# Obstacles to Learning Definitions in Mathematics

Share Cognitive Science Implications for Teaching Mathematics
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Two researchers (Rubenstein and Thompson below), have identified the following 11 categories of difficulties associated with learning the language of mathematics.

1. Meanings are context dependent
• Reciprocal of a number vs reciprocal agreement
• Foot as in 12 inches vs. the foot of the bed
• Teachers must address the confusion directly by discussing the stipulative definitions important to mathematics and discussing the difference between stipulative and lexical definitions.
2. Mathematical meanings are more precise
•  Product as the solution to a multiplication problem vs. the product of a company.
• Function as in mathematics vs my liver is functioning normally.
• The mathematics definitions are stipulative definitions.  They must be taught. Furthermore, they are precise.  Changing a single word can destroy the definition.
3. Terms specific to mathematical contexts
• Polygon, parallelogram, imaginary number
• Factorial, perfect number, sine, latus rectum
• Clearly definitions of these words must be taught. Do not lose sight of the fact that these definitions are very precise and they are stipulative.
4. Multiple meanings
• Hypothesis in Formal Logic & Mathematics vs Hypothesis in Scientific Method
• Even as in even number vs even as in to make level
• Plane in geometry vs plane in aeronautics
• The remedy is to teach the conflicting definitions and be consistent in their use.  Use them as frequently as possible.
5. Discipline-specific technical meanings
• Cone as in the shape vs. cone as in what one eats
• Square as a geometric figure vs carpenter’s tool
• The remedy is to teach the conflicting definitions and be consistent in their use.  Use them as frequently as possible.
6. Homonyms with everyday words
•  Pi vs. pie
• Graphed vs graft
• Plane vs plain
• The remedy is to teach the conflicting definitions and be consistent in their use.  Use them as frequently as possible.
7. Related but different words
• Circumference vs. perimeter
• The teacher should always be alert to this situation and address it directly when students are having difficulty.
8. Specific challenges with translated words
•  mesa vs. table
• This is a rare circumstance but when it happens, the teacher should address it with appropriate instruction.
9. Irregularities in spelling
• obelus vs. obeli
• This is a rare circumstance but when it happens, the teacher should address it with appropriate instruction.
10. Concepts may be verbalized in more than one way.
• 15 minutes past vs. quarter after.
•  An even integer vs an integer divisible by two.
• Vocabulary instruction must address all acceptable verbalizations.