# Arc of Learning Research Project

**Project Start Date**: December, 2013

**Project Team: **Elizabeth Phillips, AJ Edson, Yvonne Grant, Jacqueline Stewart, Nic Gilbertson, Funda Gonulates, Jen Nimtz, Molade Osibodu, & Rani Satyam

### Background

The curriculum development of Connected Mathematics has been guided by the following premise:

*All students should be able to reason and communicate proficiently in mathematics. They should have knowledge of and skill in the use of the vocabulary, forms of representations, materials, tools, techniques, and intellectual methods of the discipline of mathematics, including the ability to define and solve problems with reason, insight, inventiveness, and technical proficiency.*

*Connected Mathematics* is a problem-centered curriculum promoting an inquiry-based teaching-learning classroom environment. The mathematical ideas are identified and embedded in a carefully sequenced set of tasks and explored in depth to allow students to develop rich mathematical understandings.

### Description

The Arc of Learning framework is a resource for curriculum design and use that makes explicit the intentions of the curriculum designers about how students engage in the learning of mathematics over time. The Arc of Learning framework highlights the way students’ mathematical learning can evolve from informal knowledge to more sophisticated reasoning over time. It moves the learning focus beyond the analysis of isolated tasks to consider the role of a problem and its location in an instructional sequence that promotes mathematical learning. The Arc of Learning framework grew out of a need to highlight the mathematical learning embedded in problems in problem-based curricula like Connected Mathematics.

The framework consists of five stages that describe the development of understanding of important mathematical concepts and practices within CMP:

*Introduction (Setting the Scene)*: Students explore problems that reveal the mathematical theme and informally highlight the key mathematical concepts. The problems also provide an opportunity to assess what students bring to the lesson in terms of the goals of the Unit.*Exploration**(Mucking About)*: Students explore problems that establish a platform for developing key aspects of concepts and strategies. Students consider and explore a context that students can use to build, connect, and retrieve mathematical understandings.*Analysis**(Going Deeper)*: Students solve problems with a variety of contextual situations and examine nuances in key aspects of the core mathematical ideas. Students make connections between concepts and representations.*Synthesis*(*Looking Across*): Students consolidate and refine their emerging mathematical understanding(s) into a coherent structure. They recognize core ideas across multiple contextual or problem situations. Students begin to generalize their mathematical ideas and strategies.*Abstraction**(Going Beyond)*: Students make judgments about which representations, operations, rules, or relationships are useful across various contexts. Students look back on prior learning to generalize, extend, and abstract the underlying mathematical structure and provide teachers with opportunities for assessing student understandings at a more sophisticated level.

### Purpose

The Arc of Learning can inform:

- how student thinking and learning is targeted and how that thinking and learning might unfold within and across mathematics sequences of problems,
- how teachers understand the development of long-term mathematical goals embedded in a coherent sequence of problems,
- the mathematical, pedagogical, and assessment decisions teachers make when planning or enacting lessons that respond to students’ mathematical conceptions,
- professional development to support problem-based learning,
- research on problem-based curricula, and
- the design of focused, problem-based curriculum.

### Resources

- Arc of Learning General Framework
- Grade 6 Unit:
*Comparing Bits and Pieces* - Grade 6 Unit:
*Let’s Be Rational* - Grade 7 Unit:
*Stretching and Shrinking* - Grade 7 Unit:
*Comparing and Scaling* - Grade 7 Unit:
*Moving Straight Ahead* - Grade 8 Unit:
*Looking for Pythagoras* - Grade 8 Unit:
*Growing, Growing, Growing* - Publications and Presentations