# Exploring Mathematical Content & Determining a Mathematical Goal

Updated: Feb 5

After meeting to discuss __hopes and dreams for students__, narrowing in on a “north star” research area, and determining __a research question and theory of action__, lesson study teams are ready to choose __a specific content area for their research lesson__. As a team it is helpful to discuss the following prompts to choose an area:

**What content do students find it challenging to understand?****What content***do the team members find it challenging*to understand and/or to teach well?**What content is coming up in the curriculum plan?**

Our __10th grade research team chose Quadratics__, our __8th grade team__ decided to explore negative exponents, and our __3rd grade team__ chose comparing fractional quantities.

There are a couple key parts of the lesson study inquiry process where significant learning can occur. Researching a content area thoroughly is one of them. During this ‘study’ stage of the planning process the research team explores *how students develop their understanding of the important concepts throughout the unit* and *how these understandings (or mental schemas) grow over time from one year to the next*. They also research *how the key concepts are connected to other core ideas within the discipline *and how to support students in making these connections.

We have found that there is a tendency for first time research teams to perfunctorily complete this part of the inquiry process, copying over the standards from whichever resource they use (their curricular guide, the state standards, etc...), without attending to *how students might make sense of the core concepts. *In contrast, most seasoned research teams thoroughly explore these questions during this portion of their inquiry cycle. First time research teams can deepen their learning during this phase by consulting resources that provide examples of student thinking. Great sources include:

**Student work!***This is always a fantastic source for understanding how students think about mathematical concepts.***Young Children’s Mathematics***–**details how students develop an understanding of quantity and develop counting skills – the foundation for addition, subtraction, multiplication, and division.***Children’s Mathematics: Cognitively Guided Instruction***– details how students develop deep numeracy and problem solving skills***Extending Children’s Mathematics: Fractions and Decimals***– details how students develop an understanding of fractional quantities and how to solve problems that involve them*series**Developing Essential Understanding of...***– there are concept specific books for both middle and high school core mathematics content*

Once the lesson study team has researched how students might grapple with the mathematics they wish to teach, they select some options for a lesson task and * do the math together*. Each member of the research team first solves the problem as an adult, then, thinking about the focus student they will observe, they try to anticipate how that student might approach the problem. From these responses the team examines what the

**mathematical**

**understanding****goal**might be. What

*new understanding*will students develop from doing this lesson?

It is important to recognize that a mathematical understanding goal is distinct from a performance goal. A performance goal indicates something students* do*: “students will write a linear equation to represent a pattern,” but this differs from what students *understand. *Mathematical understanding goals attend to *how a student is making sense of a particular concept*. For example:

*Students will understand that 10 can be used as a benchmark number in order to create numbers up to 20. For example, in calculations such as 10 +5 is 15 and 17-7 is 10**Students understand that the constant multiple, m, in the equation y=mx+b is the amount that y changes when x changes by 1 unit*

In addition, mathematical understanding goals often represent a small piece of a larger content standard goal. For example, the 3rd grade research team wanted to focus on the following standard:

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

However, the research team knew that there were many different types of fractions that students could compare, and that students often didn’t realize that fractions could be more than one whole. They decided to focus their lesson on the following mathematical understanding goal as one part of exploring the broader standard:

*Students understand that fractional quantities can be represented as less than a whole or more than a whole and can use this as a source of comparison*

A great post about the difference between performance goals and understanding goals can be found here for those wanting to dig deeper. With this key understanding goal in place the team was now ready to think about their equity goal.

*For educators looking to try lesson study in their own contexts, we found that:*

*Lesson study teams were most engaged when***they chose a content topic that they wanted to know more about themselves, or that their students found challenging to understand****Rich learning can occur during the content research phase if the team focuses on how students think and make sense of the content topic***by looking at student work, or resources that contain examples of student thinking.***Mathematical understanding goals are distinct from performance goals**and are more useful for teams to determine the depth of student learning during the lesson

*Previous week:* __Identifying a Research Question and Theory of Action__

*Next week: *__Determining an Equity Goal__

*Daisy Sharrock works at the *__Center for Research on Equity and Innovation__* at the *__High Tech High Graduate School of Education__*, and is part of a *__Student-Centered Learning Research____Collaborative-sponsored__* research team that is currently engaged in the following study: Leveraging the Power of Improvement Networks to Spread Lesson Study. Read more about their current study *

__here__*. We are grateful to*

__JFF__*,*

__KnowledgeWorks’__*, and the Student-Centered Learning Research Collaborative and its funders for their support. Learn more at*

__sclresearchcollab.org__