You learn what you get opportunities to experience, explore, write, and talk about
Teachers are asked to regularly plan for instruction. Frequently, the lesson plans are organized around the activity that will be completed for the lesson, rather than around goals for what they will teach students (Clark, 1978). Often the plans focus on a list of events to occur during the lesson and rarely is thought given to a necessary change in the lesson based on student thinking. (Kagan & 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, D. and Bill, V., NCTM in press). Stein argues that setting mathematical learning goals provides teachers with guidance on how to design and structure their lesson, making it clear to students what they are to glom onto and make use of from the lesson (Stein, NCTM in press). 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 that they would engage their students in deep and specific math discussions (Russell, Stein, Correnti, Bill, 2016, IES Grant).
Are all goals created equal?
We recently examined the math 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. We found ourselves setting two different types of goals: performance goals and mathematical learning goals.
- Performance goals are simply the activities students will engage in during the lesson.
- Mathematical learning goals identify the deep underlying mathematics that students will understand and also name how the students will demonstrate their understanding. (Huinker, D. and Bill, V. NCTM in review)
These goals are NOT created equal. We were shocked by the difference in our planning and our lessons when we set mathematical learning goals, instead of just setting performance goals. Additionally, in a 3-year IES study with the state department of education in TN, we found that coaches also saw a difference in the effectiveness of lessons by changing from simply stating performance goals to unpacking math learning goals with teachers.
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 you are making groups of an amount less than one OR you 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 1 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 mathematical learning goals state what is true mathematically when (1) a whole number is divided by a fraction less than 1 and (2) a fraction less than 1 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 math 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 that student “grapple” with the underlying math ideas targeted in the lesson. When a student is solving 4 ÷ 1/3 and 1/3 ÷ 4 we now know, based on the math learning goals, what we are looking for in the responses below that show understanding of each of these ideas.
The diagram of 4 ÷ 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 in order to press them to talk about the mathematical learning goal of the lesson. By posing questions, we will be able to gather information and check on student understanding of the mathematical ideas (Boaler and Brodie 2004). Take a minute and analyze the responses to the questions below and see what we can learn about students understanding of mathematics.
Shannese understands the difference between the quotients for each division problem. She notes that the amount being partitioned in each situation differs and 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 her response is grounded in an algorithm and we do not yet know if Susan knows how to represent the mathematics or knows what the amounts represent.
The math 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. Knowing that Keisha’s reasoning was incomplete, notice how we pressed the class to add greater clarity
- T: Some of you say that you cannot divide 1/3 by 4 because you cannot divide a smaller number by a number that is greater. Is that true?
- S1: No, you can divide any number by any number.
- S2: Yeah, look at my drawing—it shows that you can divide 1/3 by 4.
- T: Come share your drawing with us. Tell us how you drew a model of 1/3 divided by 4 and what it means. (Student shares diagram above of 1/3 divided by 4.)
- S2: I know that division is breaking into parts, so I have to take 1/3 and break it into 4 parts. So I did that and got each piece is 1/12 .
- T: James, I see you nodding that you agree with Shonda. Come on up and tell us what you understand about Shonda’s model.
- S3: Shonda drew 1 whole and then she marked 1/3 because it talks about only 1/3 . Then she divided it into 4 equal parts. So those are the small pieces and each one is 1/12 because they have to be smaller than 1/3 .
- T: I want to hear that back from two more students. First, Shannese and then Keisha. How did he know to name the size of the piece as 1/12 ?
- S4: The question is asking for 1/3 divided into 4 equal parts and so when you take 1/3 and make 4 equal parts out of it, then each of those pieces are 1/12 in size, because if you cut up all of the thirds into three pieces, there would be 12 pieces. You have to do this to figure out that the pieces are twelfths.
- T: Okay, Keisha?
- S3: Well at first I thought you couldn’t do this but you can take 1/3 and cut it into 4 equal pieces and each one is 1/12 because you can fit 12 pieces of them in 1 whole.
The learning goal acts as a gauge for us when we monitor student responses. We knew by Keisha’s initial response that she did not recognize that it was possible to divide fractions less than 1 by whole numbers. By the end of the conversation, as you see in the transcript, the class was able to finally say, “ 1/3 divided by 4 is 1/3 divided into 4 equal parts, and each of those pieces is 1/12 in size.”
We also worry about Susan’s progress in understanding the learning goal. It is clear that she has a procedure for determining the solution, but we do not know if she understands the underlying meaning of each division problem. So we asked her to share her thinking with the class. See how the discussion makes it possible for Susan to add reasoning herself.
- T: Susan, you used what we call the traditional algorithm for dividing with fractions. Can you please come show the class what you did?
- S4: I flipped the second number so that turns into 3 over 1, so really 3, then you change the division to multiplication and you multiply 4 by 3 and get 12.
- T: Joel, you also had 4 x 3 in your work. Can you please share your diagram? (Student shares diagram above of 4 divided by 1/3 .)
- S6: I did have 4 times 3 but it’s because I was figuring out how many thirds are in each 1.
- T: I’m going to have you stop there. Susan, take a look at Joel’s number line. Can you find 4 x 3 in his diagram or from what he said?
- S4: Umm, I think I see it. He said he was trying to find how many thirds are in each one. Can I come up and count? (Teacher nods and invites Susan up to Joel’s diagram.) Let me count. I think there are 3 thirds in one, two, three, four wholes. So there’s three four times.
- S7: Oh, I get it! Three thirds are in each one whole and there are four wholes!
The questions we planned required students to find and discuss the meaning of the operations in relationship to the size of the pieces. Each of the situations was presented in the context of a situational problem in order to help student make sense of the amounts. Students were also encouraged to make a diagram. Students in this classroom are used to being challenged to explain the meaning of the quantities, as well as to explain how to model their thinking visually. As a result, students come to view this way of working as the norm for doing mathematics. Although the students are used to explaining their reasoning, without probing for students’ mathematical reasoning we cannot be sure what students are and are not understanding related to the mathematical learning goal. We want to ensure that students all walk away from a given lesson knowing the important underlying mathematics or that we at least know where they are in their understanding.
If the teacher had only set performance goals for the lesson, then the teacher would have simply focused on students’ ability to perform the computation. The teacher would not have pressed students for the meaning of the operation. The teacher would not have cared if students could draw conclusions about the size of the partitioned portion when dividing a fraction less than one. Learning goals push for depth of understanding and students are better mathematicians when meeting math learning goals, rather than performance goals.
How do you write mathematical learning goals that help you know explicitly what you should hear from students when they understand the underlying mathematics or help you write questions that 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. It would be great to create a bank of these math learning goals. Please share in the comments section below or reach out to Victoria Bill or Laurie Speranzo
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