**By Victoria Bill and Laurie Speranzo, posted July 31, 2017 —**

Teachers are asked to regularly plan for instruction. Frequently, the lesson plans are focused more on the activities that will be completed for the lesson rather than on the mathematical goals of the lesson (Clark 2003). The plans often focus on a list of events to occur during the lesson; rarely do teachers give thought to a necessary change in the lesson on the basis of student thinking (Kagan and Tippins 1992). By setting clear goals as the basis of the lesson, teachers can plan for and then assess student learning during instruction and make corrections to better meet the needs of students (Huinker and Bill 2017). Stein argues that setting mathematical learning goals provides teachers with guidance on how to design and structure their lesson, making clear to students what they are to grasp and make use of from the lesson (Stein 2017). In the recent IES study of math coach-teacher discussions in Tennessee, when coaches focused on having deep and specific discussions of mathematical goals, teachers had an increased chance of engaging their students in deep and specific math discussions (Russell et al. 2015).

**Are all goals created equal?**

We recently examined the learning goals we have been setting for our students. We realized that the types of mathematical learning goals that we were associating with our lessons made a difference in how we planned and carried out our lessons. Mathematical learning goals identify the deep, underlying mathematics that students will understand; they also name how students will demonstrate their understanding (Huinker and Bill 2017).

Let us give you two examples of what we describe as mathematical
learning goals. Students will discover, when making a diagram or drawing a
number line model, that when dividing by a number less than one, the quotient
will be greater than the dividend, because either they are making groups of an
amount less than one *or* they are
making less than one group.

Students will discover, when making a diagram or a number line model, that when dividing a fraction less than one by a whole number, the quotient will be less than the dividend, because the dividend is being partitioned into additional parts. Therefore, the quotient is less.

These math learning goals state what
is true mathematically when (1) a whole number is divided by a fraction less
than one and (2) a fraction less than one is divided by a whole number. With a
clear sense of the specific mathematical understandings that are being targeted
in the lesson, we know what to listen and look for in student responses.
Because the mathematical learning goals also name *how* students will demonstrate their understanding of the math,
teachers know what to look for as well as what to listen for.

With the goals set, we can select a task that requires students to grapple with the underlying math ideas targeted in the lesson. When a student is solving 4 divided by 1/3 and 1/3 ÷ 4, we now know, on the basis of the mathematical learning goals, what we are looking for in the responses below that show understanding of each of these ideas.

The diagram of 4 divided by 1/3 shows 4 wholes partitioned into groups that are each 1/3 in size, and there are 12 of those groups of 1/3. The diagram of 1/3 ÷
4 shows 1/3 partitioned into 4 equal pieces, each
of which is 1/12 in size. With these two mathematical
learning goals in mind, we are positioned to identify questions that we can ask
students to press them to talk about the mathematical learning goal of the
lesson. ** **

Shannese understands the difference between the quotients for each division problem. She notes that the amount being partitioned in each situation differs, and she recognizes that one situation is a partitioning of the four wholes, whereas the other is a partitioning of one-third into four parts.

Keisha’s response indicates that she does not recognize situations in which the dividend can be less than the divisor. Susan has the correct answer; however we do not yet know if Susan knows how to represent the mathematics or knows what the amounts represent.

The mathematical learning goal can serve as our guide during the lesson, prompting us to continue to press students to talk about the size of the amounts and the underlying reason why the magnitude of the quotient differs in each situation. As you can see, the mathematics learning goal can act as a gauge for us when we monitor student responses.

**Your
Turn**

How do you write mathematical learning goals that help you know explicitly what you should hear from students when they understand the underlying mathematics? How do you write mathematical learning goals that help you write questions to focus the discussion on the mathematics? Please share your mathematical learning goals, because it will help all of us know exactly what students should understand about a concept. Creating a bank of these mathematical learning goals would be useful. Please share in the comments section below or reach out to Victoria Bill or Laurie Speranzo.

**REFERENCES**

Boaler, Jo, Karin Brodie, Tobin White, Emily Shahan, Jennifer DiBrienza, and Nick Fiori. “The Importance, Nature, and Impact of Teacher Questions.” In Proceedings of the 26th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education 2, pp. 773–81, Toronto, Ontario, Canada, October 21, 2004.

Clarke, Shirley. 2003. *Enriching Feedback in the Primary Classroom*. London: Hodder and Stoughton.

Huinker,
DeAnn, and Victoria Bill. 2017. *Taking Action: Implementing Effective
Mathematics Teaching Practices in Grades K–5*. Reston, VA: National Council
of Teachers of Mathematics.

Kagan, Dona M., and Deborah J. Tippins.
1992. “The Evolution of Functional Lesson Plans among Twelve Elementary and Secondary
Student Teachers. *The Elementary School Journal* 92, no. 4 (March): 477–89.

Russell, Jennifer Lin, Richard Correnti, Mary Kay Stein, Maggie Hannan, Victoria Bill, Nate Schwartz, Laura Booker, and Nicole R. Pratt. “Using Data for Improvement to Support Implementation at Scale: Adaptive Integration in the TN Mathematics Coaching Project.” National Center on Scaling Up Effective Schools Second National Conference, Nashville, TN, October 2015.

Stein, Mary
Kay, and Erin Meikle. 2017. “The Nature and Role of Goals in and for Mathematics
Instruction.” In *Enhancing Classroom Practice with Research behind
"Principles to Actions,"* edited
by Denise A. Spangler and Jeffrey J. Wanko, pp. 1–11. Reston,
VA: National Council of Teachers of Mathematics.

**Victoria Bill**
is a resident fellow with the Institute for Learning (IFL) at the Learning
Research and Development Center, University of Pittsburgh, Pennsylvania. She
also the co-author, with DeAnn Huinker, of the 2017 NCTM publication *Taking
Action: Implementing the Effective Teaching Practices in Grades Pre-K–5*. **Laurie Speranzo** is a resident fellow
with the Institute for Learning (IFL) at the Learning Research and Development
Center, University of Pittsburgh, Pennsylvania. In addition to her work at the
IFL, she has recently been appointed to serve on the editorial panel of NCTM’s *Mathematics
Teaching in the Middle School* journal.

## Leave Comment

## All Comments

Steven Khan- 12/10/2017 4:28:19 PMThank you for the blog post. I work with pre-service teachers and re-wrote your learning goals to a smaller grain size that is useful for novices:

1a. Students will DISCERN that: when dividing a whole number by a number less than one (rational fraction), the quotient will be greater than the dividend THROUGH making a diagram or drawing a number line model. [now set up the learning experiences]

1b. Students will EXPLAIN the reason for this as either making groups of an amount less than one or they are making less than one group. [in writing, orally?]

2a. Students will DISCERN that: when dividing a fraction less than one by a whole number, the quotient will be less than the dividend THROUGH making a diagram or a number line model.

2B. Students will EXPLAIN the reason for this as the dividend is being partitioned into additional parts. Therefore, the quotient is less.

I put the mathematical idea first BEFORE the representation or model else the model/representation becomes the focus. I use the word DISCERN in place of DISCOVER, a personal preference, but I feel the idea of sense-making is better captured in the word discernment than discovery (and it is coming from the research on Variation Theory). I also include an EXPLAIN learning goal (for new teachers this needs to be explicit).

I am looking forward to such a bank being created. If someone knows of a good starting point, maybe post a link here.

## Reply processing please wait...