The document discusses several studies on teaching ratios and proportions. It finds that symbolic representations can cause difficulties for learning disabled students, while conceptual understanding is stronger. Gender differences did not impact proportional reasoning skills. Instruction should start concretely and develop multiplicative thinking over time. Allowing students to manipulate objects and verbalize their thinking deepens understanding. Students can develop their own successful problem-solving strategies. Discovery learning and group work improves comprehension more than direct instruction. Implementing brief vocabulary reviews improved student performance. Developmental differences in proportional reasoning are not based on age. Teachers also lack full understanding of proportions and strategies for teaching the concept.

1. Misconceptions Involving Ratio and Proportion By: Jason Poulin

2. Analyzing the Proportional Reasoning Skills of LD Students Learning Disabled Students More successful at problem solving. Less successful when using symbols. Non-Learning Disabled Students Successful at symbolic form of fractional problems. Demonstrated several errors in mathematics. Teaching method should de-emphasize mathematical symbols and investigate through abstract models to build deeper understanding of concept (Mick & Sinicrope, 1983).

3. Does gender matter? Hypothesis: Males are more likely to succeed in complex mathematical tasks than females. SURPRISE! Both sexes showed similar misconceptions with evidence not supporting gender difference as a reason for proportional misunderstandings. Teaching method should emphasize investigative approach to develop deeper understanding of proportions and ratios (Linn & Pulos, 1983).

4. Pre-instruction Understanding Symbolic competencies are weak. Strong conceptual competencies. Need time to develop multiplicative understanding. Ratio and proportion instruction should begin with concrete pictorial or manipulative based context and the proportion between dilations of a figure should be saved until students have had time to develop a multiplicative frame of mind (Lamon, 1993).

5. Building Conceptual Knowledge Case study involving fifth grade student revealed three important strategies. Rote computational problems involving cross-multiplication are not productive. Allowing student to manipulate physical objects strengthens proportional understanding. Developing students ability to verbalize their thought process deepens their understanding. Lo & Watanabe, 1997

6. Apples to Oranges Student A Exceptional student Taught per unit concept by teacher Could not manipulate proportions beyond teacher taught method Student B Average student Per unit concept not taught Developed composite unit method to solve proportions When allowed the opportunity, students can develop successful strategies to solve proportional strategies (Singh, 2000).

7. Instruction vs. Discovery Over 400 students participated in investigation to see which method of instruction lead to deeper understanding. Significant results were found to support student discovery and collaborative group work improved students understanding (Ben-Chaim et al, 1998).

8. Vocabulary…but I teach Math! Identical materials, texts and lesson plans were used in eight classrooms. Half of the teachers spent the first 5-10 minutes doing vocabulary activity the other half taught math concepts the entire time. The study revealed that implementing vocabulary and symbolism review for a few minutes each day significantly improved student understanding of proportion and ratio (Jackson & Phillips, 1983).

9. College = Smarter Students Working rate, mixture, and physical problems created difficulty for students. Physical similarity between objects caused problems recognizing proportional relationship. It would be valuable to teach students to focus on the quantitative relationship between the units of each object in the problem because this approach appears to lead to an innate understanding of proportional reasoning in most students (Lawton, 1993). Right?

10. Age = deeper understanding Myth: Older students are more apt to understand the multiplicative nature of proportional reasoning. Fact: Developmental differences are not based on age. Regardless of age, students are able to predict an outcome than to computationally prove. Further research needs to be reveal how computational understanding can be improved (Dixon et al, 1991).

11. Teachers Understand Proportion Inadequate understanding of inverse proportion. Lacked reasoning skills to verify incorrect answer. Teachers lacked strategies to solve as well as teach proportions. Teachers need better instruction on strategies of solving direct and inverse proportion and should also be made aware of common misconceptions made by middle and secondary students (Fisher, 1988).