Misconceptions involving ratio and proportionPresentation Transcript
Misconceptions Involving Ratio and Proportion By: Jason Poulin
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).
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).
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).
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
Apples to Oranges
Taught per unit concept by teacher
Could not manipulate proportions beyond teacher taught method
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).
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).
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).
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?
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).
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).