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Teaching and Assessing Mathematics Using Learning Progressions

Focus on R&D

Issue 3

August 2016


By: Hans Sandberg

Learning progressions are tools for teaching as well as tools for designing educational assessments. In this issue of Focus on ETS R&D, ETS experts Gabrielle Cayton-Hodges, Caroline Wylie and Jeff Haberstroh explain what mathematics learning progressions are and how they can be used.

What do we mean by learning progressions?

Gabrielle Cayton-Hodges: Learning progressions, also known as learning trajectories, are getting more attention in teaching and assessment, so ETS is not alone in doing research on this topic. They play a major role in ETS's Cognitively Based Assessment of, for, and as Learning (CBAL®) initiative, which has developed learning progressions for mathematics, English Language Arts, and science. There are several definitions of learning progressions. Here is the one we use in the CBAL initiative:

"… a description of qualitative change in a student's level of sophistication for a key concept, process, strategy, practice, or habit of mind. Change in student standing on such a progression may be due to a variety of factors, including maturation and instruction. Each progression is presumed to be modal — i.e., to hold for most, but not all, students. Finally, it is provisional, subject to empirical verification and theoretical challenge." (Deane, Sabatini, & O'Reilly, 2012)¹

Why are we working on learning progressions for mathematics? How do they relate to the Common Core State Standards (CCSS)?

Gabrielle Cayton-Hodges: The standards state what students need to learn and the learning progressions describe typical paths students take to get there. Many good mathematics teachers are already using something similar to learning progressions. A recent volume by Clements and Sarama (Learning and Teaching Early Math: The Learning Trajectories Approach, New York, 2014)² showed that mathematics teachers whose in-class discussions prove most valuable to learning see their job as helping students move through a progression, or through increasingly more sophisticated understandings, rather than just moving through the curriculum.

ETS sees learning progressions as important for understanding the domains we are assessing for both summative and formative assessment, for teacher professional development and for enhanced score reporting. Recently, ETS scientists began to develop learning progressions for mathematical practices, i.e., ways that people use mathematical content and do mathematics in the real world.

Some may wonder if the CCSS and the learning progressions are in conflict with one another. In fact, this concern is far from the truth. Several authors of the standards are experts on learning progressions or conceptual development of mathematics (i.e., moving toward comprehension of mathematical concepts, operations, and relations). The standards articulate specific core ideas that students are expected to master. In some cases learning progressions can be seen as connecting standards across and within grade levels and in other cases providing a "ladder" toward achieving a specific standard. We should also remember that some standards are more specific, while others cover a more general idea and that the interactions between the learning progressions and standards will reflect that quality.

We have, through CBAL and other initiatives, mapped most of the CCSS mathematics standards to one of our core learning progressions, documenting which level or levels of the learning progression are represented by each standard. In this way, we are able to design assessment items and tasks not only to a standard or to a learning progression, but to both simultaneously.

Can mathematics learning progressions support formative assessment?

Caroline Wylie: Think of formative assessment practice as built around a series of questions that teachers and students can ask to figure out how far the students have come: Where are we going? Where are we now? How do we close the gap? In other words, what are the learning goals? How are students doing against those goals? And what are the next instructional steps that can move student understanding closer to the goals? Learning progressions provide content-specific responses to those questions.

For example, a learning progression on fractional and decimal understanding could begin with students' initial understanding of how to divide something in a fair way (e.g., in half) although they may not be able to express that understanding mathematically. The learning progression lays out the typical developmental sequence, which starts with that intuitive understanding, and shows how it matures: students can divide both a unit and a group into fractional parts; students later understand a fraction as a number that can be placed on a number line; students begin to integrate fractional understanding with decimals; students develop initially basic and then more sophisticated ways to manipulate both fractions and decimals.

A teacher who understands this learning progression may develop a mathematical unit focused on translating between fractions and decimals, which (the teacher recognizes) maps to the higher levels of the learning progression. The learning progression helps the teacher set both short-term and long-term learning goals (to think about where learning is going). It can also inform the teacher about the kind of evidence needed to understand students' current level of learning before, during, and after the unit. The teacher realizes that observing what students say, write or do can help identify any students who are still struggling with basic understanding of fractions. The teacher can go beyond interpreting student work as "right or wrong," and can understand why students make the mistakes they often do. Understanding where they are with respect to the learning progression can then help the teacher plan instruction and learning activities to close the gap between current and intended learning.

Teachers can use learning progressions to help guide both the development of assessment tasks and of classroom discussions. The progressions can also help teachers interpret evidence and make informed judgments about how students in a class might differ in terms of their understanding of key concepts and practices. Instructional leaders, coaches, and administrators can use the learning progressions as a foundation for content-based professional development, to deepen teachers' understanding of how students' learn and to improve teachers' formative assessment practice.

How can these progressions be used to develop assessment questions and to report assessment results?

Jeff Haberstroh: We in ETS's Assessment Development area look for evidence of both students' understanding of the mathematical content and how well they apply their knowledge in different situations. Learning progressions help us think a little differently about how to write assessment questions. That is, we consider how to develop questions that can assess a certain standard, while also giving insights into how a student progresses towards understanding that standard. This approach may be particularly helpful in identifying what students who struggle in their study of mathematics do understand, and not just what they do not.

This approach means that we need to think early on in the development process about the interplay of learning progressions and standards, while keeping in mind that one goal is to create questions that give teachers and students useful feedback. To work toward that goal, we need to analyze how the learning progression relates to the intent of a particular assessment standard.

In some instances, it is possible to connect a learning progression to more than one standard, which can result in a richer assessment experience, since questions then can help us assess similar levels of understanding, but of different mathematical content. These multiple connections can also add depth to the reporting of assessment results and further guide the process of instruction.

What are we doing to evaluate how well the learning progressions reflect student development?

Jeff Haberstroh: Our research and development work includes several iterative steps that contribute to this goal: 1) completing reviews of the existing literature to inform the development of the progressions, 2) conducting cognitive interviews with students to test the initial assumptions behind the progressions, 3) obtaining reviews of the progressions by internal and external experts, and 4) developing tasks that map to the progressions and carrying out larger-scale data collections to further examine whether the progressions provide a meaningful lens through which to interpret student responses.³

What impact could learning progressions have on mathematics education?

Caroline Wylie: I believe that the knowledge we and other researchers are developing about how students learn will be beneficial for beginning and experienced teachers, and for the students too. Deborah Ball and her colleagues at the University of Michigan have, for example, studied how specialized knowledge about teaching intersects with what they call high-leverage teaching practices. They have been leading research around what they call mathematics knowledge for teaching (MKT), which goes beyond knowing and being able to use mathematics flexibly and fluently. MKT refers to the specific mathematical knowledge teachers need to teach students mathematics, and requires both knowledge of the students and the content. Such knowledge allows teachers to recognize how students think when they are doing mathematics — their errors, justifications, misconceptions, developmental sequences and so on. It relates to what Ball and others have described as "horizon knowledge" — that sense of where learning goes beyond what might be relevant to a particular grade level. While MKT includes more than we have described here, learning progressions capture an important aspect of MKT. The work around MKT has been continued and extended by colleagues at ETS, in particular in the area of secondary mathematics.

Is there one thing that you would want people to remember about learning progressions?

Caroline Wylie: Standards describe learning outcomes by grade level or band. Progressions provide insight into what learning might look like when students have not yet mastered the full understanding. They give students, teachers, administrators, researchers and assessment developers insights into important stages of developing understanding on the way to mastery.

Gabrielle Cayton-Hodges is a Research Scientist in ETS's R&D division, Jeff Haberstroh is a Principal Assessment Designer in ETS's Assessment Development area and Caroline Wylie is a Research Director in ETS's R&D division.

Learn more

Learning Progressions: Maps to Personalized Teaching (Education Week, November 9, 2015)

Using Argumentation Learning Progressions to Support Teaching and Assessments of English Language Arts (R&D Connections, No. 22, November 2013)

Formative Assessment – Supporting Students' Learning (R&D Connections, No. 19, June 2012)

1 Deane, P., Sabatini, J., & O'Reilly, T. (2012). The CBAL English language arts (ELA) competency model and provisional learning progressions. Retrieved from

2 Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach. Routledge.

3 Graf, E. A., van Rijn, P. W. (2015). Learning progressions as a guide for design: Recommendations based on observations from a mathematics assessment. Handbook of test development, Edition: 2nd, Chapter: 9, Publisher: Routledge, Editors: Suzanne Lane, Mark R. Raymond, Thomas M. Haladyna, pp.165–189.