and providing activities that further children’s development than teachers who are unaware of the development process. “Consequently, children have a much richer and more successful math experience in the primary grades.”
Comparing one of the mathematics learning trajectories to a provincial curriculum standard illustrates its efficacy and ease of use. One of the “specific expectations” (standards) for Geometry and Spatial Sense on page 38 of The Ontario Curriculum, Grades 1-8: Mathematics is that by the end of Grade 1 students should be able to identify and describe common two-dimensional shapes (e.g., circles, triangles, rectangles, squares) in concrete and pictorial form, as well as to sort and classify them by their properties (e.g., size, number of sides and vertices).
This sounds simple enough. However, based on Clements’ and Sarama’s website, before children can sort and classify shapes meaningfully, children first have to be able to:
- match and fit basic shapes into congruent puzzle spaces or container openings using their physical properties,
- recognize whether two objects are the same or different shape,
- match identical familiar shapes (circle, square, typical triangle) of the same size (congruent) and orientation, then match familiar shapes of different orientations, and finally match familiar shapes of different sizes,
- recognize and name familiar typical shapes (circle, square, typical triangle) of different sizes and orientations,
- match a wider variety of shapes, then less typical squares and triangles and possibly some rectangles (not rhombi),
- use objects to represent parts of shapes (e.g., popsicle sticks for sides) and to make approximations of goal shapes, and finally,
- recognize sides and angles as separate geometric objects.
When set out on a linear continuum, part of the Shapes learning trajectory for four-year-old children looks like this: Shape Matcher—Identical, Orientations, Sizes ➡ Shape Recognizer —Typical Shapes ➡ Shape Matcher – More Shapes, Sizes, & Orientations ➡ Constructor of Shapes from Parts ➡ Shape Recognizer – All Rectangles ➡ Side Recognizer ➡ Corner Recognizer… .
There are some misconceptions about learning trajectories. Many of these learning paths were developed by U.S. researchers and publishers. Canadian teachers may be reluctant to use math learning trajectories or the LT2 site, unsure whether they are compatible with Canadian provincial curriculum documents. Mathematics, however, is an international language. It crosses national borders.
The adoption of learning trajectories is not a panacea for improved mathematics learning in the early years. Other research by Clements and Sarama indicates that intensive, job-embedded professional development is also important for improving educator efficacy and student learning. “Without training, teachers may fail to implement new approaches faithfully. For example, teachers may reduce the cognitive demand of instructional tasks after their initial introduction. As another example, teachers need training in understanding, administering, and especially using data from new assessment strategies, essential strategies in the effective use of learning trajectories.”
References
Clements, Douglas H., and Sarama, Julie. (2014). Learning and Teaching Early Math: The Learning Trajectories Approach, 2nd edition. New York City: Routledge.
Clements, Douglas. (2013). Math in the Early Years: A Strong Predictor for Later School Success. The Progress of Education Reform. October, Vol 14, No. 5. Denver, CO: Education Commission of the States. ecs.org/clearinghouse/01/09/46/10946.pdf
Fosnot, Catherine Twomey. Contexts for Learning Mathematics (CFLM): Investigating Number Sense Grades K-3, Level 1. (2008-2017). Portsmouth, NH: Heinemann.
Lawson, Alex. (2016). Student Continuum of Numeracy Development: What To Look For: Understanding and Developing Student Thinking in Early Numeracy. Toronto: Pearson Canada.
McGarvey, Lynn. (2018). Mathematics Learning Progressions. Toronto: Pearson Canada.
New Zealand Ministry of Education. (2017). The Learning Progression Frameworks – Mathematics. Retrieved online November 12, 2017 from: lpfm.brackenlearning.com/Live/ frameworks/mathematics.aspx
Ontario Ministry of Education. (2016). The Kindergarten Program 2016, pp 218-248. Toronto: The Queen’s Printer. edu.gov.on.ca/eng/curriculum/elementary/kinderprogram.html
Ontario Ministry of Education. (2005). The Ontario Curriculum, Grades 1-8: Mathematics. Toronto: Queen’s Printer. edu.gov. on.ca/eng/curriculum/elementary/math18curr.pdf
Sarama, J., Clements, D. H., Wolfe, C. B. & Spitler, M. E. (2016). “Professional development in early mathematics: effects of an intervention based on learning trajectories on teachers’ practices.” Nordic Studies in Mathematics Education, 21(4), 29–55. researchgate.net/project/Evaluating-the-Efficacy-ofLearning-Trajectories-in-Early-Mathematics
Van de Walle, John A.; Lovin, LouAnn; Karp, Karen; and Bay-Williams, Jennifer. (2014). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades Pre-K-2, 2nd edition. Toronto: Pearson.
Willis, Sue. (2013). First Steps Mathematics. Perth, Australia: Department of Education, Western Australian. Creative Commons Licence. Retrieved online from det.wa.edu.au/ stepsresources/detcms/navigation/first-steps-mathematics/.
Edward Schroeter
Edward Schroeter, B.J., B. Ed., OCT, Reading Specialist, Kindergarten Specialist, Special Education Part 2, FSL-1, Library Resource-1, was trained and worked as a newspaper reporter and documentary video producer. He is a six-time national award-winning playwright and producer of theatre for young audiences. He has taught Core French, French Immersion SK, Grade 3 French Immersion, Grade 4 and kindergarten for the Kawartha Pine Ridge District School Board and the Peterborough County Board of Education, for the past 24 years.
This article is from Canadian Teacher Magazine’s Spring 2018 issue.