By Holly Henderson Pinter, Temple A. Walkowiak, and Robert Q. Berry, III
Analyses of classroom practices reveal useful methods of engaging students in mathematics.
Teaching mathematics is a complex process that requires teachers (and researchers as well) to understand how to move from specific mathematics teaching contexts to more general principles of teaching mathematics. We view the goal as being able to successfully construct and immerse students in opportunities to learn mathematics.
Two documents — Principles to Actions by the National Council of Teachers of Mathematics (2014) and the Common Core State Standards in mathematics — offer research-based outlines for creating such opportunities to learn for students. These guiding documents emphasize the type of engagement with mathematics that promotes problem solving, reasoning, and sense making. In fact, the Standards for Mathematical Practice in the Common Core describe opportunities for all students to engage in mathematical practices and processes to learn mathematics (See Table 1).
Researchers have documented effective teaching strategies for mathematics such as creating opportunities for mathematical discourse (Hufferd-Ackles, Fuson, & Sherin, 2004) and providing cognitively demanding tasks for students (Hiebert et al., 2005; Lambert & Stylianou, 2013; Stein et al., 2009). That has guided our work in crafting an opportunity-to-learn framework, which comprises three domains that will give teachers a way to analyze mathematics teaching. The domains, which we call the “three Ts,” are the use of time, the selection and implementation of tasks, and the facilitation of talk during mathematics lessons.
This framework emerged from two research studies that thoroughly and systematically analyzed 57 mathematics lessons from 18 elementary teachers’ classrooms of students in grades 3 and 4 (Pinter, 2013; Walkowiak, 2010). The process involved taking detailed notes on features of the lessons, transcribing discussions, and ultimately looking for patterns across the set of lessons. Through these analyses, we found the following: patterns within a subset of the teachers’ mathematics lessons that set those lessons apart in terms of the opportunities for students to engage with and learn mathematics in a deep, conceptual way and differences in the use of time, tasks, and talk among the mathematics lessons, thereby providing different types of learning opportunities for students.
When NCTM released Principles to Actions, we noticed connections between our three-domain framework and their suggested mathematics teaching practices, which are based on research about the types of teaching that produce high-quality experiences for students. In Table 2, we have grouped teaching practices to show how they correspond to the three domains of our framework. For teachers making small changes in the domains of time, tasks, and talk, the consequence is a broader effect on mathematics teaching practices and on students’ opportunities to engage in the mathematical practices outlined in the Common Core.
Consider the following content standard from the Common Core in 3rd grade (3.OA.3): Students are to represent and solve problems using multiplication and division. Specifically, the standard requires that students use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities. Below, we compare two contrasting examples through the lens of the opportunity to learn domains of time, tasks, and talk.
Scenario #1: Ms. Taylor
This lesson began with a warm-up problem where students multiply two two-digit numbers using the standard algorithm, an already learned procedure. Students spent about five minutes working individually on the problem. Ms. Taylor then gave students the correct answer and asked them to correct any mistakes. Next, students spent 15 minutes checking homework as Ms. Taylor called out the answers and subsequently turned in their homework papers. The lesson for the day then began with Ms. Taylor writing the following word problem on the board: A box of raisins contains 90 raisins. Scott has five friends he wants to share the raisins with. How many raisins will each friend get? Students had several minutes of private think time to solve the problem and then Ms. Taylor asked one student to share his thinking with the group (10 minutes). The students did another problem together with the same content and structure as a whole group (five minutes), and then Ms. Taylor assigned students to work individually on 10 problems from their textbook (15 minutes). They finished the lesson with a quiz covering the content objectives from the previous week: multiplying two two-digit numbers (10 minutes).
Scenario #2: Ms. Thomas
At the beginning of this lesson, each student received a small box of raisins. Ms. Thomas asked students to estimate the number of raisins in the box without looking inside. When students had their estimates in mind, Ms. Thomas encouraged them to open the box and look at just the top layer of raisins and use this information to adjust their estimates. The class discussed how their estimates changed or stayed the same. This activity and discussion took about 20 minutes of the allotted time. Then, Ms. Thomas told students they would be checking their estimates, but they had to count the raisins by grouping the raisins in some way. The students spent about 15 minutes counting the raisins by grouping them, circling a group, and writing the number inside the circle in order to keep a written record of the number of raisins. After students completed the task, the class came back together for a 15-minute discussion in which students shared their strategies for grouping the raisins. Ms. Thomas then modeled how to write a division sentence to represent grouping, and students wrote their own division sentences based on how they grouped their own raisins to bring closure to the activity.
Time is a fixed construct in classrooms; therefore, it is imperative that time be used to give students optimal opportunities to learn and engage in meaningful mathematics. Maximizing time within a lesson first depends on a clearly articulated and coherent mathematical learning goal for students (Hiebert et al., 2007). Principles to Actions also points to the importance of setting goals to focus learning and making goals explicit to students (Table 2).
The lessons in our research that maximized students’ opportunities to learn tended to maximize time within instruction by maintaining a focus on mathematical ideas central to the learning goal. These teachers kept all lesson components connected to the mathematical goal for the day and kept transitions short. Additionally, the lesson segments were sequenced so they built toward understanding the mathematical goal. For example, if a lesson had three smaller components, those components connected and built upon each other to enable students to move toward the mathematical goal. We also noted the importance of the amount of time allotted for each lesson component. In cases of maximizing students’ learning opportunities, time was used to support students’ opportunities for conceptual understanding or to build procedural fluency from conceptual understanding. If a lesson devoted five minutes to modeling and/or exploration and 40 minutes on practicing rote procedures, students had a different experience than when they spent 30 minutes in modeling/exploring and 15 minutes on practice.
The two above examples have distinct differences in how the lesson uses time. Ms. Taylor’s lesson had five distinct components: warm-up, homework, model, practice, and quiz. This structure does not lend itself to a conceptual progression of understanding for several reasons:
- A lesson with more components equates to more transition time, which reduces academic learning time. While it is developmentally appropriate to have several components to support engagement, teachers must find the delicate balance between too many transitions and keeping things interesting.
- The lesson components are not connected in a way that builds toward student understanding.
- The content objectives are presented rather than developed.
- The connections between tasks are not made explicit. Students’ opportunities to learn are limited by the disjointed nature of lesson segments that may or may not be mathematically related.
In contrast, Ms. Thomas’s lesson included one central task — grouping raisins — to meet the objective of modeling division sentences. Students predicted and counted the number of raisins in their boxes, and the task was structured so the counting was purposefully connected to division. Students used raisins as a tool to build two representations of division — the grouped raisins and their corresponding division sentences. The teacher structured the timing of lesson components to give students opportunities to learn by building understanding through inquiry. Table 3 summarizes the amount of time used in each lesson as well as an indication of how that time was used and each component’s connectedness to the lesson objective. Note the correlation of the time used toward the instructional goal in each lesson. Ms. Thomas’s lesson provides a coherent and connected lesson where all activities support the content objective, while Ms. Taylor’s lesson has disjointed components that disrupt the flow of the lesson and take instructional time from the objective. It is a work of balance to find the perfect blend of time usage and content coherence.
Researchers have emphasized the importance of choosing cognitively demanding tasks for over a decade (Stein et al., 2009). It is not merely the task that is important but also how the teacher enacts it. Learning most often occurs in settings where students are using higher-level thinking rather than engaging in more procedural tasks. Our research findings show that task enactment is vital in implementing mathematics instruction and that even seemingly procedural tasks can be taught in ways that encourage students to make conceptual connections.
Notice that the tasks in each example lesson were essentially identical. Both tasks aim for students to engage in division. The difference is in the facilitation of the tasks. In the first scenario, Ms. Taylor presents the task, students solved the problem, one student shared his thinking, another task was presented, and students continued to practice the skill. Developing understanding in this lesson was somewhat rushed toward practice.
In the second scenario, Ms. Thomas presented the task, and students were first asked to make predictions. After students made initial predictions, they used data to adjust their predictions by looking at the top layer of raisins. During the exploration with raisins, Ms. Thomas’s students were not told how to group their raisins in order to make them easy to count. Students were problem solving as they determined how to group the raisins. They performed a procedure when they grouped the raisins, but they connected the division of the raisins to their own symbolic division sentences, an indication of a higher level of cognitive demand (Stein et al., 2009). Many students put their raisins into groups of five or 10, likely due to their early exposure to skip counting by five or 10 in the elementary curriculum. Because of the way Ms. Thomas structured this task in a more exploratory fashion, students had opportunities to explore and make sense of mathematics on their own.
The presence of talk alone in classroom instruction is certainly not enough to promote student understanding. The act of talking, however, is a key first step to a much more complicated process of fostering mathematical discourse, where the teacher must pose purposeful questions, elicit student thinking, and facilitate meaningful discourse (Hufferd-Ackles, Fuson, & Sherin, 2004; NCTM, 2014). Teachers must find the balance between allowing students to express thinking freely while keeping the focus on the mathematical goal. Mathematical discourse is a multidimensional practice where students and the teacher share ideas, questions, and other mathematical wonderings to build mathematical understanding. The analyses of our lessons indicated two distinctions related to discourse practices: Effective teaching uses probing questions to dig deeper into children’s thinking, and effective teaching gives students a sense of autonomy in directing mathematical discussions.
In the first scenario, talk is virtually nonexistent, and most of it is directed from Ms. Taylor to her students. This lesson did not afford students the opportunity to share their thinking with peers or negotiate mathematical understanding through talk. The only exception to this is when one student was given the opportunity to share his thinking about the raisin problem.
In the second scenario, however, students had multiple opportunities to contribute through student talk. For example, Ms. Thomas asked students to explain why they changed their raisin estimates after opening the boxes:
Ms. Thomas: Would someone like to share why you changed your estimate?
Mitchell: I counted the layer on top and added the number of layers.
Ms. Thomas: Cara, why did your estimate change?
Cara: My estimate went up because the raisins were smaller than I thought. So, I thought there would be a lot more.
Ms. Thomas: Corinne, how did you change your estimate? What was your strategy?
Corinne: I counted the layer on top with 10 raisins and counted nine layers. So, I did 10 x 9, which is 90.
Ms. Thomas’s use of “how” and “why” questions promoted explanation and justification among her students. This allowed for sense making to occur not only for the individual student, but it also allowed other students to think about their own strategies compared to their peers. When a teacher interacts even with one student by asking the student how or why he chose a specific strategy, other students have the mental space to compare their strategy to their peer’s strategy. When students do this, they must decide if their strategy is the most effective or if they want to abandon it and adopt the strategy of a peer. In other lessons in our research, this happened frequently when a teacher explicitly asked students to explain or elaborate on a peer’s thinking. This process of talk facilitation overlaps well with the Standards for Mathematical Practice (Table 1) by asking students to construct arguments and critique the reasoning of others. Other research, such as Cross’ (2009) study of student use of argumentation in mathematics, also suggests that students who have opportunities to explain and justify their answers will show gains in knowledge.
Honing our craft as teachers takes focused time and effort. However, small changes in teaching practices can make large effects on structuring mathematics teaching to broaden students’ opportunities to learn. We have outlined a framework with three domains of mathematics instruction based on careful observation of elementary mathematics lessons. Table 4 presents each domain of the framework along with two related key questions as starting points for analyzing mathematics lessons.
The three key domains of instruction can influence the use of the Mathematics Teaching Practices (Table 2) and consequently increase students’ opportunities to engage in the mathematical practices outlined in the Common Core (Table 1). We encourage teachers to start reflecting on their use of these three domains in their teaching in order to identify potential areas for growth. Teachers may learn that through the purposeful structuring of time they find more opportunities to engage students in meaningful tasks and talk.
Cross, D.I. (2009). Creating optimal mathematics learning environments: Combining argumentation and writing to enhance achievement. International Journal of Science and Mathematics Education, 7, 905-930.
Hiebert, J., Stigler, J.W., Jacobs, J.K., Givvin, K.B., Garnier, H., Smith, M., . . . Gallimore, R. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 video study. Educational Evaluation and Policy Analysis, 27 (2), 111-132.
Hiebert, J., Morris, A.K., Berk, D., & Jansen, A. (2007). Preparing teachers to learn from teaching. Journal of Teacher Education, 58 (1), 47-61
Hufferd-Ackles, K., Fuson, K.C., & Sherin, M.G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35 (2), 81-116.
Lambert, R. & Stylianou, D.A. (2013). Posing cognitively demanding tasks to all students. Mathematics Teaching in the Middle School, 18 (8), 500-506.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
Pinter, H.H. (2013). Patterns of teachers’ instructional moves: What makes mathematics instructional practices unique? (Unpublished doctoral dissertation). University of Virginia, Charlottesville, Va.
Stein M.K., Smith M.S., Henningsen M.A., & Silver, E.A., (2009). Implementing standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press.
Walkowiak, T.A. (2010). Third grade teachers’ mathematics instructional quality, mathematical knowledge for teaching, and mathematics teaching efficacy: A quantitative and qualitative analysis (Unpublished doctoral dissertation). University of Virginia, Charlottesville, VA.
HOLLY HENDERSON PINTER (@hhpinter) is an assistant professor at Western Carolina University, Cullowhee, N.C. TEMPLE A. WALKOWIAK is an assistant professor at North Carolina State University, Raleigh, N.C. ROBERT Q. BERRY, III is an associate professor at the University of Virginia, Charlottesville, Va.
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