Math Forum: TRAILS: Difficult Concepts List

Difficult Concepts List

It occurred to me that one way to start thinking about this was to look through the Ask Dr. Math archives. Below you'll find links to questions from students with corresponding replies from the math doctors.

My next step in developing this list will be to correspond with teachers in classrooms at these three levels to get their opinion of what the concepts are that are difficult for their students to understand. As I receive that information, I'll add it to the areas below.

-- Suzanne

Twenty-one Misconceptions in Mathematics from the Count On site are listed including PDF files to download to learn more.

K-2 Primary: Difference Between Zero and Nothing; Explaining Addition; Explanation of Place Values; How to Tell Time; Tricks for Learning Addition; the value of coins and money; place value (getting to the point where they think in "chunks"); time (both the skill of telling time and elapsed time); subtraction (when to subtract, especially comparison situations; also the traditional "regrouping" algorithm actually seems to get in the way of developing the skill needed); measurement
3-5 Intermediate: 3rd Grade Math Struggle (subtraction); Adding and Subtracting Measurements; Adding Fractions; Adding Three Digit Numbers; Borrowing in Subtraction; Least Common Multiple Puzzle; Learning Only 36 of the Times Tables; Numbers between 0 and 9999; Reading a Ruler; Subtraction: Decomposition (Regrouping/Borrowing); What is Length in a Rectangle?; concept of rate
the notion of dealing with two different units; understanding fractions and decimals
long division; Place value (it doesn't help that they don't master the primary concepts first...); Subtraction (issues carry over from primary); Multiplication and division (what makes a situation a multiplication or division situation, skills - place value problems are an issue here); Rational numbers, especially fractions (lack of understanding of basic concepts gets in the way of skill development); Estimation and number sense (lack of number sense interferes with EVERYTHING else); Area and perimeter; Measurement
6-7 Middle School: Area and Perimeter
Kids can regurgitate formulas for them and even apply them as long as the problem is straightforward, but they rarely understand them in their guts, or understand that they are independent of each other. Their understanding is also pretty limited to squares and other rectangles. As a matter of fact most children (and adults?) will tell you that a square is not a rectangle.; Comparing Fractions; Distributive Property, Illustrated; Introduction to Negative Numbers; Order Of Operations in Four Steps; Multiples and Factors; Prime Numbers; Subtracting Numbers by Walking a Number Line; Turning a Perimeter into a Scale Factor; Why does PEMDAS work?; understanding the relationship among decimals, fractions and percents; fraction operations
plane geometry in the sense of angle relationships with lines and transversals
ratio and proportion, mostly because it looks too much like a fraction so they tend to resist
I think the pre-Algebra concepts are difficult for many teachers to help students understand. I teach with manipulatives so they understand it a lot more but if taught traditionally it is very confusing for students.
combining like terms, the distributive property, formal equation solving
long division It is very hard to get them to look at alternative strategies if this isn't working for them. They are so ingrained from elementary school in the algorithm that they don't want to give it up and look at other methods.
percent applications; concept of rate
the notion of dealing with two different units; unit conversions; fractional anything (excluding algorithms); fractions and decimal fractions are all between 0 and 1 (or -1 and 1); multipliers and divisors; scale factor/scale; percents are not numbers; exponents are not numbers; differences between length (perimeter), area, and volume and their units

This material is based upon work supported by the National Science Foundation under Grant No. 0205625.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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