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Build a Bridge: Provide Access to Grade-Level Content for All Students Struggling students engage in a scaffolded series of problems and learning experiences that provide access to grade-level content.
LauraMarie K. Coleman
Significant positive changes sometimes begin with a hallway conversation. Last year, a sixth-grade teacher approached me and exasperatedly asked, “My students can barely add and subtract whole numbers; how can I teach these kids division with fractions?” As a special education teacher and this teacher’s instructional coach, I recognized that this common challenge was one that required more than empathy—it needed a solution. I wondered, What options are there to support student learning on grade level when students are missing the prerequisite content? Two came to my mind. Option 1: Provide remediation. This option begins at the students’ last point of success, which often means that students are learning mathematics concepts they should have mastered two or three years earlier. The problem is that to catch up students quickly, concepts might be reduced to basic procedural elements, ignoring critical conceptual foundations, which means that students will continue to struggle (Paulo et al. 2019, p. 20). Students may also experience embarrassment and frustration while learning content from significantly lower grades. Option 2: Teach grade-level content. In this option, students jump directly into grade-level content without obtaining prior knowledge from earlier grades. This often leads to instruction focused on procedures and skills (Zager 2017, p. 239). For example, students may
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learn the “keep, change, flip” procedure when dividing fractions without understanding why or how it works. Lacking a foundation of conceptual understanding, students may mix up the order of steps, forget a procedure completely, or not know when to apply the procedure. Picturing the look of desperation on that teacher’s face, I thought about how this challenge is addressed in literacy: Struggling students are met at their current reading levels and provided opportunities to engage with grade-level texts by reading aloud. Teachers read high-interest, age-appropriate books to all students and use scaffolding and modeling to support students making sense of the text, even when they cannot read it independently. I wondered, What could this instruction look like in mathematics? Is there a third option? I considered how teachers could use what students already know and provide scaffolding and modeling to help them apply that knowledge to grade-level math ematics content. Is it possible to stop providing remediation in isolation and create opportunities to embed support in grade-level instruction? Could all students gain access to deep mathematics content knowledge built through experiences that help struggling learners discover understanding instead of just memorizing steps? The answer to both questions is yes. Option 3: Build a bridge. I imagined building a bridge. The foundation, at one end, is students’ most
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basic understanding of an intentionally selected gradelevel topic and, at the other end, a strong foundation in the grade-level topic. To move across the bridge, students engage in a scaffolded series of problems and learning experiences that provide access to grade-level content. Instruction for Option 3 would incorporate research-based practices including the following: 1. Build fluency from conceptual understanding. This is one of the effective teaching practices in Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014, p. 42). To go beyond skill review, instruction focuses on helping students understand the concepts behind the operation being addressed. Learning the concept of division begins with breaking sets of objects into a given number of equal groups or determining how many groups of a given size make up the whole. Fluency with conceptual understanding allows students to make sense of more abstract calculations, such as division of fractions. 2. Progress from concrete to semiconcrete to abstract (CSA). This progression connects concepts using manipulatives and visuals so students can make sense of a topic (Van de Walle, Karp, and Bay-Williams 2019, p. 108). Cubes and tape diagrams represent real-world situations, helping students make sense of more abstract mathematical notations. Concrete and semiconcrete representations provide multiple access points for students of all levels. Some students may need to draw tape diagrams to represent division with fractions, whereas others may be able to represent their thinking using only equations. 3. Use parallel problems for access to complex concepts. The use of parallel problems can provide context without added complexity (Johanning and Mamer 2014, pp. 346–48). Exploring concepts and properties of operations with whole number problems can help students apply those understandings to related, but increasingly complex, mathematics. For example, solving 7 ÷ 2 first could support students as they make sense of 7/8 ÷ 1/4.
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4. Make connections explicit. Hiebert and Grouws (2007, pp. 383–90) found that two classroom practices supported student understanding: (1) making connections explicit and (2) engaging students in productive struggle. Being explicit does not mean telling students the answer. Intentionally selected problems create opportunities for productive struggle as students discover their own connections between concepts and problems. Both the problems, 7 ÷ 2 and 7/8 ÷ 1/4, have solutions where three equal groups can be made with a remainder. Dividing by a group of two sets students up to make the connection 2/8 = 1/4.
BUILD A BRIDGE IN ACTION The following example is how one sixth-grade classroom teacher and I (his instructional coach) put this research into action. We built a bridge for students in a grade 6 self-contained classroom to support their conceptual understanding and procedural fluency with dividing fractions. Baseline data showed that most of the students in this class had mastered only the identification of unit fractions and had very little, if any, understanding of operations with fractions. Each step of the process is described below. This process has also been adapted by other teachers during pull-out intervention and push-in support.
Step 1: Study Standards The planning process began as the classroom teacher and I studied the standards and alignment to the curriculum. Knowing that addressing gaps in learning takes time, we focused specifically on content aligned to major cluster standards (Student Achievement Partners n.d.). We identified that dividing fractions by fractions was a focus for this classroom’s module of study. We selected a problem aligned to this learning goal as our first target: “How many 1/4 teaspoon doses are in 7/8 of a teaspoon of medicine?”
Step 2: Look at Learning Progressions The next step was to study the learning progression for this target. The teacher and I used the coherence links in Eureka Math. Other tools that could be used for this include resources embedded in curricula and state
LauraMarie Coleman, LauraMarie.Coleman@greatminds.org, coaches teachers, creates resources, and delivers professional development for Great Minds, the creator of Eureka Math. Previously, she was a special education classroom teacher and an instructional coach, working with alternative and special education students and teachers in K–grade 8, in Binghamton, New York. doi:10.5951/MTLT.2019.0200 MATHEMATICS TEACHER: LEARNING & TEACHING PK–12
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Bridge Plan
Day of the Week
Monday
Concepts to be covered
Introduce the target problem
*All division work this week will focus on measurement division to align with the target problem. Next week, we will be looking at similar concepts but with partitive division.
Review fraction meaning and notation—grade 3
Strategies and models
Concrete objects: counters, cubes, candy, and playdough w/plastic knives
*Equations/abstract representations will be used daily.
Whole number division word problems—grades 2 and 3
Tuesday
Wednesday
Represent fractions (drawings and fraction notation)— grade 3
Fractions and equivalent fractions—grades 3 and 4
Whole number division word problems—grade 3
Continue to work on whole number division word problems and add complexity of a remainder—grade 4
Tape diagrams on whiteboards (concrete objects if needed)
Tape diagrams, drawing dots, or numbers inside the tape to represent objects
Thursday Whole number divided by a fraction—grade 5
Tape diagrams (concrete objects as needed)
Friday Solve the Target Problem: 7/8 ÷ 1/4 (measurement division word problem)—grade 6
Apply all strategies and models from the week as needed. Work as abstractly as possible!
Note: *Denotes an explanation for some of the instructional decisions when planning instruction on division.
standards. Working backward from the target, division of a fraction by a fraction, we identified work with division of a whole number by a fraction in grade 5, division with a remainder and additional work with fraction operations in grade 4, and the initial conceptual understanding of basic fractions and division in grade 3. Throughout our study, we also noted many of the concrete and pictorial representations used and, specifically, that tape diagrams were used in all four grades.
understanding across the grades and moved from concrete to more abstract representations. The week culminated with students applying their learning to the target problem. The planning time to complete this process and build this bridge plan varied from a few preparation periods to full-day professional development sessions.
Fig. 1
Step 3: Identify Student Conceptions The teacher and I intentionally focused on what students did know and built on these understandings. With the curriculum and progressions in mind, we turned our attention to students’ last point of success. Observations and baseline assessments helped us determine that this point was an understanding of unit fractions and whole number division—grade 3 concepts.
Step 4: Design the Bridge The next step was to write a plan for the week. Each day’s experiences took students one step farther across the bridge, making connections, supporting students’ movement through the learning progression, applying and expanding their knowledge, and ultimately solving the grade-level target problem. The final weekly plan (see table 1) shows the progression of lessons and how students developed conceptual
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Sixth graders connected the whole number and fraction division with (a) a whole number division problem, and (b) a target problem for fraction division.
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As we gained knowledge of the learning progressions, our process became more streamlined.
BRIDGING TO THE TARGET PROBLEM After four days of working through the planned progression, on the fifth day, students completed their journey across the bridge and applied the work of the week to the grade-level target problem. The sixth-grade class began the day with a whole number problem (see figure 1a). Students used cubes to represent each piece of candy. The idea of “eating” one piece of candy helped students understand that there was one less piece or unit. Moving on to the target problem, cubes represented 1/8 of the whole. Tape diagrams for both problems provided a semiconcrete way to identify the problem structure and helped students make sense of the whole as they divided it into equal parts and modeled 2/8 = 1/4. Ultimately, students showed their thinking abstractly by writing equations. The class applied what they knew to the grade 6 target problem (see figure 1b). Referring to the whole number problem as students worked supported their process as they solved the target problem.
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LESSONS IN TEACHING Throughout the week, originally hesitant and disengaged students came alive. They excitedly shared connections and became confident in their ability to discover new understandings. Very quickly, and with little scaffolding, many students were able to solve the grade-level problem on Friday. There was an immediate shift in the classroom mindset from “I don’t know” to “I can use what I know.” For the classroom teacher, this was a pivotal moment. He had previously been humoring me, waiting for students to fail so that we could go back to option 1 and provide remediation. After this week, followed by week after week of success, he and his students fully embraced that they could access grade-level content. This experience and similar weeks in other classrooms proved to me that my idea to Build a Bridge can work. Students with significant gaps in content knowledge can access grade-level problems with support and scaffolding, just as in literacy. Using what students know to build a bridge shifts the focus of both the teacher and students from one of deficit to one of success. _
REFERENCES Hiebert, Jason, and David A. Grouws. 2007. “The Effects of Classroom Mathematics Teaching on Students’ Learning.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester Jr., pp. 371–404. Charlotte, NC: Information Age Publishing. Johanning, Debra I., and James D. Mamer. 2014. “How Did the Number Get Bigger? Developing Number and Operation Sense Associated with Fraction Division is Viewed from Multiple Perspectives: Modeling, Equivalence and Symbolism.” Mathematics Teaching in the Middle School 19, no. 6 (February): 344–51. https://doi.org/ 10.5951 /mathteacmiddscho.19.6.0344 National Council for Teachers of Mathematics (NCTM). 2014. Principals to Actions: Ensuring Mathematical Success for All. Reston, VA: NCTM. Paulo, Tan, Alexis Padilla, Erica N. Mason, and James Sheldon. 2019. Humanizing Disability in Mathematics Education. Reston, VA: National Council of Teachers of Mathematics. Student Achievement Partners. n.d. Mathematics Focus by Grade Level. Accessed October 2019. https://achievethecore.org /category/774/mathematics-focus-by-grade-level. Van de Walle, John A., Karen S. Karp, and Jennifer M. Bay-Williams. 2019. “Providing for Students Who Struggle and Those with Special Needs.” In Elementary and Middle School Mathematics: Teaching Developmentally. 10th ed., pp. 104–12. New York: Pearson Education. Zager, Tracy. 2017. Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms. Portland, ME: Stenhouse Publishers.
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