Vol. 10 #4 Beyond Key Words: The Use of Schemas to Solve Word Problems and Promote Conceptual Unders
Beyond Key Words: The Use of Schemas to Solve Word Problems and Promote Conceptual Understanding
Joan Gujarati, Ed.D.
Walk into any elementary school classroom during mathematics instruction and chances are that you will see students presented with some type of word or story problem to solve given the importance of problem-solving in today’s mathematics classrooms (CCSSO, 2010; NCTM, 2014). Not only do computational strategies come into play when solving a word problem, but so do reading comprehension strategies. In many instances, students are instructed to rely on a key word approach to solve a problem where “all,” “altogether,” or “combined” suggest addition; “left” indicates subtraction; “times” suggests multiplication; and “share” indicates division. However, since there are frequent exceptions to these “rules”, students only scanning for key words often miss out on a deeper understanding of the problem. Key words may reinforce procedural fluency in early years, but hinder conceptual understanding (Clement & Bernhard, 2005). For example, in the problem “Pablo took the 15 pieces of candy that he was not going to eat and gave them to Susanna. Now, Pablo has 27 pieces of candy left. How many pieces of candy did Pablo have to begin with?” Relying solely on the key word approach, students may focus on the word “left” and decide to subtract where the operation required is really addition.
In another approach to problem-solving, a general heuristic is used such as the four step problem-solving process posed by Pólya (1945), a renowned mathematician, where students first, have to understand the problem; second, devise a plan; third, carry out the plan; and fourth, look back and reflect. Although this approach does foster multiple strategies to tackle a problem, for those students with learning disabilities, coming up with a broad plan with many possibilities might tax their cognitive memory load, and therefore, they benefit from a more focused approach (Jitendra & Star, 2011; Powell, 2011) and more explicit instruction to solving word problems (Parmar, Cawley, & Frazita, 1996).
Many students with learning disabilities have mathematics as well as reading difficulties, which can make solving word problems more challenging (Fuchs, L., Fuchs, D., Stuebing, Fletcher, Hamlett, & Lambert, 2008). Schema-Based Instruction (SBI) by Jitendra (2007; 2010; 2011) and colleagues is a focused and organized approach to problem-solving to make mathematics instruction accessible to all students (Jitendra & Star, 2011). Using the SBI model, teachers emphasize the structure of the problem and teach students to categorize problems by their function (Jitendra, George, Sood, & Price, 2010). This instructional strategy has yielded particular success with low-performing subsets of students in both academic performance and mathematical attitudes (Jitendra, Sczesniak, Griffin, & Deatline-Buchman, 2007).
Research has shown that students with learning disabilities can benefit from more direct instruction using schemas (Jitendra & Star, 2011; Powell, 2011). Through the use of schematic diagrams which help the student to categorize various problem types and organize information, Schema-Based Instruction (SBI) highlights a problem’s structure. SBI is teacher-mediated instruction followed by paired partner work and then independent work. This approach scaffolds student learning through the use of visual diagrams and checklists that eventually fade as students become independent learners. A self-monitoring plan is used in preparation from teacher-mediated instruction to independent learning. Jitendra (2010; 2011) and colleagues have devised this structure as FOPS:
F = Find the problem type. Students read the problem and paraphrase it in their
own words to understand the problem’s context (i.e., what is known and what
O = Organize the information in the problem by using a schematic diagram;
P = Plan to solve the problem; solve for the unknown by selecting the appropriate
S = Solve the problem which includes writing the complete answer (i.e., number
In this SBI structure, students are reflecting on information in the problem rather than just plugging in numbers to compute. They are using schemas to first identify a word problem as belonging to a specific problem type and then using a schema to solve the problem. Due to working memory deficits evidenced by students with disabilities, SBI explicitly teaches a small, but adequate, number of strategies to scaffold student learning by providing explicit instruction (Jitendra & Star, 2011). Teaching them to represent the situation described in the problem using schematic diagrams is critical to reduce working memory resources (Jitendra & Star, 2011).
For the early to mid-elementary grades, Jitendra (2007; 2010; 2011) and colleagues focused on three problem schema (change, group, and compare):
Change schema. The change schema, exemplified in Figure 1, generally begins with some initial quantity and a direct or implied action causes either an increase or decrease in that quantity.
Group schema. The group schema, exemplified in Figure 2, focuses on the part-part-whole relation and involves smaller groups combining together to form a new larger group.
Compare schema. The compare schema, exemplified in Figure 3, compares two distinct, disjoint sets and the relation between them is emphasized.
Action Research Project Utilizing SBI Plus Visualization and Kinesthetic Representations
In a semester-long action research study (Blackburn, 2016) during her student teaching semester, one of my teacher candidates was interested in utilizing SBI with her third grade students to target her research question: “How can I use conceptual models of teaching mathematics to better support students’ understanding of mathematical word problems?” The context of her student teaching classroom was 25 third-grade students in a low-income urban bilingual classroom with three students on IEPs and four more in the process of being referred, and overall 12 students performing below grade level in mathematics and 21 below grade level in reading based on mid-year computerized STAR assessment data. Although third grade mathematics Common Core Standards (CCSSO, 2010) stipulate that students should be able to solve multi-step, multiplication, and division word problems, the students in this classroom were still struggling to solve one-step addition and subtraction word problems.
With guidance from her mentor and myself, my teacher candidate designed an intervention unit to address student difficulty in solving one-step word problems. The unit was taught over an eight week period, approximately two-to-three times per week, as a supplement to the students’ standard mathematics instruction. The unit was designed using the SBI methodology and taught a series of word-problem categorizations. A new category was introduced every two weeks, and taught in conjunction with previously learned categories. Students participated in direct instruction, cooperative learning tasks, individual practice, and station work. To accompany the schemas (change, group, compare) presented by Jitendra and colleagues, my candidate added visualization and kinesthetic representations of the problems. When students were first reading the problem, they were instructed to visualize the problem in their heads by painting a picture of it and then explaining it in their own words. Additionally, she taught them hand motions to represent change, group, and compare. Thus when identifying the type of problem, they first represented it kinesthetically. This helped students to conceptualize types of problems before getting to the schematic representations.
For the first part of the final assessment, students were given a scenario with no numbers and asked to categorize it according to word-problem type using a multiple choice format. For example, “Troy has some pencils. Some are mechanical and some are regular lead. We want to figure out how many of each kind Troy has,” would be categorized as a part-part-whole situation. Sixteen (out of 25) students were able to correctly identify 100% of the mathematical situations. An analysis of student performance on the change and compare word-problem types showed a 14% increase in the number of students who correctly answered the change problem and a 35% increase in the number of students who correctly answered the compare problem from pre-to-post-test data.
This action research project has led my candidate to wonder if SBI and other conceptual models of teaching mathematics would be more effective if taught as an original strategy to solving mathematics word problems, as opposed to an intervention strategy once students are struggling with mathematics. This can be a fruitful line of inquiry. Findings from Jitendra (2007; 2010; 2011) and colleagues demonstrate that SBI is a promising model for students with learning disabilities. My teacher candidate’s action research study shows SBI can help English Language Learners develop conceptual understanding as well. Overall, SBI is one approach which could be useful to raise student achievement to close the achievement gap for different groups of children.