Part 1: The Power of Mathematics Learning Trajectories in Early Learning and Primary Classrooms

Originally posted on The Learning Exchange by Edward Schroeter | March 2, 2018


I vividly recall one fateful Professional Activity Day at the end of June 2016 when the Vice-Principal of my K to 8 elementary school turned to her kindergarten and grade 1 to 3 teachers and pointed out research that shows children’s preschool mathematics knowledge predicts mathematics achievement even into high school and that surprisingly it predicts later reading achievement even better than early reading skills. Doing more mathematics actually increases oral language abilities, including vocabulary, inference, independence and grammatical complexity (Math in the Early Years: A Strong Predictor for Later School Success, p 2).

My colleagues and I looked at each other in shock. How did we not know about this research? The focus of the past decade of our professional development had been reading and writing. Mathematics PD was just becoming a provincial priority. We needed to know a lot more about teaching and learning math.

Our learning journey began that day. In Part 1 of this 2 part blog, I document a portion of our exciting experience using mathematics learning trajectories and an online support tool for them.

Ava Smiley is a reserved, April-born, 4 ½ year old, first-year kindergarten student in my classroom. I noticed that she spends a lot of her free play time working with shape sorters, pattern block puzzles and pentominoes. Her interest in these materials emerged clearly during the course of her first 3 months of enrolment in Ontario’s play-based, inquiry learning approach kindergarten program (2016).

I decided to sit down with Ava to collect more information about her geometry and spatial reasoning interests and skills. When my co-teacher, Katherine Benz, RECE, and I reviewed our pedagogical documentation of Ava – photos, observations, video clips and brief transcripts of our conversations with her – we noticed that:

  • when given four typical shapes in the same colour, she could point to the shapes and name them: circle, triangle, rectangle, square
  • when answering the question, “How do you know this shape is a … ?” her answer changed from “I don’t know” to tracing the outline of a shape with her fingers to picking up my language and referring to the number sides
  • when given four typical shapes in two sizes and the same colour, she could group the different-sized shapes into pairs
  • when given a set of six yellow paper rectangles of different sizes and orientations she first called them “rectangle squares” but quickly switched to the term rectangles, was confused when I asked her to “put together the shapes that match” because they were all slightly different heights and widths and she was looking for an exact match (congruency), but knew they were all rectangles
  • she could make a square out of two red triangles, a rectangle out one red square and two red triangles, but had difficulty making a rectangle out of one yellow square and one red square, and,
  • she could use pattern blocks to make a simple “picture” and describe it orally (“a car”), as well as fill simple pattern block puzzles where each shape represents a unique role e.g., a body part

Ava sorting yellow rectanglesAs a member of the 2017-18 Kindergarten Educator Spatial Reasoning Collaborative Inquiry, a professional development initiative developed by an 11-member team of like-minded educators and financed by a Teacher Learning and Leadership Program (TLLP) from the Ontario Ministry of Education, I have discovered that geometry and spatial thinking are extremely important to mathematics development. Research shows an early link between spatial and mathematics skills as well as a strong relationship between them, suggesting that math skills can be improved by strengthening spatial and geometric skills. The highly respected early mathematics researchers Dr. Douglas Clements and Dr. Julie Sarama have conjectured that geometry may be a gateway skill to the teaching of higher-order mathematics thinking skills. The Ontario kindergarten document itself recognizes the fundamental importance of spatial reasoning for young children’s math development. It states on page 75 that “spatial thinking skills and geometric reasoning play a critical role in the development of problem-solving skills, mathematical learning, and reading comprehension.”

Katherine and I decided that spatial reasoning and geometry would be key learning for Ava and that she needed our help to develop more vocabulary and concepts in this realm of math.

The question for Katherine and I was exactly what to do next to advance Ava’s mathematical learning. Our pedagogical documentation revealed much about her thinking, but it had limitations. I wanted to know precisely what her next step in learning and mathematical thinking would most likely be. I was looking for a clear mathematical learning goal and activities which would foster this learning and Ava’s mathematical thinking. What should they be?

During the TLLP project I have learned that in order to develop mathematics concepts children need a variety of experiences, including intentional, educator guided, playful experiences. My curriculum, the 328-page Ontario Kindergarten Program 2016 document, advocates this form of direct but playful instruction. The document states on page 75 that “the presence alone of mathematics in play is insufficient for rich learning to occur” and that “intentional, purposeful teacher interactions are necessary to ensure that mathematical learning is maximized during play.”

There were a wide variety of classroom resources at our disposal: wooden unit blocks, puzzles, pattern blocks, shape sorters, pentominoes, snap cubes, tangrams, Magna-Tiles and more. I was also aware of numerous resources which could assist us:

  1. The Trent Math Education Research Collaborative
  2. Mathematics for Young Children
  3. The Robertson Program: Inquiry-Based Teaching in Mathematics and Science (lesson study teams)
  4. The Robertson Program: Inquiry-Based Teaching in Mathematics and Science (math lessons)
  5. Taking Shape: Activities to Develop Geometric and Spatial Thinking
  6. Learning and Teaching Early Math: The Learning Trajectories Approach, 2nd Edition

There was one more hurdle – my scant mathematics teacher training and minimal knowledge of geometry: 69% in grade 12 math circa 1974. I, like many of my kindergarten and early primary school educator colleagues, received inadequate professional training for effective mathematics teaching and in particular do not have sufficient geometric knowledge to adequately assess or teach geometry to children at any level – even the youngest learners.

Ultimately, we turned to mathematics learning trajectories to help us determine a mathematics learning goal for Ava and learning activities to help her achieve this goal. I owned the books Taking Shape: Activities to Develop Geometric and Spatial Thinking by Joan Moss, Catherine D. Bruce, Bev Caswell, Tara Flynn and Zachary Hawes (Pearson, 2016) and Learning and Teaching Early Math: The Learning Trajectories Approach, which I consider the quintessential book on early math instruction. Research and classroom testing have shown Mathematics Learning Trajectories (LTs) to be an effective vehicle for assessment for learning (formative), assessment as learning (reflection and self-assessment) and assessment of learning (summative). LTs are also a key component of educator professional development in mathematics instruction.

The book is comprehensive to the point of being daunting for an Ontario kindergarten educator like me. I teach a class of 27 students aged 3 to 5 years led by one teacher (OCT), me, and Katherine, an early childhood educator (RECE). Workload is always a pressing issue.

Fortunately, I happened upon the free, web-based tool for educators and parents, Learning Trajectories. It was created by none other than Clements and Sarama. It is open access thanks to the Heising-Simons Foundation and the Bill and Melinda Gates Foundation. The site is relatively new, so it is still growing and improving.

The site, known as Learning and Teaching with Learning Trajectories (or LTLT or LT2 for short), is designed to explain how children from birth to age 8 learn mathematics and how to teach mathematics to them. Learning trajectories are at the heart of this web tool.

The LT2 site addresses mathematics developmental progressions in four areas: number, operations and algebraic thinking, geometry and measurement and data. Each strand is composed of several mathematics learning trajectories. For example geometry contains LTs for Shape, Composing 2D Shapes, Composing 3D Shapes, Disembedding Geometric Figures, Spatial Orientation and Spatial Visualization.

LT2 site on Ed Schroeter blog

Each LT consists of a half dozen to two dozen developmental levels. Each developmental level has its own page which provides one or more videos of children performing playful tasks at level and tested and true large and small group activity suggestions designed to foster and consolidate the skills of that developmental level. If in comparing a student to the video you determine that he or she is not at level you can also click up or down to the next developmental level on the LT. The more you explore and use the site, the more familiar you become with the levels in each trajectory, connecting them to your experience with students, eventually becoming fluent with the mathematical goals, the LT and related activities.

In Ava’s case, I put our pedagogical documentation into the LT2 site’s Student Progress tracker. After examining a few of the developmental levels and their videos on the LT2 Shape learning trajectory, I concluded that Ava is a Shape Recognizer—Circles, Squares and Triangles. (She recognizes some less typical squares and triangles and may recognize some rectangles, but not usually rhombuses. She may call a rhombus a “diamond”).

According to the site, Ava’s next step (mathematical goal) on the Shape trajectory would be Constructor of Shapes from Parts—Looks Like. When she reaches this developmental level she will be able to use manipulatives such as straws to represent parts of shapes, such as their sides, to make a shape which looks like a goal shape. She will learn to build shapes with the correct properties. For example, the sides of squares should all be the same length and the angles would all be right angles, though she might think of angles as a corner and call them “pointy.” There are two recommended small group activities designed to help her achieve the goal of this level. You can even download a lesson plan with key mathematical points.

On the Composing 2D Shapes trajectory Ava seems to be a Piece Assembler. At this level, a child can make pictures in which each shape represents a unique role (e.g., one shape for each body part) and fill simple “Pattern Block Puzzles” using trial and error. Ava’s next mathematical goal on the Composing 2D Shape trajectory would be Picture Maker. At this level a student can put several shapes together to make one part of a picture (e.g., two shapes for one arm), use trial and error, does not anticipate creation of new geometric shape, chooses shapes using “general shape” or side length, and fills “easy” “Pattern Block Puzzles” that suggest the placement of each shape. LT2 offers two computer activities and two small group activities, including six slightly complex pattern block puzzles, all designed to help develop these skills. The puzzles and lesson plans can be downloaded.

LT2 Picture Maker from Ed Schroeter's blogLT2 Pattern Block from Ed Schroeter's blog

Armed with mathematical goals in Shape and Composing 2D Shapes for Ava, and activities to help her achieve them, Katherine and I set out with the aid of the LT2 assessment videos and activities to find students at the same developmental level as Ava. At that point, the final piece of puzzle, organizing small group instruction for them, would fall into place.

Katherine and I are making mathematics learning trajectories a fixture in our Ontario kindergarten classroom. They have proven themselves to be powerful assessment, teaching and professional development tools for early mathematics education.

In Part 2 of this blog, the prominent early mathematics researcher and reformer Dr. Douglas H. Clements reacts to our struggle to improve our teaching practices, touches on early math instruction in North America, and provides background on the topic of early mathematics as well as information about his and Dr. Julie Sarama’s new Learning and Teaching with Learning Trajectories tool (LT2, see



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.





Association of Mathematics Teacher Educators. (2017). AMTE Standards for Preparing Teachers of Mathematics. Raleigh, NC.

Clements, Douglas H., & Sarama, Julie. (2014). Learning and Teaching Early Math: The Learning Trajectories Approach, 2nd edition. New York City: Routledge.

Ontario Ministry of Education. (2016). Kindergarten Program 2016. Toronto: The Queen’s Printer.

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.

Clements, D. H., & Sarama, Julie. (2011). “Early childhood teacher education: The case of geometry.” Journal of Mathematics Teacher Education. April 2011. DOI: 10.1007/s10857-011-9173-0

Moss, J., Bruce, C. D., Caswell, B., Flynn, T., & Hawes, Z. (2016). Taking Shape: Activities to Develop Geometric and Spatial Thinking. Toronto: Pearson Canada.

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