Studies on math homework in secondary classrooms from 2010 and 2012 indicate an average of 15% -20% of class time daily is spent reviewing homework. Given the amount of time dedicated to homework review in class, many education specialists are advocating the use of **discourse in the math classroom** as an instructional strategy that can provide students with opportunities to learn from their homework and from their peers.

The National Council of Teachers of Mathematics (NCTM) defines discourse as the following:

"Discourse is the mathematical communication that occurs in a classroom. Effective discourse happens when students articulate their own ideas and seriously consider their peers’ mathematical perspectives as a way to construct mathematical understandings."

In an article from the National Council of Mathematics Teachers (NTCM) September 2015, titled Making the Most of Going over Homework, authors Samuel Otten, Michelle Cirillo, and Beth A. Herbel-Eisenmann argue that teachers should** "****Reconsider typical discourse strategies when discussing homework and move toward a system that promotes the Standards for Mathematical Practice."**

**Research on Discourse in Review of Math Homework**

Their research focused on the contrasting ways to have students engage in discourse-the use of spoken or written language as well as other modes of communication to convey meaning—in going over homework in class.

They acknowledged that an important characteristic of homework is that "it provides each individual student with the opportunity to develop skills and to think about important mathematical ideas." Spending time in class going over homework also gives students the "opportunity to discuss those ideas collectively."

The methods for their research was based on their analysis of 148 video-recorded classroom observations. The procedures included:

- Observing classroom teachers with varying degrees (novice to veteran) of classroom experience;
- Observing eight middle-grades classes in several different school districts (urban, suburban, and rural);
- Calculating total time spent in various classroom activities compared to total time observed.

Their analysis showed that going over homework was consistently the predominant activity, more than whole-class instruction, group work, and seat work.

**The Review of Homework Dominates the Math Classroom**

With homework dominating all other categories of math instruction, the researchers argue that the time spent going over homework can be "time well spent, making unique and powerful contributions to students’ learning opportunities"* only* if the discourse in the classroom is done in purposeful ways.Their recommendation?

"Specifically, we propose strategies for going over homework that create opportunities for students to engage in the Common Core’s Mathematical Practices."

In researching the kinds of discourse that happened in the classroom, the researchers determined that there were **two "overarching patterns" **:

The first pattern is that the discourse was structured around individual problems, taken one at a time.The second pattern is the tendency for discourse to focus on answers or correct explanations.

Below are details on each of the two patterns were recorded in 148 video-recorded classrooms.

### Pattern #1: Talking Over Vs. Talking Across Individual Problems

This pattern of discourse was a contrast between **talking over homework problems as opposed to** **talking across homework problems**

**In talking over homework problems, **the tendency is the focus is on the mechanics of one problem rather than the big mathematical ideas. The examples from the published research shows how discourse can be limited in talking over homework problems. For example:

TEACHER: "Which questions did you have problems with?"

STUDENT(S)calling out:"3", "6", "14"...

Talking over problems may mean that student discussion can be limited to calling out problem numbers of describing what students did on specific problems, one at a time.

In contrast, the kinds of discourse measured by** talking across problems focus** on the big mathematical ideas on connections and contrasts **between problems. **The examples from the research show how discourse can be expanded once students are aware of the purposes of the homework problems and asked to contrast problems with each other. For example:

TEACHER:

"Notice all that we were doing in previous problems #3, and #6 . You get to practice _______, but problem 14 is making you go even further. What is 14 making you do?"

STUDENT: "It’s different because you are deciding in your head which one would equal that ______ because you’re already trying to equal something, instead of trying to figure out what it equals.

TEACHER: "Would you say the question #14 is more complicated?"

STUDENT: "Yes."

TEACHER: "Why ? What is different?"

These kinds of student discussions involve specific Standards of Mathematical Practices which are listed here along with** their student friendly explanations:**

CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. Student-friendly explanation: **I never give up on a problem and I do my best to get it right**

CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively. Student-friendly explanation:**I can solve problems in more than one way**

CCSS.MATH.PRACTICE.MP7 Look for and make use of structure. Student-friendly explanation: **I can use what I know to solve new problems**

### Pattern #2: Talking about Correct Answers vs. Student Errors

This pattern of discourse was a contrast between **the focus on ****correct answers and explanations** as opposed to** t****alking about student errors and difficulties.**

In the focus on correct answers and explanations, there is a tendency for the teacher to repeat the same ideas and practices without considering other approaches. For example:

TEACHER: "This answer _____ seems off. Because...

(teacher explains how to solve the problem)"

When the focus is on **on ****correct answers and explanations**, the teacher above attempts to help a student by answering what may have been the reason for the error. The student who wrote the incorrect answer may not have the opportunity to explain his or her thinking. There would be no opportunity for other students to critique other student reasoning or justify their own conclusions. The teacher may provide additional strategies for computing the solution, but the students are not asked to do the work. There is no productive struggle.

In the **discourse about**** student errors and difficulties**, the focus is on what or how students thought in order to solve the problem. For example:

TEACHER: "This answer _____ seems off...Why? What were you thinking?

STUDENT: "I had thought _____."

TEACHER: "Well, let's work backward."

OR

"What are other possible solutions?

OR

"Is there an alternative approach?"

In this form of discourse on **student errors and difficulties,** the focus is on using the error as a way to bring student(s) to a deeper learning of the material. The instruction in class can be clarified or complemented by the teacher or student peers.

The researchers in the study noted that "by identifying and working through errors together, going over homework can help students see the process and value of persevering through homework problems."

In addition to the specific Standards of Mathematical Practices used in talking across problems, student discussions on error and difficulties are listed here along with** their student friendly explanations:**

CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.

Student-friendly explanation: **I can explain my math thinking and talk about it with others**

CCSS.MATH.PRACTICE.MP6 Attend to precision. Student- friendly explanation:** I can work carefully and check my work.**

### Conclusions about Math Homework in the Secondary Classroom

As homework will no doubt remain a staple in the secondary math classroom, the kinds of discourse described above should be geared to having students participate in mathematical practice standards that make them persevere, reason, construct arguments, look for structure, and be precise in their responses.

While not every discussion will be lengthy or even rich, there are more opportunities for learning when the teacher is intent on encouraging discourse.

In their published article, Making the Most of Going over Homework, the researchers Samuel Otten, Michelle Cirillo, and Beth A. Herbel-Eisenmann hope to make math teachers aware of how they might use the time in homework review more purposefully,

"The alternative patterns we suggested emphasize that mathematics homework—and, by extension, mathematics itself—is not about correct answers, but rather, about reasoning, making connections, and understanding big ideas."