Solving Ratio, Proportion, & Percent Problems Using Schema-Based Instruction

An evidence-based practice, schema-based instruction (SBI), can be used to develop student understanding of ratios and proportional relationships.

The instructional components of SBI include:

• Problem solving and metacognitive strategies that provide students with guidance and structure for approaching unfamiliar problems and for monitoring and reflecting on the problem solving process,
• Identifying the underlying mathematical structure of problems,
• Representing problems using diagrams that highlight the quantitative relations described in the problem, and
• Developing procedural flexibility through comparing and contrasting multiple solution methods and explaining when, how, and why to use a broad range of methods for a given class of problems.
• Aligned with Common Core State Standards

Program Features

1. Lessons are scripted and include anticipated student responses. However, we do not expect teachers to read the scripts verbatim. Instead, teachers should be familiar with the content to be able to use own explanations and elaborations to implement the SBI program.
2. This program does not cover the topics “slope” and “similar figures.” If they are included in your state standards, we suggest that you use other resources to support your instruction.
3. Included in the program are optional challenge problems – note the purpose of challenge problems is to stimulate deep thinking and hone students’ mathematical problem-solving skills and logical reasoning skills. They are not meant for all students especially low-achieving students who need to master the basic content.
4. The program focuses on 3 content-related features – (a) determining the type of problem, (b) estimation, and (c) multiple strategies. In terms of the problem type, generally students are introduced to the most basic ‘form’ of the problem, with variations and more complex problems used subsequently.
   
   1. A key part of our curriculum is getting students to recognize problem types and to use the visual-schematic diagrams to help in solving them. Using these diagrams to solve problems is only a subtle change in how teachers already solve these types of problems, but we think that this subtle change makes a big difference in student outcomes. Research suggests that “visual-schematic representations should be used to support the first phase of the word problem solving process (i.e., problem comprehension) and that arithmetical representations are only appropriate in the problem solution phase” (Boonen, van der Schoot, van Wesel, de Vries, & Jolles, 2013, p. 60).
   2. A few somewhat tricky things about generating estimates. First, there is no one set strategy for coming up with an estimate – it really depends on the quantities in the problem. Second, sometimes it might seem like an estimate is not necessary, because the numbers in the problem don’t appear to be too hard.^* But nevertheless we feel that generating and using an estimate is still very important to do. Third, each student may come up with a different estimate, and this is fine.
   3. There are three strategies that are taught in the SBI program: cross multiplication, unit rate, and equivalent fractions. We want students to know all three. We want students to be able to use the “best” one for a given problem. What does “best” mean? Usually it means the strategy that is the easiest for a particular problem. But we also know that what might be easy for one student may not be easy for another student. So, it is possible that students can have different opinions on which strategy is best for a given problem – this is fine. But we do hope that students do not “mindlessly” use cross multiplication for every problem. Note: Although we introduce cross-multiplication as the first strategy in the program (early development of the curriculum was informed by what strategies were familiar to students and cross multiplication was the one that was taught in most schools), it is fine to delay this strategy until students have learned the other strategies.
5. Problem solving and DISC: The SBI program makes extensive use of a specific problem solving process that we call DISC (Discover, Identify, Solve, Check). Our rationale for the DISC is that it is an anchor for students to reflect on the problem-solving processes.  Questions are used to scaffold a solution process and encourage students to regulate their strategy knowledge during the problem-solving processes: (a) problem comprehension (e.g., How do you know it is a Ratio, Proportion, or Percent problem? How is this problem similar or different from the previously solved problem?), (b) problem representation (e.g., What diagram best fits this problem type to represent information in the problem?), (c) planning (e.g., How can you solve this problem? What are the different ways to solve it? Which strategies are better and why?), and (d) problem solution (e.g., What is an estimated answer to this problem? Is the answer reasonable given the question asked?). A key component of the problem solving approach is to have students use carefully constructed diagrams that bridge the relationship between the word problem and the underlying quantitative relations in that problem.  Each of the three diagrams in the SBI program underscores different types of ratio or proportional relationships found in word problems.  The use of visual representations – or diagrams in this case – is a key recommendation in today’s research on problem solving.
6. Homework: Homework problems were developed to complement and reinforce critical concepts and skills taught in the SBI program. Optional challenge problems are included in each lesson. Select problems to assign for homework based on meeting the needs of your students. A caveat – it is critical to appropriately balance the time spent in class reviewing and checking homework problems (no more than 15-20% of the total instructional time) with the time spent on core instruction of critical content. The distributive practice in the homework is critical because it reinforces the interactive work conducted in class on different problem types.  By having teachers help students work carefully through new concepts in class, students are in a much better position to be successful on similar kinds of problems found in the homework.  It is expected that students should be able to solve these problems independently.

*We purposefully used nice numbers in the word problem contexts to focus student attention on problem solving skills rather than computation skills, but these nice number are at odds with generating an estimate.

Based on the findings of several randomized control studies, benefits of SBI include: (a) student gains (including students with mathematics difficulties) in proportional problem solving. Such gains (although indirect) are also associated with improved attitudes toward mathematics; (b) SBI instructional resources help teachers apply the evidence-based practice to effectively teach proportional reasoning.

References

Jitendra, A. K., Dupuis, D. N., Star, J. R., & Rogriguez, M. C. (2016). The effects of schema-based instruction on the proportional thinking of students with mathematics difficulties with and without reading difficulties. Journal of Learning Disabilities, 49, 354-367. doi: 10.1177/0022219414554228

Jitendra, A. K., Harwell, M. R., Dupuis, D. N., & Karl, S. R. (2017). A randomized trial of the effects of schema-based instruction on proportional problem solving for students with mathematics problem-solving difficulties. Journal of Learning Disabilities, 50, 322-336. doi: 10.1177/0022219416629646

Jitendra, A. K., Harwell, M. R., Dupuis, D. N., Karl, S. R., Lein, A. E., Simonson, G. & Slater, S. C. (2015). Effects of a research-based mathematics intervention to improve seventh-grade students’ proportional problem solving: A cluster randomized trial. Journal of Educational Psychology, 107, 1019-1034. doi: 10.1037/edu0000039

Jitendra, A. K., Harwell, M. R., Karl, S. R., Dupuis, D. N., Simonson, G., Slater, S. C., & Lein, A. E. (2016). Schema-based instruction: The effects of experienced and novice teacher implementers on seventh-grade students’ proportional problem solving. Learning and Instruction, 44, 53-64. doi.org/10.1016/j.learninstruc.2016.03.001

Jitendra, A. K., Harwell, M. R., Karl., S. R., Im, S-H., & Slater, S. C. (under review). Improving student learning of ratio, proportion, and percent: A replication study of schema-based instruction.

Jitendra, A. K., Harwell, M. R., Karl, S. R., Simonson, G. R., & Slater, S. C. (2017). Investigating a Tier 1 intervention focused on proportional reasoning: A follow-up study. Exceptional Children, 83(4), 340-358.

Jitendra, A. K., & Star, J. R. (2012). An exploratory study contrasting high- and low-achieving students' percent word problem solving. Learning and Individual Differences, 22, 151–158.

Jitendra, A. K., Star, J. R., Dupuis, D. N., & Rodriguez, M. (2013). Effectiveness of schema-based instruction for improving seventh-grade students’ proportional reasoning: A randomized experiment. Journal of Research on Educational Effectiveness, 6, 114-136.

Jitendra, A. K., Star, J. R., Rodriguez, M., Lindell, M., & Someki, F. (2011). Improving students' proportional thinking using schema-based instruction. Learning and Instruction, 21, 731-745.

Jitendra, A. K., Star, J., Starosta, K., Leh, J., Sood, S., Caskie, G., Hughes, C., & Mack, T. (2009). Improving students’ learning of ratio and proportion problem solving: The role of schema-based instruction. Contemporary Educational Psychology, 34(3), 250-264.

Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., Koedinger, K. R., & Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8: A practice guide (NCEE 2012-4055). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.