By focusing on the identification of significant and recognizable clusters of concepts and connections in students’ thinking that represent key steps forward, trajectories offer a stronger basis for describing the interim goals that students should meet if they are to reach the common core college and career ready high school standards. In addition, they provide understandable points of reference for designing assessments for both summative and formative uses that can report where students are in terms of those steps, rather than reporting only in terms of where students stand in comparison with their peers.
If substantially all students are to succeed at the hoped-for levels, it will not be sufficient just to meet the “opportunity to learn” standard of equitably delivering high- quality curricular content to all students, though that of course is a necessary step. Since students’ learning, and their ability to meet ambitious standards in high school, builds over time—and takes time—if they are to have a reasonable chance to make it, their progress along the path to meeting those standards really has to be monitored purposefully, and action has to be taken whenever it is clear that they are not making adequate progress. When students go off track early, it is hard to bet on their succeeding later, unless there is timely intervention.
The concept of learning progressions offers one promising approach to developing the knowledge needed to define the “track” that students may be on, or should be on. Learning progressions can inform teachers about what to expect from their students. They provide an empirical basis for choices about when to teach what to whom. Learning progressions identify key waypoints along the path in which students’ knowledge and skills are likely to grow and develop in school subjects (Corcoran, Mosher, & Rogat, 2009). Such waypoints could form the backbone for curriculum and instructionally meaningful assessments and performance standards. In mathematics education, these progressions are more commonly labeled learning trajectories. These trajectories are empirically supported hypotheses about the levels or waypoints of thinking, knowledge, and skill in using knowledge, that students are likely to go through as they learn mathematics and, one hopes, reach or exceed the common goals set for their learning. Trajectories involve hypotheses both about the order and nature of the steps in the growth of students’ mathematical understanding, and about the
nature of the instructional experiences that might support them in moving step by step toward the goals
of school mathematics.
The discussions among mathematics educators that led up to this report made it clear that trajectories are not a totally new idea, nor are they a magic solution to all of the problems of mathematics education. They represent another recognition that learning takes place and builds over time, and that instruction has to take account of what has gone before and what will come next. They share this with more traditional “scope and sequence” approaches to curriculum development. Where they differ is in the extent to which their hypotheses are rooted in actual empirical study of the ways in which students’ thinking grows in response to relatively well specified instructional experiences, as opposed to being grounded mostly in the disciplinary logic of mathematics and the conventional wisdom of practice. By focusing on the identification of significant and recognizable clusters of concepts and connections in students’ thinking that represent key steps forward,
trajectories offer a stronger basis for describing the interim goals that students should meet if they are to reach the common core college and career ready high school standards. In addition, they provide understandable points of reference for designing assessments for both summative and formative uses that can report where students are in terms of those steps, rather than reporting only in terms of where students stand in comparison with their peers. Reporting in terms of scale scores or percentiles does not really provide much instructionally useful feedback.
However, in sometimes using the language of development, descriptions of trajectories can give the impression that they are somehow tapping natural or inevitable orders of learning. It became clear in our discussions that this impression would be mistaken. There may be some truth to the idea that in the very early years, children’s attention to number and quantity may develop in fairly universal ways (though it still will depend heavily on common experiences and vary in response to cultural variations in experience), but the influence of variations in experience, in the affordances of culture, and, particularly, in instructional environments, grows rapidly with age. While this influence makes clear that there are no single or universal trajectories of mathematics learning, trajectories are useful as modal descriptions of the development of student thinking over shorter ranges of specific mathematical topics and instruction, and within particular cultural and curricular contexts— useful as a basis for informing teachers about the (sometimes wide) range of student understanding they are likely to encounter, and the kinds of pedagogical responses that are likely to help students move along.
Most of the current work on trajectories, as described has this shorter term topical character. That is, they focus on a particular mathematical content area—such as number sense or measurement— and how learning in these areas develops over a few grades. These identified trajectories typically are treated somewhat in isolation from the influence of what everyone recognizes are parallel and ongoing trajectories for other mathematical content and practices that surely interact with any particular trajectory of immediate concern. The hope is that these delimited trajectories will prove to be useful to teachers in their day-to-day work, and that the interactions with parallel trajectories will prove to be productive, if arranged well in the curriculum. From the perspective of policy and the system, it should eventually be possible to string together the growing
number of specific trajectories where careful empirical work is being done, and couple them with curriculum
designs based on the best combinations of disciplinary knowledge, practical experience, and ongoing attention to students’ thinking that we can currently muster, to produce descriptions of the key steps in students’ thinking to be expected across all of the school mathematics curriculum. These in turn can then be used to improve current standards and assessments and develop better ones over time as our empirical knowledge also improves.
With this goal in mind, I offer the following recommendations:
• Mathematics educators and funding agencies should recognize research on learning trajectories in mathematics as a respected and important field of work.
• Funding agencies and foundations should initiate new research and development projects to fill critical knowledge gaps. There are major gaps in our understanding of learning trajectories in mathematics. These include topics such as:
»» Algebra
»» Geometry
»» Measurement
»» Ratio, proportion and rate
»» Development of mathematical reasoning
An immediate national initiative is needed to support work in these and other critical areas in order to fill in the gaps in our understanding.
• Work should be undertaken to consolidate learning trajectories. For topics such as counting, or multiplicative thinking, for example, different researchers in mathematics education have developed their own learning trajectories and these should be tested and integrated.
• Mathematics educators should initiate work on integrating and connecting across trajectories.
• Studies should be undertaken of the development of students from different cultural backgrounds and with differing initial skill levels.
• The available learning trajectories should be shared broadly within the mathematics education and broader R & D communities.
• The available learning trajectories should be translated into usable tools for teachers.
• Funding agencies should provide additional support for research groups to validate the learning trajectories they have developed so they can test them in classroom settings and demonstrate their utility.
• Investments should be made in the development of assessment tools based on learning trajectories for use by teachers and schools.
• There should be more collaboration among mathematics education researchers, assessment experts, cognitive scientists, curriculum and assessment developers, and classroom teachers.
• And, finally as we undertake this work, it is important to remember that it is the knowledge of the mathematics education research that will empower teachers, not just the data from the results of assessments.
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