5 Deep Mathematics Learning Made Visible

Copyright © Erin Null

Fifth-grade teacher Mrs. Wolf is about to begin a unit on volume. Students enter the classroom to find a container of one-inch cubes and some graph paper at each set of desks. On the board is the following task.

The Build-a-Block Company sells children’s cube-shaped building blocks. Each block is 1 inch long. They want to sell the blocks in boxes of 36. Use the tools at your table groups and what you know about measurement to design a box that would hold 36 cubes.

Students set off to work with little explanation from Mrs. Wolf. When the first group comes up with a solution, she stretches their thinking by asking how they would describe their box to someone on the phone. This starts a discussion among the group members while others continue to work. Groups that finish early are challenged to find other possible boxes that will hold the thirty-six cubes. When all of the groups have found at least one solution to the problem, Mrs. Wolf has each group post its solution about the length, width, and height of their boxes in the table in Figure 5.1, which is on the front board. They are told to leave the volume columns blank for now.

As students complete the chart, Mrs. Wolf challenges them with the following prompt. “Use any patterns you see to convince me that we found all of the possible boxes that can hold thirty-six cubes.”

The discussion that continues brings up a variety of important mathematical questions. Students debate whether a 3 × 4 × 3 box is the same as or different from a 3 × 3 × 4 box. They start to organize the table at their seats to show that they have indeed found all of the possible combinations. They notice that if they multiply the dimensions, the product is thirty-six, which is the number of blocks in the box.

The next day, Mrs. Wolf introduces the learning intention, which is for students to find patterns that will help them generalize finding the volume of a rectangular solid. Students will know they have achieved understanding when they are able to find the volume of any rectangular solid based on their understanding of patterns. That is the success criterion, and students know it. Mrs. Wolf then facilitates a discussion in which students tie together the mathematical ideas from the “Box Problem.” Through their discussion, they begin to recognize that the size and shape of their boxes have something to do with how many cubes would fit inside, which creates an initial understanding of volume—a new term that Mrs. Wolf introduces and defines. Mrs. Wolf then explicitly guides students to see how the discoveries they have made can be generalized for finding the volume of any rectangular solid. Through purposeful questions and prompts, she helps the students make the connection between their patterns (multiply the three dimensions) and the formula for volume (L × W × H) as a way of demonstrating how volume can be found more efficiently than always stacking cubes in imaginary boxes. At this point, they work in groups to complete the volume column of their table. To continue deepening this learning about volume, and move students toward transfer, Mrs. Wolf will go on to other tasks that week that give students the opportunity to apply the same thinking to different rectangular solids, and more complex shapes composed of several rectangular solids.

Figure 5.1 A Table for Student Recording on the Box Problem

Mrs. Wolf intentionally began this class unit using a high cognitive demand task that required students to use their previously mastered skills about measuring three-dimensional objects in a way that resulted in deeper learning and, ultimately, extension into a new idea. Notice, there was no formal introduction to this particular exploration task other than answering questions individual groups might have. Both deep and surface learning happened over these two days in Mrs. Wolf’s class. Students built on previous experiences to see patterns and make generalizations, which enabled them to enter into a new conceptual understanding that they will continue to explore throughout the deep phase of their learning. And of course, if students began to struggle as they moved deeper, Mrs. Wolf knew she could always pull back and inject a dose of surface learning to support the students who needed it.

The Nature of Deep Learning

Deep learning focuses on recognizing relationships among ideas. During deep learning, students engage more actively and deliberately with information in order to discover and understand the underlying mathematical structure. This is particularly linked to mathematical practices 3, 7, and 8 in the Common Core and the mathematical process standards or higher order thinking standards in other states and countries that speak to displaying, explaining, and justifying mathematical ideas and arguments; communicating; interpreting; reasoning; and analyzing mathematical relationships and connections (see a nonexhaustive list of state and national practice standards in Appendix C).

By way of example, we’ll look at Common Core Math Practices (MPs) and the Texas Essential Knowledge and Skill (TEKS) process standards, as those are specifically described.

Common Core MP 3: Construct viable arguments and critique the reasoning of others.

TEKS G: Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

This practice requires students to engage in active mathematical discourse. Discourse reaches beyond discussion because it includes ways of representing, thinking, talking, agreeing, and disagreeing. It is the way ideas are exchanged and what the ideas entail. It is shaped by the tasks in which students engage as well as by the nature of the learning environment (NCTM, 1991). This might involve having students explain and discuss their thinking processes aloud, share and explain their representations, or indicate agreement/disagreement with a hand signal. Students who are using this practice are able to explain their thinking and articulate the relationships they see in mathematics. They are able to ask questions to help clarify an argument another student makes and to reason about what makes one case stronger than another.

Discourse is about the exchange of ideas, including ways of representing, thinking, talking, agreeing, and disagreeing.

In the case of Mrs. Wolf’s lessons on volume, students used this practice to justify their thinking by proving that their individual solutions were accurate, debating whether packages with the same dimensions in a different order were the same or different, and proving that they had determined all of the possible box shapes and sizes.

Video 5.1 Deep Learning: Applying Understanding to Mathematical Situations

http://resources.corwin.com/VL-mathematics

Common Core MP 7: Look for and make use of structure.

TEKS B: Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

TEKS F: Analyze mathematical relationships to connect and communicate mathematical ideas.

Students who engage in such practices focus on identifying and evaluating efficient strategies to reach a solution. For example, primary students can use a hundreds chart to count by tens starting with any one-digit number. They might recognize that the tens digit increases by one each time they count down in a vertical column on the chart. Middle school students might find patterns in a table to guess a rule and complete the table.

Common Core MP 8: Look for and express regularity in repeated reasoning.

TEKS D: Communicate mathematical ideas, reasoning, and their implications using multiple representations including symbols, diagrams, graphs, and language as appropriate.

These practices are about students noticing when a calculation or pattern happens over and over. This requires teachers to plan lessons that invite students to recognize the repetition. It calls for teachers to recognize any “a-ha” moments students have and to facilitate discourse so that students can make connections, describe patterns, and generalize. The goal of this practice is for students to look for efficient ways to describe and use repetition to develop fluency.

Figure 5.2 Representing 4 × 30

Source: Clipart courtesy FCIT, http://etc.usf.edu/clipart.

The fourth graders in Mr. Hunt’s class are working on multiplication of two-digit times one-digit numbers. They use objects on a place value chart (Figure 5.2) to physically show what happens when you multiply a one-digit number times multiples of ten. After several concrete examples, Mr. Hunt demonstrates how to record the equations using numerals. Students continue with several additional examples, when Helen is suddenly very excited and announces, “All of the answers end in zero!”

A few things are happening here. Helen notices repetition, and Mr. Hunt needs to seize the moment to help students make connections to place value and to generalize why that pattern works. It is also a good time to remind students of precision with vocabulary. He replies, “What do you notice about the place value of the one-digit factor when you are multiplying by a multiple of ten?” Students look over the examples they have completed. Students are a bit reluctant to respond, so Mr. Hunt continues with another question to guide their thinking. “What do all of the examples we have been solving this morning have in common?” His question nudges students to focus on finding similarities and connecting them to make a generalization. Billy says, “One of the factors is a multiple of ten and the other is a one-digit number.” Mr. Hunt then asks them to consider what is happening to the place value of the one-digit factor. Helen responds, “Since it is ten times greater, the place value of the factor moves to the tens place because I have ten times as many.” Students need time to explore the pattern they have found and connect it to mathematical understanding (the place value of the digit moves one place to the left), which lays the foundation for multiplying by multiples of 100 and by decimals in later grades.

Figure 5.3 Mobile Data Plans

Let’s take a look at a middle school example. In Mr. Bintz’s eighth-grade mathematics class, the students are graphing a set of data about various mobile phone plans. Mr. Bintz gives the class a data chart (see Figure 5.3) and asks the students to work in small groups to create graphs that represent the pricing for at least two plans for various amounts of data used each month.

As the students figure out how to use their data to create graphs to compare plans, their talk focuses on the wide range of both voice and text charges and data charges. As they graph (shown in Figure 5.4), students start to notice that the lines they are drawing are parallel and wonder how that could be given the data they have. Mr. Bintz encourages students to use what they know about lines and functions to figure out why this appears to be true. “Remember, you have to convince your parents and yourself that you’re choosing the most cost-effective plan. What do the parallel lines tell you about the pricing of the various plans?”

Many arithmetic algorithms take advantage of repeated reasoning. If these algorithms are explored at only a surface level, students may know about “just count up the zeroes in the factors and put that many zeroes in the product,” but they won’t know the mathematics behind that pattern or shortcut, about how multiplication relates to place value. So they will miss the beauty of this relationship, and when it’s time to apply this principle in a different context, they will get stuck. In Mr. Bintz’s class, the pricing for data was always at the same rate, just $1/1MB, although that rate was presented in different forms. Students could see that pricing can be presented in various ways to make plans appear more or less appealing depending on the area of concern. A deeper understanding of the mathematics helps students use their learning to understand the world around them.

Figure 5.4 Graphic Representation of Cell Phone Plans

All of these examples are instances of deep learning. Deep learners are able to make connections and think metacognitively—to think about their thinking, discuss ideas, take action, and see errors as a necessary part of learning. Surface learning is an essential part of the learning process by giving students tasks in which they are building a knowledge base of conceptual understanding that connects to flexible and efficient procedures. This prepares students to engage with more cognitively challenging tasks, and to use more rigorous discourse to identify patterns and make generalizations. Remember that this is a cyclical process, and students are constantly moving between surface learning and deep learning.

In the rest of this chapter, we will highlight the nature of mathematical tasks, questioning techniques, and discussion structures that teachers can use both in collaborative groups and in whole-class discussion to help students to make conceptual connections, and begin to apply their thinking to new contexts and situations in order to deepen their learning and move themselves in the direction of the transfer phase of learning.

Selecting Mathematical Tasks That Promote Deep Learning

As we saw in the case of Mrs. Wolf’s class, mathematical tasks that promote deep learning necessitate students to use their surface learning to make connections and find relationships among ideas. These tasks require greater cognitive demand, are usually more open-ended, and have multiple routes to a solution or multiple solutions. By working through these challenges, learners build their deep thinking capacity.

It is important to know how much surface learning is necessary in order to prepare for a related task that leads to deep learning. In the case of Mrs. Wolf’s class and their Build-a-Block containers, the teacher decided to provide students with an opportunity to begin to explore the study of volume. However, if she had not explicitly connected the mathematical experiences inherent in this task with important surface learning (leveraging prior knowledge of measurement and multiplication as well as developing vocabulary), it is likely this would have been an engaging activity that had no real learning outcome. Similarly, students needed to connect the patterns they discovered to the formula for finding the volume of a rectangular solid to determine how the dimensions of different solids were related to volume. The deep learning activity, connected to important surface learning, provided students with opportunities to see how and why generalizations or mathematical rules work!

In Chapter 3, we talked about exercises versus problem solving. Figure 5.5 gives examples of each using the same mathematics concepts. Think about good problems or rich tasks that can help promote deep learning that connects to surface learning.

It is not enough to provide students with rich tasks that support deep learning. Instructional protocols should include implementing whole class activities and small group activities, facilitating discourse, and knowing when to interject an idea or ask a question versus when to let productive struggle happen. A teacher’s finesse in using a combination of these protocols differentiates routine instruction from effective instruction.

Too often, it is the teacher who is doing most of the talking. It is the teacher who is posing questions. It is the teacher who is calling on students, deciding who will speak next and when.

Mathematical Talk That Guides Deep Learning

In a recent study in England, John found that 89 percent of class time was teacher talking (Hattie, 2012). We have seen similar patterns in other countries. It seems that students come to class to watch the teacher work. It is too often the teacher who is posing questions. It is the teacher who is playing traffic cop—calling on students (hands raised, of course) and deciding who will speak next and when. Only about 5 to 10 percent of teacher talk triggers more conversation or dialogue engaging the student. But listening needs dialogue, which involves students and teachers joining together in addressing questions or issues of common concern, considering and evaluating different ways of addressing and learning about these issues.

Too much teacher talk reduces opportunities for students to use their own prior achievement, understanding, sequencing, and questioning. This means less thinking and less learning for kids. Limiting teacher talk is so important for teachers, coaches, and administrators to recognize as crucial in successful student learning! There are several considerations that can help to move the talk away from the teacher and into the mouths of students.

EFFECT SIZE FOR CLASSROOM DISCUSSION = 0.82

Figure 5.5 Exercises Versus Rich Tasks

Accountable Talk

Think of high-stakes discussions you have witnessed in your life. Perhaps you visited a state legislature in session and watched the members debate the merits of a bill. You may have listened to testimony in court when you were on jury duty. Perhaps you attended a school board meeting because the board members were considering an initiative in which you were interested. In all of these cases, the level of accountability was high. Everyone who spoke was accountable to ensure that the discussion remained on topic, that accurate information was presented, and that they thought deeply about what was being said by others. If any of these principles are violated, it seriously compromises the integrity of the collective work of the institution.

Teaching Takeaway

Use accountable talk—framed as a set of discourse expectations for students—to enrich math discussions.

These same three tenets form the heart of accountable talk in the classroom, an approach to framing discourse practices of students (Resnick, Michaels, & O’Connor, 2010). They also describe the quality indicators of high levels of engagement and critical thinking among learners. Accountable talk is frequently framed as a set of expectations for students that is supported through the use of language frames that scaffold the use of language to explore a topic. These language expectations are distributed across five conversational moves shown in Figure 5.6.

However, in the rush to get students to engage in accountable talk, the responsibility of the teacher may be overlooked. Accountable talk begins with the teacher. Unless the teacher is consistently modeling how these conversational moves are used, and using appropriate prompts and questions to facilitate these moves, students will not integrate them into their own discussion. And we don’t mean simply modeling them once or twice and then moving on. Students need to be drenched in accountable talk; it should flood the classroom. The daily enactment of these principles by the adult in the room communicates volumes about the expectations for classroom discourse. Examples of these teacher prompts (Resnick et al., 2010) include the following:

• Marking the conversation—“That’s an important point.”
• Challenging students—“What do you think about that question Vanessa asked?”
• Keeping everyone together—“Who can repeat what Pedro just said, using your own words?”
• Keeping the channels open—“Did everyone hear that? Devon, can you say that again?”
• Linking contributions—“Allie, can you put your idea together with the one Oliver just suggested?”
• Pressing for accuracy—“Where can we find that?”
• Pressing for reasoning—“Why do you think so?”
• Building on prior knowledge—“How does your idea connect, Tonya, with what we’ve been studying?”
• Verifying and clarifying—“I want to make sure I understand. Are you saying . . . ?”

Students need to be drenched in accountable talk; it should flood the classroom.

Figure 5.6 Accountable Talk Moves

Imagine how your conversational moves can elevate the discourse in the math classroom. Students who are routinely exposed to a teacher who engages as an active participant will absorb some of these same practices. Having said that, students need much more in terms of the norms and structures of classroom discussion.

Supports for Accountable Talk

There are a couple of habits we suggest teachers and students adopt to work accountable talk into their daily routines.

Language Frames

Language frames are scaffolds that prompt the type of talk we want from our students. These language frames might be the tap on the shoulder your students need to engage in productive conversation around a math task. Figure 5.7 contains several language frames that we have found useful for students to use to jumpstart their conversation. These should be provided as an optional support and not a mandate for students.

Language frames are scaffolds that prompt the type of talk we want from our students.

Revoicing and Restating

Talking about mathematical ideas with clarity is not always easy, even for adults. Revoicing and restating are two of five math talk moves, along with wait time, prompting, and reasoning (Chapin et al., 2009). The teacher revoices a student comment or embeds the thinking in a question to help the student clarify thinking for herself and for others in the class. Revoicing opens a conversation in which a student can shed more light on her thinking. This should value the thinking of every student, not solely those who understand. Students are more willing to take a chance to share ideas when they know they will have the support of the teacher in making those ideas clear. Even when students are thinking deeply, it is not always easy to voice their ideas.

Revoicing is when a teacher restates a student comment or embeds the thinking in a question to help the student clarify thinking.

Figure 5.7 Sample Language Frames for Mathematics

Restating, which is similar to revoicing, is a move that is extended to other students by asking them to rephrase or repeat what a peer has said. This empowers students to know that their thinking is valued by the teacher and by their peers, and it allows them to listen to classmates and verify that the interpretation of their thinking is accurate. Teaching students how to restate comments made by others in their groups grants permission for students to challenge each other and request clarification. Without this explicit expectation, students are often reluctant to correct their peers, and the collaborative conversations falter. However, be aware that modeling, practice, respect, and trust are important components to successful mathematical arguments and critiquing the reasoning of others.

Restating is a way for students to rephrase or repeat what a peer has said.

EFFECT SIZE FOR SELF-VERBALIZATION AND SELF-QUESTIONING = 0.64

Note that student thinking does not necessarily need to be accurate in order to be revoiced or restated. In fact, incorrect thinking can often lead to powerful small group or whole class conversations. Revoicing and restating are valuable protocols for helping students to self-verbalize (explain ideas to themselves and others) and self-question (ask themselves and others questions about their thinking).

Teach Your Students the Norms of Class Discussions

You will want to make sure your whole class and small group discussions are worthy of precious instructional minutes, and high-quality discussions hinge on effective classroom management. This means that norms and rules of class discussions should be explicit at the beginning of the school year, and these norms need to be consistently maintained, revisited, and discussed when necessary. The social norms of discussion will not on their own guarantee that the discourse promotes thinking. However, they form the essential foundation that accountable talk is built upon.

Norms and rules are not the same. Norms are the agreements of a group about how the members will work together, and they usually describe four dimensions: trust, belonging, sharing, and respect (Center on Disability & Community Inclusion, 2014). The rules and procedures that follow are meant to align with these agreements. But before you can bring your norms for class discussion to students, you will need to be clear on what you believe is important in order to have students succeed. For instance, how important is a growth mindset in your classroom? How should the community respond to mistakes? Is it safe and trustworthy for students to admit they do not know? How should people be treated and supported? The rules and procedures created should align with the norms you have identified. Yet too often, they contradict expressed norms.

Fourth-grade teacher Darren Hardy realized this when he filmed his own class as they discussed a problem he had posted on the board. “I set up the video camera because I wanted to see what was going on during discussions. I knew that I kept hearing from the same handful of kids, but couldn’t figure out what was getting in the way of the others being able to participate,” he said.

What Mr. Hardy saw surprised him. “It’s my classroom. You think I’d have been aware of it. But because I had the camera at the back of the room, I got a completely different view.” He saw students sitting in the back of his classroom who didn’t participate and didn’t attend to the conversation. “I realized that I had clustered a number of students who are English learners in a place in the classroom that didn’t offer them a decent sight line. It was harder to hear back there.” Mr. Hardy made a number of changes, including rearranging the desks in a U-shape to afford a better view for everyone, reassigning seats to pair quiet students with more participatory ones, and especially in altering his practices. “I saw that I needed to move around the room more, and give students a chance to talk to one another first before opening it up to the class. This has caused the ones who want to answer everything to spend their time talking first with their partner. It also gives those quiet ones a chance to check in with someone else before talking to the large group. There’s some rehearsal taking place, which has been a big help for the English learners.”

Norms are the agreements of a group about how the members will work together, and they usually describe four dimensions: trust, belonging, sharing, and respect.

Teaching Takeaway

Clarify norms first for yourself, then for your classroom, making sure that rules and procedures reinforce the norms and are developmentally appropriate.

Mr. Hardy altered his procedures to better reflect one of his classroom norms: “Everyone belongs to this classroom community.” The rules you develop should be developmentally appropriate and allow for you to manage discussion without thwarting it. Will students in your class need to raise their hands and wait to be called on? Many teachers of intermediate students begin to fade this rule so that students learn how to yield and gain the floor. Will you expect your students to speak in complete sentences? Teachers of younger students teach their students how to stretch their statements so that others can better understand their thinking. Do you want them to ask questions if they don’t understand something another student said? Make sure that there are expectations that students talk directly to one another during discussion, rather than addressing only you.

Teachers at an elementary school adopted some simple signals during discussion that are used in every classroom to increase participation and provide support:

• A hand cupped around the ear means “Please speak more loudly because I am having trouble hearing you.”
• A thumbs-up means “I agree with what you’re saying, and I have a reason I can explain.”
• Hands crossed over one another means “I disagree, and I have a reason I can explain.”
• Hands rolling end over end are meant to encourage the speaker to “Keep going with your idea. Tell us more.”

In addition, the primary students stand next to their seat or place on the carpet and face the class to talk. The principal explained that this makes it easier for small people to track the discussion because they can see who is talking. As well, the gestures provide the classroom teacher with a way to monitor the thinking that is occurring within the group, and not just with the speaker. When they see signs of disagreement or agreement, they know who to engage in further discussion. They report that the hand signals are especially valuable for supporting mathematical discourse, as it interrupts the conventional expectations that while one person answers, the rest are simply passive observers rather than active thinkers. This makes the difference between a few students being engaged in a discussion, and having an entire class of active learners.

Mathematical Thinking in Whole Class and Small Group Discourse

Let’s apply some of the ideas of the previous section to facilitating worthwhile mathematical discourse. Recall that discourse encompasses both the way ideas are exchanged and what the ideas entail.

Video 5.2 Student Collaboration and Discourse for Deep Learning

http://resources.corwin.com/VL-mathematics

Without intending to, discourse can devolve into the same kind of Initiate-Respond-Evaluate (I-R-E) questioning cycle used with individual students. Instead of mathematical thinking, students are primarily engaged in simple recall of information and reproduction of processes, such that accuracy is valued over the thinking that went into it. Unlike I-R-E, classroom discourse that values mathematical thinking allows students to speculate, hesitate, change their viewpoints, and take risks. This means that space must be created for students to engage with quandaries that may initially confound them. Yackel and Cobb (1996) described norms that promote true mathematical discourse, and they called these sociomathematical norms:

• Explanations are mathematical arguments, not procedural summaries of the steps that were used to arrive at an answer. Explanations include justifications.
• Errors are opportunities to reconsider a problem from a different point of departure. Even when the answer is correct, there is further discussion about more efficient and more sophisticated pathways.
• Mathematical thinking requires that teachers cultivate a sense of intellectual autonomy that prizes participation in the discussion about possible solutions.

What are the talk moves of a skilled mathematics teacher that promote conceptual thinking? When looking for explanations, skilled teachers follow up with questions and prompts that ask for elaborations and justifications. They draw other students into the discussion to probe their thinking, especially in seeking to locate areas of disagreement or contradiction. They use errors as opportunities to reboot the discussion by requiring students to wrestle with a discrepancy rather than furnishing them with an easy answer. We’re not romanticizing mistakes; we all like to be right, and feeling like you’re wrong all the time is discouraging. But skilled teachers hunt for the partial understandings that lead students down an incorrect path. Keep in mind that an erroneous answer has its own logic. When these errors occur, take the time to find out how that occurred by unearthing the logic behind it. Figure 5.8 includes the conversational moves of a skilled math teacher.

Small Group Collaboration and Discussion Strategies

Discourse can bloom out of observed small group collaborative learning. When students use tools in different ways to approach a task, small group discussion is the perfect time for students to notice and relate one another’s techniques. As they compare various approaches to the same problem, they may notice an underlying structure that is useful.

Remember Brian Stone, the high school math teacher from Chapter 2 who so effectively collaborates with students on his success criteria? He is always on the lookout for connections to the practice standards, and he regularly has students construct viable arguments and critique one another’s reasoning. The small groups in Mr. Stone’s math class were considering which of three representations they thought would be most useful in a lab report on radioactive dating: parts per million, percentage, or fraction.

Figure 5.8 Conversational Moves of a Skilled Mathematics Teacher

“These are all representations you could use in a lab report if you’re a scientist—it’s up to you to decide which is best for this situation. Most importantly, be sure to justify your reasoning and say why your choice is the best. You have ninety seconds to decide with your group.”

As he circulated around the room to eavesdrop on group discussion, Mr. Stone heard one group having a particularly thoughtful and passionate debate, so he brought the class back together to listen in.

“All right, I need everyone’s attention. I need everyone except for Mohamed and Musab to quiet down, and I would like Mohamed and Musab to continue with their debate, but just turn up the volume. Let’s all listen to their arguments.”

Mohamed continued arguing that parts per million made more sense because the numbers wouldn’t take up so much room on the page, and would be easier to write, while Musab made the case that everyone was used to working with percentages. After about a minute, Mr. Stone interjected, “Now I want to pause this discussion so we can hear from Addison. Addison, you were saying you wanted to use fractions for a representation. Why did you say you wanted to use fractions?” The two-way debate turned into a three-way debate, until a fourth student stepped in and made the case for scientific notation. Notice how the classroom discourse started in small groups, and how Mr. Stone attended to the talk that was happening in those small groups and then used that talk to ignite a whole class discussion.

Learning can be a social endeavor. Humans learn better when they interact with other humans. In addition, students learn a lot more language when they are required to produce language. Mathematics is a language, foreign to some and familiar to others. One of the best ways to apprentice students into the language of mathematics, which then facilitates their mathematical thinking and reasoning, is to have them collaborate with their peers in solving complex, rich tasks.

EFFECT SIZE FOR COOPERATIVE VERSUS INDIVIDUALISTIC LEARNING = 0.59

EFFECT SIZE FOR CLASSROOM DISCUSSION = 0.82

When Is Collaboration Appropriate?

Our recommendation, based on personal experiences and reviews of evidence, is that 50% of classroom time (averaged over a week) be devoted to student discourse and student interactions with their peers. To our thinking, effective collaborative and cooperative learning tasks must do the following:

• Be complex enough so that the students need to work together. Less complex tasks entice learners to divide and conquer the task. Also, the task should be complex enough for the number of students in the group. If there are too many students, someone might not participate.
• Allow for argumentation in which students agree and disagree with one another, negotiate understanding, make claims supported by reasons and evidence, and reach consensus or agree where they disagree.
• Include sufficient language support so that students know how to say what they want to say. That may mean teaching vocabulary, providing sentence frames, using teacher modeling, or mobilizing peer language brokers.
• Provide both individual and group accountability so that learning is visible for students and teachers. In addition, these types of accountability prevent one student from completing the task for the other members of the group.

When these conditions are met, students spur each other’s thinking. In order for them to do so, they need to be taught how to collaborate and how they will be held accountable for their own learning as well as that of the group.

Grouping Students Strategically

We all made the mistake early in our teaching careers of letting students choose their own groups, and then got upset when students didn’t focus on, or complete, the task. We came to realize that socializing when in the company of friends was inevitable and developmentally appropriate, and that most of our students did not have the self-regulation skills to focus on academic work while face-to-face with their afterschool buddies. Importantly, in time, each of us came to the further realization that we needed to actually teach those self-regulation skills as carefully as we taught the subject content.

EFFECT SIZE FOR CONCENTRATION/PERSISTENCE/ENGAGEMENT = 0.48

EFFECT SIZE FOR ABILITY GROUPING = 0.12

Another error we made in our grouping practices was in grouping by math ability. Researchers are in conflict about whether or not grouping students by their perceived ability is appropriate in a math class. Some researchers have found a small but positive effect on student math outcomes in the short term, probably due to teachers having an easier time differentiating instruction (Gentry & Owen, 1999; Kulik & Kulik, 1992). But a larger number of researchers have found that grouping students by their abilities within a class hurts student motivation and makes them dislike math (Boaler, 1997; Zevenbergen, 2005). Whether or not your students like math is important, because long-term learning of mathematics is dramatically related to how many math courses people take in their lives, and only slightly related to how well they do in the courses they take (Bahrick & Hall, 1991; Ellis, Semb, & Cole, 1998). It might seem a bit counterintuitive, but fixed ability grouping does not help students to understand the math they’re learning. John’s analysis of 129 studies on in-class ability grouping found a small effect size of 0.12, leading him to describe this practice as “a disaster.”

As you would expect, this is a controversial and touchy topic. We recommend that when taking ability into account, you make sure that these groupings are flexible and balanced, and allow for a moderate but not extreme range of skill levels. Sitting the students who need more time and repetition together, or the ones who are already ahead of the curriculum, should not be fixed, rigid, or permanent. Meeting with a small group of students for some needs-based, teacher-guided instruction is valuable and allows for more robust and responsive differentiation of instruction. But permanently tracking students contributes to a destructive and fixed mindset, enough to counteract any growth mindset activities such as number talks you may have planned. The minor convenience of having similarly achieving students seated at the same table is vastly outweighed by the destructive effects on students’ mindsets. Most importantly, the students placed most at risk by the practice of permanently assigned ability groups within the classroom are those that can least afford it: students who struggle with math concepts, are learning English as a subsequent language, or have disabilities.

Teaching Takeaway

The most effective grouping strategy is one that is flexible and balanced, and that allows for a moderate but not extreme range of skill levels.

Permanently tracking students contributes to a destructive and fixed mindset, enough to counteract any growth mindset activities such as number talks you may have planned.

We have seen some of the most effective teachers make good use of flexible groupings that shift partnerships throughout the week to foster collaborative and cooperative learning. In Natalia Smith’s fourth-grade class, students sometimes enter the room to find that the tables have been rearranged, and their name placard is somewhere different from where they usually sit. Ms. Smith tailors her table arrangements and groupings to her preassessments, learning intentions, and tasks of the day. She always plans these seating arrangements ahead of time using an alternate ranking system. Using an appropriate metric, such as the results of a diagnostic screening assessment, the last administered assessment, or her evaluation of each student’s communication skills, the members of the class are rank ordered from most to least skilled. The list is then split in half, and the two students from the first half of the list are paired with the corresponding two students on the second half of the list. Therefore, in a class of twenty-four, students ranked first and second are paired with those ranked thirteenth and fourteenth.

Figure 5.9 The Alternate Ranking Method for Grouping

We call this method alternate ranking, and it ensures that there is a range of skills within each group, but not such a broad gap that they will just furnish the answers rather than truly work together. In other words, highly skilled students are paired with those possessing an average level of skills (see Figure 5.9). In turn, those with an average skill level are paired with those who are the least skilled. It doesn’t work to pair the top two students in the class with the two at the bottom. Bennett and Cass (1989) studied the work accomplished in small groups, noting that when the balance of high-achieving students within a group outweighed the number of low-achieving students, those in the majority took control and performed the task without input from those who needed more time, explanation, or repetition. This phenomenon disappeared when the groups were balanced, and the skills the group possessed were moderately, but not extremely, different.

The most frequent group size is four students, but we have seen groups of three or five work well, too. Groups of six tend to be too big for all group members to contribute, and groups of two tend to lack the cognitive diversity necessary for interesting academic discourse. However, groups of two work for brief conversations, even if they are less effective for rich tasks that require deeper discussion and dialogue. Teachers sometimes have one or two students who will make the case that they should be allowed to just work alone. Often, this is because they have been in groups gone bad and ended up doing all of the work for the other members of the group. Fortunately, they have a caring adult in their lives (that’s you) who understands that students not only learn math better in groups but also learn key collaborative and interpersonal skills that will be required of them in college and the workplace. And they have a teacher who knows how to include individual and group accountability into the tasks.

Video 5.3 Grouping Strategies for Deep Learning

http://resources.corwin.com/VL-mathematics

If you dig a little deeper into the experiences and skills of students who request to be excused from collaborative learning, you’ll often find it’s their interpersonal skills, not their mathematical ones, that are lacking. The prospect of having to manage all of those social relationships while performing cognitive tasks may be overwhelming for them. But these soft skills are vital for their success. So consider reducing the size of the group just a bit for that child. Perhaps he’ll work in a group of three, while the other groups have four or five students. Over time, as social skills strengthen, you can increase the number of partners the student will be working with.

Above all, don’t fall into the trap of considering the least common denominator when making decisions about collaborative work. Just because one or two students will be more challenged by collaboration doesn’t mean that it should be eliminated from your classroom. You wouldn’t do this with the curriculum: “There are two students who really struggle with fractions, so that means I won’t teach fractions to anyone this year.” Why would you do so with collaborative learning?

What Does Accountable Talk Look and Sound Like in Small Groups?

Consider the following scenario. Seventh-grade students were working with ratios and proportions. Their teacher, Bianca Rayos, had selected a task from Illustrative Mathematics (www.illustrativemathematics.org/content-standards/7/RP/A/1/tasks/470).

The task begins . . .

Travis was attempting to make muffins to take to a neighbor that had just moved in down the street. The recipe that he was working with required $\frac{3}{4}$ cup of sugar and $\frac{1}{8}$ cup of butter.

Travis accidentally put a whole cup of butter in the mix.

1. What is the ratio of sugar to butter in the original recipe? What amount of sugar does Travis need to put into the mix to have the same ratio of sugar to butter that the original recipe calls for?

2. If Travis wants to keep the ratios the same as they are in the original recipe, how will the amounts of all the other ingredients for this new mixture compare with the amounts for a single batch of muffins?

3. The original recipe called for $\frac{3}{8}$ cups of blueberries. What is the ratio of blueberries to butter in the recipe? How many cups of blueberries are needed in the new, enlarged mixture?

Sakina starts with her group, saying, “I think we need to set up the ratio first. I think it’s $\frac{3}{4}$ to $\frac{1}{8}$ . Do you all agree?” Mrs. Rayos is observing the conversation and asks Angelina to restate what Sakina just said. “The original recipe was $\frac{3}{4}$ cup of sugar and $\frac{1}{8}$ cup of butter, so I wrote it as $\frac{3}{4}$ : $\frac{1}{8}$ .”

Francisco speaks next, adding, “I think we have to get the denominators the same. If we write the fraction equivalent $\frac{3}{4}$ as $\frac{6}{8}$ , then they’re the same. I think that’s the next step.” Cassandra doesn’t agree, so she restates Francisco’s comment, adding her own: “I think you’re saying that we need to make the denominators the same so we can add or subtract, but the problem is really about a ratio, so I think we need to get them to whole numbers.”

Francisco responds, “I’m not sure how to get them to whole numbers. How do we do that?” Everyone in the group goes back to work thinking about Francisco’s question. Cassandra says, “I think I know how to do it, but I’m not sure. Let me explain my thinking and let me know what you think. I think we have to multiply the sugar times eight. Eight times $\frac{3}{4}$ equals six. And we multiply the butter times eight, which is eight times $\frac{1}{8}$ or one. But I’m not sure that’s right.” Sakina adds, “I think you have it. You multiplied both of them by eight so we could work with whole numbers to get a ratio. Just like you wrote, the ratio of sugar to butter is 6:1.”

Notice how students pressed each other for clarification and explanations, and asked each other for support. See how together they worked out any misconceptions and made sure that their final answer was understood by everyone in the group. In other words, through the conversation, they were holding themselves and each other accountable for deep learning. But these types of student conversations do not just happen on their own. Great teachers do a lot of behind-the-scenes work to make them happen.

Supports for Collaborative Learning

It would be unfair to hold students accountable for academic conversations without providing them the supports they need to be successful. One crucial support is to teach students how to work collaboratively during the first month of school. But savvy teachers know that they need to revisit teaching collaborative skills and routines throughout the year. It’s certainly worth reviewing expectations for collaborative work, but sometimes a teacher will need to completely reteach these expectations as well. A return from a weeklong break from school can be the perfect time to refresh everyone on how things are done in your classroom.

Contribution Checklists

One support you can offer is a readily available checklist of ways that students can contribute. These lists can include student behaviors like asking questions, checking others’ work, keeping the team on task, encouraging others respectfully, making sure the answer makes sense, making sure everyone can explain the reasoning, and drawing connections to things they’ve already learned. These lists can also include more task-specific roles, like “checking the units.” The behaviors can be listed on a poster, projected onto a screen, taped to their tables, or inserted into table tents, which are inexpensive pieces of plastic that hold paper in a vertical position on a table so that it is in students’ lines of sight. A sample checklist is included in Figure 5.10.

Figure 5.10 Contribution Checklist

Allow Students to Move

Study after study encourages regular physical movement in the classroom (Jensen, 2005; Parks, Solmon, & Lee, 2007). Orlowski and her colleagues (Orlowski, Lorson, Lyon, & Minoughan, 2013) summed up the research well:

Classroom-based physical activity is an instructional tool teachers can use to boost mood, energy level, and student learning. . . . Evaluations of active environments have demonstrated positive changes in student classroom behavior, word recognition and reading fluency, math scores, time on task, and concentration levels. (p. 48)

Depending on your school culture, the grade level you teach, and your relationship with students, you will likely be able to just explain the benefits of movement and then have students join you for stretches or jumping jacks every twenty minutes or so as a break from sitting in their chairs. The benefits to their learning would be well worth the time invested in these routines. An added bonus is that group exercise can have a positive impact on your classroom culture.

EFFECT SIZE FOR CLASSROOM COHESION = 0.53

Peer Support

You can get students to move around the room to interact with different peers or ideas. If you have already numbered the students in their groups, you can have all the number ones stand in one corner of the room, twos in another, threes in a third, and fours in a fourth corner. Once your students are in their corners, they can engage in dialogue with their peers from other groups, comparing and contrasting their teams’ different approaches to the math task at hand, asking each other for help, or engaging in some other type of meaningful dialogue.

EFFECT SIZE FOR PEER TUTORING = 0.55

Some teachers have students make appointment calendars before beginning a lesson. In this format—also known as clock partners—teachers take time before the lesson gets started for students to find three or four different partners around the room. At spaced intervals, students go and find, for example, their “one o’clock appointment,” “two o’clock appointment,” and so on. Students like the freedom to choose their partners (and it is easier to allow them to choose partners for short conversations than for longer tasks or projects), but we have also seen teachers just assign appointment calendars to students. You can also instruct your students to go and meet with a student at another table whom they haven’t talked to yet that day, appointment or not.

Movement isn’t just for students, and you probably wouldn’t like sitting in a school chair for forty or eighty minutes at a time either. As we are researching and writing this book, all of us authors are periodically getting up out of our seats and stretching, going for brief walks around the house, or doing some other form of light exercise. We encourage you to do the same if you will be writing, grading, or constructing lesson plans for long periods of time. Our experience lines up perfectly with the research: we feel more energized, sharper, and happier when we treat our minds to these movement breaks. Ultimately, we get more work done.

EFFECT SIZE FOR TACTILE STIMULATION PROGRAMS = 0.58

Supports for Individual Accountability

Some teachers are resistant to assigning a lot of collaborative work because they were scarred by their own experience with poorly designed group work when they were in school. They remember doing all the work, while their classmates coasted by without doing much at all. (We call them worker bees and hitchhikers.) That wasn’t fair, and teachers don’t want their students to have the same experience. What was missing from so many of these group tasks was individual accountability. It’s important to find ways to ensure that students are individually accountable for deep learning and doing their best to contribute to a group’s math work.

Importantly, not all tasks require individual accountability. Sometimes, a quick collaborative conversation is all that is required. When this is the case, teachers can use a think-pair-square to encourage conversations. Rather than having some students share out to the whole class after their partner conversation, we prefer to have partnerships join forces (squaring up) to continue their conversation.

Think-Pair-Square

For example, in Hector Espinoza’s second-grade class, students were focused on the following problem:

Annaleah measured a piece of string for a game she was playing in her backyard. She thought that it was too long, so she cut off 42 inches. Then her string was 44 inches. How many inches was Annaleah’s string before she cut it?

The students in Mr. Espinoza’s class were used to think-pair-square and started talking about the problem with their partners. In one group, Paul said, “Why did she cut it? Now it’s just two inches.” His partner Jessie responded, “Why do you think it’s two inches? Show me the way you got that.” Paul picked up his dry-erase pen and wrote 44 – 42, saying, “You take away forty-two because she cut that off. So forty-four minus forty-two is two. See?”

Jessie, looking confused, said, “But that’s too little to use. That doesn’t make sense. Let’s read it again.” They read the problem together. Jessie says, “It says then, so it was after that it was forty-four. So it was bigger before, right?” Paul, studying the words, says, “Maybe if we draw a picture of what is going on, it will help.” Jessie responds, saying, “First it says she had a long string. So let’s start with a long line for the string.” Paul adds, “Then the problem says she cut off forty-two inches. So let’s put that we cut off forty-two inches.”

Their picture looked like the following:

Paul notices and comments, “But it says she had forty-four inches left, and the amount we cut off is more than what’s left. Forty-four is more than forty-two so it should be longer.” Jessie excitedly says, “Wait! I think I got it. Since what is left is forty-four inches, our picture should look like this.”

“That’s it! Now all we have to do is add forty-four and forty-two to find out how long it was before she cut it,” adds Paul.

As they finish their conversation, Mr. Espinoza invites each partnership to meet with another partnership, saying, “Let’s square up, but please don’t start with telling the other team your answer. Please tell them how you went about solving the problem. We’ll get to the answer in a bit.” Jessie and Paul joined Arturo and Brian, engaging in a conversation about their respective problem solving. In collaborative tasks like these, students see the value in their voice and in listening to the voices of others.

Writing Rules

When individual accountability is warranted, the first (and possibly easiest) layer of accountability you can offer students is to ensure that if anyone writes, everyone writes. While having everyone in a group record the group’s thinking won’t guarantee they all learn, it does increase the likelihood of positive outcomes. One technique we use is collaborative posters during which time we assign each student a different colored marker. At a glance, you can see who contributed ideas, to what extent, and whether they need more guidance. Students are likely to be thinking about what they’re writing and to critique or accept one another’s reasoning as they’re writing it down, and are less likely to tune out if they have to produce something. Something about writing helps to focus the mind.

Figure 5.11 Conversation Roundtable

Rich problems often include performance-based tasks. These tasks involve more than just solving an equation to reach an answer. Students must go through several steps and often use multiple strategies such as making a model and finding a pattern, or making a table and generalizing. In these situations, writing helps students not only to make sense of a given situation, but also to think about how to show their work and explain how they reached the solution. Writing increases opportunities for students to think about their thinking (metacognition), which contributes to the phase of deep learning.

Conversation Roundtable

Another method we rely on is conversation roundtable. Students fold a sheet of paper into four quadrants, and then fold down the inner corner of the folded paper to form a rhombus. They unfold the paper and outline each section on the folds. The top left quadrant is reserved for their thinking, while the other three quadrants are reserved for recording the contributions of each group member. The rhombus in the middle is reserved for their own thinking at the end of the task. Each member hands in his or her own conversation roundtable, making all students individually accountable for the task. A sample can be found in Figure 5.11.

Individual Exploration

We have also seen teachers get better results by giving their students time to wrestle with a task individually before engaging with their teammates. In these cases, the teacher can structure students’ think time so that they can answer four questions for themselves:

1. What is the question asking?
2. What useful information is given?
3. What other information would be helpful?
4. What might an answer look like?

Whether or not students are able to answer all four questions, the time spent reflecting on these questions increases the likelihood that they’ll be able to bring something to the table when they join their teammates. Students may respond to the questions in writing or by creating models, charts, or diagrams. This access to multiple representations gives more ready access to each learner and encourages later conversation about how various representations relate to one another. This individual exploration time sets the stage for individual accountability by ensuring each group member has ideas to share.

Whole Class Collaboration and Discourse Strategies

Too often, whole group classroom discussion quickly slides back into a lecture, with the teacher occasionally posing a question and allowing a few students to respond before once again returning to his or her monologue.

When Is Whole Class Discourse Appropriate?

Discourse is not about brief interjections of students’ voices. It should be designed to promote and extend mathematical thinking. Your guiding questions, prompts, and cues can either foster this or shut it down. Remember the difference between a funneling question and a focusing question? You will want to stick with focusing questions for your class discussions, just like you do with small groups. You want your students to own the cognitive work, to figure things out as they reason with one another. In particular, you want students to ask questions about their work that they do not know the answer to. Sadly, students are rarely invited to ask these types of questions. John notes that the mode is about two questions per class per day, and that driving up the number of these questions underlies one of the most effective practices we know.

EFFECT SIZE FOR QUESTIONING = 0.48

One of the differences between guided instruction with a small group and with the whole class is that discourse in a whole class should have students assume as much of the question posing as they can.

One of the differences between guided instruction with a small group and with the whole class is that discourse in a whole class should have students assume as much of the question posing as they can. This is the nature of true discourse; student learning becomes visible to them as they witness their own thinking and that of their peers. Students need encouragement from the teacher to question one another’s thinking, to ask others how they arrived at a solution or why they chose one representation over another. You’re looking for those times when the conversation soars, and you can step back for a few minutes as students engage with one another and carry the conversation themselves. The teacher can step to the side, waiting for students to clarify each other’s thinking and question each other’s thinking. Through these types of interactions, students come to see themselves as shaping the thinking of their peers, and learn to accept the same support from others (Hufferd-Ackles, Fuson, & Gamoran Sherin, 2004). But in order to do so, students need to be carefully taught. The following actions and guidelines will help build productive classroom discussions.

What Does Accountable Talk Look and Sound Like in Whole Class Discourse?

As with the example of Mrs. Rayos’s class earlier in the chapter, accountable talk during whole class discussion should focus on having students ask questions of one another and press for clarification and explanations from each other. The teacher’s role is to initiate the discussion with an engaging task or prompt. This might be a question or an invitation for one group to share a discussion with the entire class, allowing everyone to listen and join in the conversation. Students should be talking with one another while the teacher serves as a guide. As in small group work, student discourse should dominate whole class discussions; the teacher is just a facilitator.

Supports for Whole Class Discourse

Have you attended a meeting in a room that wasn’t suited for the conversation? Perhaps you couldn’t see the speaker because of a pillar in a large meeting hall or were trying to talk with someone seated behind you on an airplane. The physical space where we work impacts the quality of the work we do. For whole class discourse to occur, the teacher must set the stage by creating an environment that supports the work.

Physical Layout

The physical layout of a classroom can foster or inhibit discussion across the entire group. Some teachers have desks attached to chairs; others have tables. Sometimes the furniture is in fixed locations, and other times it can be moved around. We can’t change the reality of the furniture, and really, the type of furniture a teacher has should not limit students’ interactions. Think creatively about ways to use the classroom environment, irrespective of the type of furniture provided.

Inside-Outside Circles

Inside-outside circles promote listening, because only a subset of students is engaged in active discussion while the rest of the group is observing the conversation. The easiest way to accomplish this is to have students arrange their chairs (or stand) in two concentric circles, with approximately half of the class in the inner talking circle and the other half of the class observing. Make sure the outer group has a task, such as cataloging when a particular mathematical concept is being utilized. After facilitating the discussion with the inner circle for five to ten minutes, ask the members of the outer circle to share their observations. This is an effective way to address worked examples, especially those that are incorrect.

Strategy Selection

Develop and deepen argumentation and reasoning by starting a discussion with a problem to be solved; then ask students to gather in areas of the classroom based on their chosen strategy for solving it. After allowing time for similarly minded students to discuss their approach, invite the groups into a larger discussion about the merits of each.

Jazmine Green uses this technique when posing questions about math concepts that her fifth-grade students are still learning, but have not completely mastered. “Multiplying fractions is a really tough concept because they have a hard time getting their heads wrapped around the fact that it’s going to yield a smaller number, not a bigger one,” she said. “Unfortunately, many students have been taught that multiplying makes everything bigger.” She posed the following word problem.

There is $\frac{3}{4}$ of a pan of brownies sitting on the counter. You decided to eat $\frac{1}{3}$ of the brownies in the pan. What part of the whole pan of brownies did you eat?

Then she invited students to make a prediction about the answer given three possibilities: $1\frac{1}{12}$ , $\frac{3}{12}$ , and $\frac{5}{12}$ . “No pencils and paper. I don’t want you calculating. Just look at the three possible solutions; then stand near the sign on the wall that has your best choice. When you get there, talk with the other people who chose that answer about why you think that answer is correct.” Ms. Green gave them three minutes of discussion time, and then brought their attention back as a whole group. “Now let’s talk about this. Let’s hear the reasoning for each possible answer, and I want you to listen to these opinions. If you change your mind, you can vote with your feet and join a new group.” The teacher used accountable talk conversational moves to facilitate thinking, posing focusing questions and prompting justifications. Most of the students went to the corner with $1\frac{1}{12}$ as the product because they thought the product should be bigger than the factors. However, as they dove into the problem, they realized that if they ate $\frac{1}{3}$ of $\frac{3}{4}$ , it just didn’t make sense that they ate more than a whole pan of brownies when there was only $\frac{3}{4}$ of the pan to begin with. Now, they were working to make sense of the situation! As students were swayed by different arguments, they crossed the room to join a new group. As a student moved, Ms. Green asked him or her to state why he or she was changing the answer. Within several minutes, the entire class had arrived at the correct answer and, more importantly, a rationale for why multiplying fractions yields a smaller number. “I’ve taught this concept for many years, but many of them never really get it. Hearing the thinking of their classmates makes a world of difference. They get to consider errors and listen to faulty and accurate reasoning. The concepts behind this seem to stick so much better because students really made sense of the situation rather than trying to work with the numbers.”

Figure 5.12 Important Connections Among Mathematical Representations

Source: National Council of Teachers of Mathematics (2014, p. 25).

Using Multiple Representations to Promote Deep Learning

Deep learning is about making connections. In mathematics, one of the powerful forms of connection is noticing how different representations are related or similar. Students demonstrate stronger problem-solving abilities and deeper mathematical understanding when they have experience representing mathematics in a variety of ways and showing the connections they see across those representations (Fuson, Kalchman, & Bransford, 2005; Lesh, Post, & Behr, 1987). Depth of understanding in mathematics, the connections that are central to deep learning, comes from the process of students internalizing various representations and connections among them (Pape & Tchoshanov, 2001; Webb, Boswinkel, & Dekker, 2008). Five types of mathematical representations are shown in Figure 5.12.

Physical representations are tangible objects, both formal manipulatives and objects from everyday life, used to represent mathematics. Visual representations are pictures and sketches, sometimes based on a physical representation. Symbolic representations, such as numerals and operation signs, are the abstract form of mathematics that we see and use most often as adults. For students, these symbols often appear as “squiggles on the page” until they are able to connect the symbols to representations that have more meaning. Deep learning is further enriched by the addition of contextual and verbal representations. Contextual representations situate the mathematics in a realistic scenario so students can see how the mathematics describes what is happening in the situation. Verbal representations are the language we use to talk about a physical model, describe a visual sketch, or speak to symbolic notation. For example, if a young learner is thinking about the number sentence 5 – 3 = 2, a contextual representation might be a simple story.

Lorenzo saw five tortoises in the field. After three walked into the woods, two were left in the field.

The verbal representation might be “five subtract three is the same as two” or “three less than five is equal to two.”

Strategic Use of Manipulatives for Deep Learning

The use of manipulatives can be as instructive at the deep phase of learning as it is at the surface phase. Employing these physical tools shouldn’t be limited to building the initial understanding of a concept, but teachers and students can and should use them to make concepts concrete and visible, look for patterns, make connections, and form generalizations. They can likewise be used when constructing viable arguments and critiquing others’ reasoning. To extend that thought, much research suggests that the instructional sequence of moving from physical representations through visual representations to symbolic representations leads to significant gains in mathematics learning and understanding, particularly for students struggling in mathematics (Gersten et al., 2009). As with surface learning, the deep learning phase benefits from having physical, verbal, and contextual representations incorporated as students engage in discourse about their thinking and make sense of the mathematics they see. In the deep phase of learning, teachers should select mathematical tasks and tools that allow students to see the progression from physical representations (e.g., building a model with base ten blocks to represent the notion that 8 times 14 is the same as 112) to visual representations of the same problem (e.g., drawing an area model), to the symbolic representation of the partial products (e.g., the algorithm 8 × 14 = 112). This multiplication task might have arisen from an exploration of area, and the teacher and students will employ the language of multiplication and place value throughout their discussion.

Conclusion

In this chapter, we have focused on developing deep learning through rich tasks and accountable talk. This phase of learning particularly helps students to leverage their surface learning to more deeply see patterns, look for and express regularity in repeated reasoning, make generalizations, and engage in discourse to help them construct viable arguments and critique the reasoning of others. Responsibility for accountable talk is shared by teachers and students. Meaningful mathematical discourse happens when students work on engaging tasks in thoughtfully selected small groups, so they can push each other to be accountable for deep learning about the mathematics at hand. Whole class discourse provides opportunities for students to share and debate both misconceptions and solution strategies. In this chapter, we shared a variety of strategies that promote student-to-student interaction in small groups and in whole class discussion. In the next chapter, we will discuss transfer learning, the application of surface and deep learning to new situations.

Reflection and Discussion Questions

1. Recall that discourse reaches beyond discussion to include ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged. Consider the mathematical discourse in your classroom. What opportunities do students have to share, explain, and justify their thinking? What questioning strategies could you use to provide additional opportunities for students to show their deep learning through explanation and justification?
2. Consider the grouping practices in your classroom. What strategies do you use to form mixed-ability groups and to ensure both group and individual accountability? What new ideas from this chapter could you use to ensure more rich and rigorous collaborative work in your mathematics classroom?
3. How do you use manipulatives in your mathematics instruction? Are students encouraged to use multiple representations as they work on mathematics collaboratively? Mathematical practice 5 calls for students using appropriate tools strategically. How do you allow students to make decisions about the tools they use in their work? What strategies can you use to make these tools more available to your students?