Part 2: Responding to The Power of Mathematics Learning Trajectories in Early Learning and Primary Classrooms
Edward Schroeter’s article is simply lovely—this is exactly what we had hoped learning trajectories (LTs) would offer teachers/caregivers and their children. Ed and his co-teacher, Katherine Benz used the LTs to learn more about the mathematics (clicking on the video icon next to the LT’s title provides an overview), the levels of thinking in the developmental progressions, and the instructional activities and teaching strategies that help a child operating mainly at one level develop the next one.
In Part 2 of this 2-part blog, I provide additional reactions to Ed and Katherine’s commendable teaching and provide background and more information about our Learning and Teaching with the Learning Trajectories tool (LTLT, or LT2, see Learning Trajectories).
In part 1 of this 2-part blog, you met Ed Schroeter and Katherine Benz, two Ontario Kindergarten educators who are on a mission to improve their ability to teach math. They are representative of many early primary teachers and early childhood educators who believe they have not received sufficient training in mathematics. This is the situation in which most teachers across the continent find themselves. The in-service preparation in mathematics that most teachers receive is lacking1.
We could see Ed and Katherine started using learning trajectories and our Learning and Teaching with the Learning Trajectories tool (LTLT, or LT2) because of their growing awareness of the need for both teachers and children to learn more about mathematics.
My colleague and wife, Dr. Julie Sarama, and I created the Building Blocks curriculum (Clements & Sarama, 2007/2013) with funding from the National Science Foundation. Through a series of evaluations (Clements & Sarama, 2007, 2008); Clements, 2011 #4177} and scale-up projects (Clements, Sarama, Wolfe, & Spitler, 2013), we became increasingly aware of the importance—and difficulty—of helping teachers learn about learning trajectories. LT2 is a major upgrade of the internet-based tools we created to successfully do so (Clements, Sarama, Wolfe, & Spitler, 2015; Sarama, Clements, Wolfe, & Spitler, 2016).
Ed and Katherine’s experiences illustrate two important benefits of using learning trajectories and the LT2 tool. First, the approach opens up worlds of learning for teachers like Ed and Katherine and of course for the children in their care. Did you see Ava productively struggling with the notion that “a shape that matches” could mean congruence (same shape, same size) or same geometric figure category? She moved from just learning shape names to constructing shapes and even composing shapes to make new shapes. Did you see Ava using problem-solving, communicating, and reasoning? Critical mathematical processes.
Second, this approach provides master teachers like Ed and Katherine with a valuable resource for the powerful teaching strategy of formative assessment. There are three key questions in using formative assessment—and they line up with LTs nicely:
|Formative Assessment Questions||
Learning Trajectory’s Part
|Where are you trying to go?||LT Goal|
|Where are you now?||LT’s developmental progression—where are children now and what are next levels?|
|How can you get there?||LT’s activities; feedback and links to strategies for improvement|
No matter what curriculum you use, research shows that understanding children’s thinking and learning in this way helps you be a more effective and reflective teacher.
Children delight in learning mathematics when it is taught as an active, playful way of thinking.
This brings up an important point for teachers of the youngest children (infants to preschoolers); what about the position that some advocate that any “curriculum” is developmentally inappropriate and children should just play and not learn mathematics? We suspect a lot of the passion behind that viewpoint is many peoples’ belief that children should have fun rather than suffer through mathematics! That is, many people remember their own mathematics education as tedious and painful. Who wants that for children, especially the youngest? The good news here is that children delight in learning mathematics when it is taught as an active, playful way of thinking. By opening up new windows to seeing young children and the inherent curiosity and creativity behind their mathematical reasoning, learning trajectories ultimately make teaching more joyous.
Further, opportunities to have high-quality mathematics experiences is a critical equity issue. Opportunities to learn early mathematics are more frequent in some communities and families than in others (for a review, see Clements, Fuson, & Sarama, 2017). This opportunity gap negatively affects children who live in poverty and who are members of linguistic and ethnic minorities. Again, the good news is that high-quality learning experiences based on learning trajectories help close that gap in opportunities, resulting in greater school readiness. To do so, teaching must go beyond incidental experiences and be intentional. Guidance is provided in Table 1 below.
…it’s a surprisingly rich and joyful domain for children’s playful explorations and learning of other domains such as executive functions and language.
As stated previously, most of us also have not had many opportunities to learn about early mathematics, and my colleague Julie Sarama and I are trying to help those interested in doing so. Research is clear that early mathematics is surprisingly important. For example, it is the main predictor of later school success! And it’s a surprisingly rich and joyful domain for children’s playful explorations and learning of other domains such as executive functions and language.
How can teachers and caregivers use LT2? To begin, go to Learning Trajectories and view the introductory videos and download a guide (see the link, “LT2 User Instructions”). If you wish to explore- sign in! It’s free and open to all. Then choose a topic that you are thinking about as Ed and Katherine did, and see what level of thinking your children are operating at and what level they then need to develop next.
We are still working on LT2—adding connections to standards and assessments, providing supports for Dual Language Learners, and adding software which will eventually help identify children’s levels of thinking.
We are also adding many videos. But we need videos of teaching and learning from the field—please, contact us if you would like to contribute some!
Finally, we have found that learning trajectories help teachers communicate within age groups/grades, and also to administrators and parents. And learning trajectory-based teaching is much more powerful if done in communities, including across ages from infants through the primary grades. Join us! We would love to hear from you.
Table 1. Developmentally Appropriate Intentional Teaching Practices
(Adapted from Clements, Fuson, & Sarama, 2017; NCTM, 2010a, 2010b, 2010c, 2011)
A. For each math topic, the teacher leads children through research-based learning trajectories. This allows the teacher to differentiate instruction within whole-class, small group, center-based, and individual activities.
B. The teacher expects and supports children’s ability to make meaning and mathematize the real world by:
- Providing settings that connect mathematical language and symbols to quantities and to actions in the world.
- Leading children’s attention across these crucial aspects to help them make connections.
- Supporting repeated experiences that give children time and opportunity to build their ideas, develop understanding, be creative, and increase fluency.
C. The teacher creates a nurturing and helping math talk culture:
- To elicit thinking from students, and to help students explain and help each other explain and solve problems.
D. For later pre-K, Kindergarten, and primary grades, the teacher follows up activities with real 3-dimensional objects by working with math drawings and other written 2-dimensional representations that support practice and meaning-making with written mathematical symbols. Children of all ages also need to see and count groups of things in books; they need to experience and understand 3-dimensional things as pictures on a 2-dimensional surface. Working with and on 2-dimensional surfaces as well as with 3-dimensional objects, supports equity in math literacy because too many children have not had experiences with 2-dimensional representations in their out-of-school environment.
Dr. Douglas Clements, Kennedy Endowed Chair in Early Childhood Learning and Professor at the University of Denver, is a major scholar in the field of early childhood mathematics education, one with equal relevance to the academy, to the classroom and to the educational policy arena. At the national level, his contributions have led to the development of new mathematics curricula, teaching approaches, teacher training initiatives and models of “scaling up” interventions. He has served on the U.S. President’s National Mathematics Advisory Panel, the Common Core State Standards committee of the National Governor’s Association and the Council of Chief State School Officers, the National Research Council’s Committee on Early Mathematics, the National Council of Teachers of Mathematics national curriculum and Principles and Standards committees, and is and co-author each of their reports. He has directed more than 35 funded projects.
Clements, D. H., Fuson, K. C., & Sarama, J. (2017). The research-based balance in early childhood mathematics: A response to Common Core criticisms. Early Childhood Research Quarterly, 40, 150–162.
Clements, D. H., & Sarama, J. (2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 38(2), 136–163.
Clements, D. H., & Sarama, J. (2007/2013). Building Blocks, Volumes 1 and 2. Columbus, OH: McGraw-Hill Education.
Clements, D. H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal, 45(2), 443–494. doi:10.3102/0002831207312908
Clements, D. H., Sarama, J., Wolfe, C. B., & Spitler, M. E. (2013). Longitudinal evaluation of a scale-up model for teaching mathematics with trajectories and technologies: Persistence of effects in the third year. American Educational Research Journal, 50(4), 812 – 850. doi:10.3102/0002831212469270
Clements, D. H., Sarama, J., Wolfe, C. B., & Spitler, M. E. (2015). Sustainability of a scale-up intervention in early mathematics: Longitudinal evaluation of implementation fidelity. Early Education and Development, 26(3), 427–449. doi:10.1080/10409289.2015.968242
NCTM. (2010a). Focus in Grade 1: Teaching with the Curriculum Focal Points. Reston, VA: National Council of Teachers of Mathematics.
NCTM. (2010b). Focus in Kindergarten: Teaching with the Curriculum Focal Points. Reston, VA: National Council of Teachers of Mathematics.
NCTM. (2010c). Focus in Prekindergarten: Teaching with the Curriculum Focal Points. Reston, VA: National Council of Teachers of Mathematics.
NCTM. (2011). Focus in Grade 2: Teaching with the Curriculum Focal Points. Reston, VA: National Council of Teachers of Mathematics.
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.
1This is documented in a recent report my colleagues and I wrote, which also clearly states what we need to do, and stresses the need for early childhood teachers to learn about learning trajectories! Association of Mathematics Teacher Educators. (2017). AMTE Standards for Mathematics Teacher Preparation. Raleigh, NC: AMTE. https://amte.net/standards
The research and development reported here were supported by The Bill & Melinda Gates Foundation, Grant #OPP1118932, and the Heising-Simons Foundation Grant #2013-79 and 2015-157. The opinions expressed are those of the authors and do not represent views of the Foundations.