Misconceptions and Errors in Mathematics: A Guide, Exams of Mathematics

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Misconceptions and Errors

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Place Value

. When counting tens and ones (or hundreds, tens, and ones), the student misapplies the procedure for counting on and treats tens and ones (or hundreds, tens, and ones) as separate numbers.

When asked to count collections of bundled tens and ones, such as |||••, student counts 10, 20, 30, 1, 2, instead of 10, 20, 30, 31, 32.

Example

2. The student has alternative conception of multidigit numbers and sees them as numbers independent of place value.

Student reads the number 32 as “thirty-two” and can count out 32 objects to demonstrate the value of the number, but when asked to write the number in expanded form, she writes “3 + 2.”

Student reads the number 32 as “thirty-two” and can count out 32 objects to demonstrate the value of the number, but when asked the value of the digits in the number, she resonds that the values are “3” and “2.”

Example

3. The student recognizes simple multidigit numbers, such as thirty (30) or 400 (four hundred), but she does not understand that the position of a digit determines its value.

Student mistakes the numeral 306 for thirty-six.

Student writes 4008 when asked to record four hundred eight.

Example

4. The student misapplies the rule for reading numbers from left to right.

Student reads 81 as eighteen. The teen numbers often cause this difficulty.

Example

Misconceptions and Errors

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0. Student misapplies the rule for “rounding up” and changes the digit in the designated place while leaving digits in smaller places as they are.

Student rounds 127,884 to 128,884 (nearest thousand).

Example

Student rounds 62.38 to 62.48 (nearest tenth).

Example

. Student overgeneralizes that the comma in a number means “say thousands” or “new number.”

Student reads the number 3,450,207 as “three thousand four hundred fifty thousand two hundred seven.”

Example

Student reads the number 3,450,207 as “three, four hundred fifty, two hundred seven.”

Example

2. Student lacks the concept that 10 in any position (place) makes one (group) in the next position and vice versa.

If shown a collection of 12 hundreds, 2 tens, and 13 ones, the student writes 12213, possibly squeezing the 2 and the 13 together or separating the three numbers with some space.

Example

0.72 + 0.72 = 0.144 or

3 5 or 3, 1 1 1, 195

Example

3. Student lacks the concept that the value of any digit in a number is a combination of the face value of the digit and the place.

When asked the value of the digit 8 in the number 18,342,092 the student responds with “8” or “one million” instead of “eight million.”

Example

Place Value

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Addition and Subtraction

. The student has overspecialized his knowledge of addition or subtraction facts and restricted it to “fact tests” or one particular problem format.

Student completes addition or subtraction facts assessments satisfactorily but does not apply the knowledge to other arithmetic and problem-solving situations.

Example

2. The student may know the commutative property of addition but fails to apply it to simplify the “work” of addition or misapplies it in subtraction situations.

Student states that 9 + 4 = 13 with relative ease, but struggles to find the sum of 4 + 9.

Student writes (or says) “12 –50” when he means 50 – 12.

Example

3. Thinking that subtraction is commutative, for example 5 – 3 = 3 – 5

Example

4. The student may know the associative property of addition but fails to apply it to simplify the “work” of addition.

Student labors to find the sum of three or more numbers, such as 4 + 7 + 6, using a rote procedure, because she fails to recognize that it is much easier to add the numbers in a different order.

Example

Misconceptions and Errors

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9. The student knows how to subtract but does not know when to subtract (other than because she was told to do so, or because the computation was written as a subtraction problem).

Student cannot explain why she should subtract or connect subtraction to actions with materials.

Example

0. The student has overspecialized during the learning process so that he recognizes some addition situations as addition but fails to classify other addition situations appropriately.

Student recognizes that if it is 47° at 8 AM, and the temperature rises by 12° between 8 AM and noon, you add to find the temperature at noon.

However, he then states that the situation in which you know that the temperature at 8 AM was 47° and that it was 12° cooler than it is now is not addition.

Example

. The student has overspecialized during the learning process so that she recognizes some subtraction situations as subtraction but fails to classify other subtraction situations appropriately.

Student recognizes that if there are 7 birds in a bush and 3 fly away, you can subtract to find out how many are left.

However, she may be unable to solve a problem that involves the comparison of two amounts or the missing part of a whole.

Example

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2. The student can solve problems as long as they fit one of the following “formulas.”

a + b =? a b – a =? b b – a

• c He has over-restricted the definition of addition and/or subtraction.

Given any other situation, the student responds, “You can’t do it,” or resorts to “guess and check.”

Example

3. The student sees addition and subtraction as discrete and separate operations. Her conception of the operations does not include the fact that they are linked as inverse operations.

Student has difficulty mastering subtraction facts because she does not link them to addition facts. She may know that 6 + 7 = 13 but fails to realize that this fact also tells her that 13 – 7 = 6.

Student can add 36 + 16 = 52 but cannot use addition to help estimate a difference, such as 52 – 36, or check the difference once it has been computed.

Example

4. When adding or subtracting, the student misapplies the procedure for regrouping.

1 1 1 1

Example

5. When subtracting, the student overgeneralizes from previous learning and “subtracts the smaller number from the larger one” digit by digit.

Example

Addition and Subtraction

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5. The student has overspecialized during the learning process so that she recognizes some multiplication and/or division situations as multiplication or division and fails to classify others appropriately.

Student recognizes that a problem in which 4 children share 24 grapes is a division situation but states that a problem in which 24 cherries are distributed to children by giving 3 cherries to each child is not.

Example

Student recognizes “groups of” problems as multiplication but does not know how to solve scale, rate, or combination problems.

Example

6. The student knows how to multiply but does not know when to multiply (other than because he was told to do so, or because the computation was written as a multiplication problem).

Student cannot explain why he should multiply or connect multiplication to actions with materials.

Example

7. The student knows how to divide but does not know when to divide (other than because she was told to do so, or because the computation was written as a division problem).

Student cannot explain why she should divide or connect division to actions with materials.

Example

8. The student does not understand the distributive property and does not know how to apply it to simplify the “work” of multiplication.

Student has reasonable facility with multiplication facts but cannot multiply 12 × 8 or 23 × 6.

Example

Multiplication and Division

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9. The student applies a procedure that results in remainders that are expressed as “R#” or “remainder #” for all situations, even those for which such a result does not make sense.

When asked to solve the following problem, student responds with an answer of “10 R2 canoes,” even though this makes no sense: There are 32 students attending the class canoe trip. They plan to have 3 students in each canoe. How many canoes will they need so that everyone can participate?

Example

0. The student sees multiplication and division as discrete and separate operations. His conception of the operations does not include the fact that they are linked as inverse operations.

Student has reasonable facility with multiplication facts but cannot master division facts. He may know that 6 × 7 = 42 but fails to realize that this fact also tells him that 42 ÷ 7 = 6.

Student knows procedures for dividing but has no idea how to check the reasonableness of his answers.

Example

. The student undergeneralizes the results of multiplication by powers of 10. To find products like 3 × 50 = 150 or 30 × 50 = 1,500, she must “work the product out” using a long method of computation.

× 500

Example

Misconceptions and Errors

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4. The student misapplies the procedure for regrouping as follows:

The first step (multiplying by ones) is done correctly, but the same numbers are used for regrouping again when multiplying by 10s whether it is appropriate or not.

3 37 × 65 185

• 2120 2305

1 4 128 × 75 640

• 8860 9500

Example

5. The student overgeneralizes the procedure learned for addition and applies it to multidigit multiplication inappropriately.

Original process for addition: When performing addition with regrouping, the student first adds the amount that is regrouped to the appropriate amount in the topmost addend and then continues by adding the remaining amounts in that place value column.

Inappropriate generalization: When performing multiplication, the student first adds the amount that is regrouped to the amount in the multiplicand and then multiplies (instead of multiplying first and then adding the amount that was regrouped).

Students regroup the 4 tens. They then add 4 to 0 to get 4 and multiply that by 8 (4 • 8) to get 32, instead of multiplying 8 • 0 and then adding the 4, to get 4.

3 4 206 × 18 4028

• 2060 6088

Example

6. The student generalizes what she learned about single-digit multiplication and applies it to multidigit multiplication by multiplying each column as a separate single-digit multiplication. This can also be looked at as an example of Misconception 5.

× 62

1 128 × 71 848

Example

Misconceptions and Errors

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7. Thinking that division is commutative, for example 5 ÷ 3 = 3 ÷ 5

5 ÷ 3 = 3 ÷ 5

Example

8. Thinking that dividing always gives a smaller number

9. Thinking that multiplying always gives a larger number

20. Always dividing the larger number into the smaller

4 ÷ 8 = 2

Example

2 . Thinking that the operation that needs to be performed (+, –, ×, ÷) is defined by the numbers in the problem

Multiplication and Division

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4. Student does not understand that fractions are numbers as well as portions of a whole.

Student recognizes

in situations like these:

one-half of the area is shaded one-half of the circles are shaded

but cannot locate the number

on a number line, or says that “one-half is not a number, it is a part.”

Example

5. Student thinks that mixed numbers are larger than improper fractions because mixed numbers contain a whole number part and whole numbers are larger than fractions.

Student says that 1

because whole numbers are larger than fractions.

Example

6. Student is confused about the whole in complex situations.

Anna spent

of her homework time doing math. She still has

hour of

homework left to do. What is the total time Anna planned for homework?

When looking at this problem, the student can easily become confused about the whole. Is the whole the total time Anna planned for homework, or is the whole one hour?

Example

7. Student has restricted her definition and thinks fractions have to be less than 1.

When confronted with an improper fraction, the student says it is not a fraction because in a fraction the numerator is always less than the denominator.

Example

Fractions

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8. Student counts pieces without concern for whole.

Student says that

of a circle is shaded.

Example

When asked to measure the line segment to the nearest

inch,

(^0) inches 1 2 3 4 5

student says that the line segment is

inches in length.

Example

9. Student thinks that when finding fractions using area models, the equal-sized pieces must look the same.

Student says this diagram does not show fourths of the area of the square because the pieces are “not the same (shape).”

Example

0. Student overgeneralizes and thinks that “all

s (for example) are equal”; she does not understand that the size of the whole determines the size of the fractional part.

Amir and Tamika both went for hikes. Amir hiked 2 miles and Tamika hiked 8 miles.

Student thinks that when both students had completed

of their hikes, they

have each walked the same distance because

Example

Misconceptions and Errors