# 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.
• Diamond vs. rhombus.
• In the house vs. in the division bracket.
• Donut vs torus.
• Instruction should focus on the correct term. The teacher should not permit those verbalizations which are not correct and proper mathematics.

Definitions (vocabulary) in mathematics have two unique characteristics which must be addressed in any form of vocabulary instruction.

• Definitions in mathematics are very precise.
• Definitions in mathematics are stipulative definitions.

Rubenstein, R., & Thompson, D. (2002). Understanding and supporting children’s mathematical vocabulary development. Teaching Children Mathematics, 9, 107–112.