Fixing Common 27 Myth-Takes & Myth-Conceptions in 43 Myth-Tical Minutes: From Elementary through High School to Col

Fixing Common 27 Myth-Takes & Myth-Conceptions in 43 Myth-Tical Minutes: From Elementary through High School to College Mathematics Alan Zollman Northern Illinois University April 23, 2010 National Council of Teachers of Mathematics 2010 Annual Meeting – San Diego, CA

Counting • Myth: When is 5 five?

Counting • Myth: When is 5 five? • Truth: It’s not the object, but the total collection of objects

Counting • Myth: Since “twenty-one” begins with 2 “thirty-one” begins with a 3 “forty-one” begins with a 4, then “fifteen” must begin with a 5

Counting • Myth: Since “twenty-one” begins with 2 “thirty-one” begins with a 3 “forty-one” begins with a 4, then “fifteen” must begin with a 5 • Truth: The numbers 11 through 19 are mis-named, perhaps they should be onety-one, onety-two, … It’s only a tradition, not a pattern.

Number Lines • Myth: When is 5 five? • Myth: 3/5th and 1/2nd are equivalent.

Number Lines • Myth: When is 5 five? • Myth: 3/5th and 1/2nd are equivalent. • Truth: “Points” on the line are total distance travelled.

Time • Myth: Reading the time (on a clock) is a measure.

Time • Myth: Reading the time (on a clock) is a measure. • Truth: You must be able to identity and distinguish two measures to “read” the time. There here are at least two number lines (a 0-12 hour number line and a 0-60 minute number line).

Money • Myth: Using real coins helps students learn money.

Money • Myth: Using real coins helps students learn money. • Truth: The attributes of real coins contradict the attribute of “value” of the coin. Grids of 1’s, 5’s, 10’s, 25’s, 50’s can help students see the corresponding values of coins. Saying “5-cent piece” is relates to value better than “nickel”.

Place Value • Myth: A number has one value.

Place Value • Myth: A number has one value. • Truth: Numbers have many possible representations for different situations, e.g. 125 is 125 ones; 1 hundred 2 tens 5 ones; 12 tens 5 ones; 1 hundred 25 ones.

Operations • Myth: We read numbers left-to-right but we do operations right-to-left.

Operations • Myth: We read numbers left-to-right but we do operations right-to-left. • Truth: We read numbers just like reading text, from left-to-right, but we can do operations in various order (as in division) – as long as we keep track of place value of the numbers.

Operations • Myth: Key words identify the operation in story problems.

Operations • Myth: Key words identify the operation in story problems. • Truth: A word’s meaning varies with the context. “More” can imply add, subtract, multiply or divide in an application. Identifying the key concept identifies the operation(s)

Division • Myth: “Gozinda” is a math operation.

Division • Myth: “Gozinda” is a math operation. • Truth: In 15 ÷ 5, we are not trying to find out how many times 5 “goes into” 15, but how many times we can subtract 5 from 15. The operation we do repeatedly in long division is subtract.

Fractions • Myth: 2/3rds is larger than 1/3rd

Fractions • Myth: 2/3rds is larger than 1/3rd • Truth: The most important “number” in a fraction is not the numerator nor the denominator but the unit, e.g., 2/3rds of a smaller pizza may be less than 1/3rd of a much larger pizza.

Fractions • Myth: In fractions, all the pieces must be the same size.

Fractions • Myth: In fractions, all the pieces must be the same size. • Truth: There are two uses (and different operations) for fractions: parts-of-a-whole (area model), and parts-of-a-total (ratio model). If 18/30ths of the class of 30 students is boys, we don’t care that the boys are different sizes, and we add ratios differently than area models.

Fraction Division • Myth: If 5 ÷ 2/3 = 7 ½ , what does the ½ represent? ½ of 1? • Myth: In fraction and decimal division, the answer cannot be larger that what you divide.

Fraction Division • Myth: If 5 ÷ 2/3 = 7 ½ , what does the ½ represent? ½ of 1? • Myth: In fraction and decimal division, the answer cannot be larger that what you divide. • Truth: Repeatedly subtract servings of 2/3 pizza from the total of 5 pizzas in order to find out how many servings can be prepared – which is ½ of a serving ( ½ of 2/3). An equal sign shows equivalence, thus the unit of measure must change from “5” to “7 ½ ”.

Fraction Division • Myth: The decimal point separates the whole numbers from the decimal fractions, but there are no “oneths”.

Fraction Division • Myth: The decimal point separates the whole numbers from the decimal fractions, but there are no “oneths”. • Truth: In place value the decimal point “points out” the ones place value.

Decimal Division • Myth: In 2.5 ÷ 0.05 you move the decimal point.

Decimal Division • Myth: In 2.5 ÷ 0.05 you move the decimal point. • Truth: In 2.5 ÷ 0.05 you make equivalent fractions of 2.5/0.05 = 25./0.5 = 250./5. = … to get whole numbers.

Division by Zero • Myth: You cannot divide by zero.

Division by Zero • Myth: You cannot divide by zero. • Truth: If you look at division as repeated subtraction, it is indefinite exactly how many times you can subtract zero from a number, e.g. it is “undefined.”

Algebra and Geometry • Myth: A “square” in algebra is different than a “square” in geometry.

Algebra and Geometry • Myth: A “square” in algebra is different than a “square” in geometry. • Truth: They are the same, but with a different representation, useful in different contexts. Drawing three squares (with side of the radius) inside a circle approximates the area of the circle, e.g. A=πr2.

Algebra • Myth: Algebra is solving for x. • Myth: Algebra is the intense study of the last three letters of the alphabet.

Algebra • Myth: Algebra is solving for x. • Myth: Algebra is the intense study of the last three letters of the alphabet. • Truth: Algebra are many things: 1. patterns (Study of Generalized Arithmetic ), 2. solving for an unknown (Study of Procedures ), 3. studying relationships (Study of Relationships Among Quantities ), and 4. proving mathematical ideas (Study of Structure ).

Algebra • Myth: Math is consistent.

Algebra • Myth: Math is consistent. • Truth: We change math to fit the context, e.g., give six ideas of what x means in algebra.

Algebra • Myth: In algebra, we “plug in” for x.

Algebra • Myth: In algebra, we “plug in” for x. • Truth: We change math to fit the context, e.g., for 8x, we do not mean to substitute 7 for x, as 8x does not mean 87 but 8 times 7.

Algebra • Myth: To solve application problems in math we substitute letters for words.

Algebra • Myth: To solve application problems in math we substitute letters for words. • Truth: In algebra there is a difference between variables and labels, e.g. “There are 3 feet equals 1 yard” does not transfer to the equation 3f=1y. (Put in 6 feet for f and see if you get 2 yards.)

Infinity • Myth: Infinity is a really big number. You can add, subtract, multiple and divide it.

Infinity • Myth: Infinity is a really big number. You can add, subtract, multiple and divide it. • Truth: Infinity is not a “thing” but a concept. We can determine a limit as a number approaches infinity.

Calculus • Myth: 0.9 and 1 are “almost” equivalent

Calculus • Myth: 0.9 and 1 are “almost” equivalent • Truth: If 0.3 and 1/3rd are equivalent, so are 0.9 and 3/3rd

Calculus • Myth: and are the same function.

Calculus • Myth: and are the same function. • Truth: They have the same limits as x approaches 1, but one is continuous and one is not defined at x=1. Their domains are different.

Calculus • Myth: The derivative of the function f(x)= |x| at x=0 is 0.

Calculus • Myth: The derivative of the function f(x)= |x| at x=0 is 0. • Truth: The function is continuous at x=0 but the left-hand limit of the difference quotient defining the derivative does not equal the right-hand limit of the difference quotient.

Mathematicans • Myth: If an Northern Illinois University mathematics faculty member writes 26 myths in his title there will be 26 myths presented.

Fixing Common 27 Myth-Takes & Myth-Conceptions in 43 Myth-Tical Minutes: From Elementary through High School to Col