Common
Mathematical
Misconceptions
Kitty Rutherford and Denise Schulz
NC Department of Public Instruction
Welcome!
“Who’s in the Room?”
maccss.ncdpi.wikispaces.net
NC EOG/EOC Percent
Solid or Superior Command (CCR)
Grade
2012-2013
2013-2014
2014-2015
3
46.8
48.2
48.8
4
47.6
47.1
48.5
5
47.7
50.3
51.3
6
38.9
39.6
41.0
7
38.5
39.0
40.0
8
34.2
34.6
35.8
Math I
42.6
46.9
48.5
http://www.ncpublicschools.org/accountability/reporting/
Why is this important?
4
Proficiency Rate for Grade 4 End-of-Grade Assessment in Mathematics
8
Proficiency Rate for Grade 8 End-of-Grade Assessment in Mathematics
2008 - 2012
2007 - 2011
2006 - 2010
Turn-and-talk…..
“Why do we see a drop in
achievement scores between
4th grade and 8th grade?”
“Too often, mathematics instruction gives
students the erroneous notion that learning
math is all about learning procedures, rather
than making sense of ideas.”
Marilyn Burns
Butterfly Method
This video clip shares an
experience of a student who
was taught how to use a "trick"
in mathematics. Needless to
say the "trick" didn't teach
mathematical understanding!
http://maccss.ncdpi.wikispaces.net/K-12+Resources
Research: 7 + 52 + 186
Young Children Reinvent Arithmetic
Constance Kamii
Math Tasks
“There is no decision that teachers make that has
a greater impact on students’ opportunities to
learn and on their perception about what
mathematics is, than the selection or creation of
the tasks with which the teacher engages students
in shaping mathematics.”
Lappan & Briars, 1995
Let’s look at some of these
misconceptions…..
“When you multiply a number
by ten, just add a zero to the
end of the number.”
7.47 x 10 ≠ 7.470
Disproven in 5.NBT.7
Addition and multiplication
make numbers bigger.
Works from Kindergarten onwards!
¼ x ¾ = 3/8
Disproven in 5.NF.4.a
0.5 x 0.2 = 0.1
Disproven in 5.NBT.7
Multiplication of Fractions
Two-fifths of the employees at a very large
company has Type A blood. If ½ of the
company’s employees donate blood what
fraction will donate type A blood.
Blue = company
1/2
1/2
Multiplication of Fractions
Two-fifths of the employees at a very large
company has Type A blood. If ½ of the
company’s employees donate blood what
fraction will donate type A blood.
Blue = company
1/5
1/5
1/5
1/5
1/5
Multiplication of Fractions
Two-fifths of the employees at a very large
company has Type A blood. If ½ of the
company’s employees donate blood what
fraction will donate type A blood.
Blue = company
Yellow = Employees with Type A blood
1/5
1/2
1/2
1/5
1/5
1/5
1/5
Multiplication of Fractions
Two-fifths of the employees at a very large
company has Type A blood. If ½ of the
company’s employees donate blood what
fraction will donate type A blood.
Blue = company
Yellow = Employees with Type A blood
1/5
1/2
1/2
1/5
1/5
1/5
1/5
Multiplication of Fractions

1/5
1/5
1/5
1/5
1/5
Multiplication of Fractions

1/3
1/5
1/5
1/5
1/5
1/5
1/3
1/3
Multiplication of Fractions

1/3
1/5
1/5
1/5
1/5
1/5
1/3
1/3
Multiplication of Fractions

1/3
1/5
1/5
1/5
1/5
1/5
1/3
1/3
Multiplication of Fractions
1
2
1
7
1
7
1
7
6
7
•
1
7
1
7
1
7
1
7
Multiplication of Fractions
1
2
1
7
1
7
1
7
•
6
7
1
7
1
7
1
7
Three-fourths of the class is boys. Two-thirds of the
boys are wearing tennis shoes. What fraction of the
class are boys with tennis shoes?
This question is asking what is 2/3 of 3/4
or what is 2/3 x 3/4.
Subtraction and division make
numbers smaller.
6 ÷ ½ = 12
Disproven in 5.NF.7.b
Division of Fractions
5÷⅓=?
Division of Fractions
5÷⅓=
Division of Fractions
5÷⅓=
1
2
4
3
10
11
12
5
6
7
13
14
15
8
9
You always divide the larger
number by the smaller number.
Works with 3.OA.3
½ ÷ 6 = 12
Disproven in 5.NF.7.a
4 ÷ 6 = 2/3
Disproven in 5.NF.3
Division of Fractions
⅓÷5=
Fractions are a rich part of mathematics, but
we tend to manipulate fractions by rote
rather than try to make sense of the
concepts and procedures. Researchers
have conclude that this complex topic
causes more trouble for students than any
other area of mathematics.
Bezuk and Bieck 1993
Use keywords to solve word problems.
• Key words are misleading.
• Many problems have no key words.
• The key word strategy sends a terribly
wrong message about doing mathematics.
A sense making strategy will always work.
Van de Walle & Lovin (2006)
A rule that expires:
Use keywords to solve problems.
• Keywords encourage students to strip
numbers from the problem and use them
to perform a computation outside of the
problem context.
• Many keywords are common English
words that can be used in many different
ways.
Karp, Bush, & Dougherty, 2014
Key Word Strategies
Keywords become particularly
troublesome when students begin to
explore multistep word problems, because
they must decide which keywords work
with which component of the problem.
Karp, Bush, & Dougherty, 2014
Student’s math reasoning…
http://edublog.scholastic.com/post/be-wary-using-key-words-your-math-instruction
Key Words
“Math is not about decoding
clues but about reasoning and
making sense of situations.”
“Flexibility in thinking about
operations is essential.”
(Mathematical lies we tell students, Graybeal).
Math Problem Types
How do you think students would respond to these
questions if they’ve been taught a key word strategy?
Key words don’t work…
A. 27%
B. 8 %
C. 6%
D. 60%
Key words don’t work…
A. 12%
B. 66 %
C. 9%
D. 14%
Key words don’t work…
A. 46%
B. 9%
C. 29%
D. 17%
Key words don’t work…
A.
B.
C.
D.
22%
2%
73%
3%
When students are taught the underlying
structure of a word problem, they not only
have greater success in problem solving
but can also gain insight into the deeper
mathematical ideas in word problems.
Peterson, Fennema, & Carpenter, 1998)
Teaching students to distinguish
superficial from substantive information
in problem also leads to marginally or
statistically significant positive effects on
measure of word problem solving.
Fuchs et al., (2003)
The equal sign means “find the
answer” or “write the answer.”
Problem Types:
• Result Unknown (2 + 3 = 5)
• Change Unknown (2 + ? = 5)
• Start Unknown (? + 3 = 5)
Math Problem Types
8+4=[ ]+5
8+4=[ ]+5
Percent Responding with Answers
Grade
7
12
17
12 & 17
1st - 2nd
3rd - 4th
5th - 6th
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003
8+4=[ ]+5
Percent Responding with Answers
Grade
7
12
17
12 & 17
1st - 2nd
5
58
13
8
3rd - 4th
5th - 6th
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003
8+4=[ ]+5
Percent Responding with Answers
Grade
7
12
17
12 & 17
1st - 2nd
5
58
13
8
3rd - 4th
9
49
25
10
5th - 6th
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003
8+4=[ ]+5
Percent Responding with Answers
Grade
7
12
17
12 & 17
1st - 2nd
5
58
13
8
3rd - 4th
9
49
25
10
5th - 6th
2
76
21
2
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003
3th Grade – 3.OA.3
O
O
O
O
O\
A.
B.
C.
D.
70%
8%
3%
18%
Causing Misconception
• When you multiply a number by ten, just add a zero to
the end of the number.”
• Addition and multiplication make numbers bigger.
• Subtraction and division make numbers smaller.
• Use keywords to solve word problems.
• The equal sign means “find the answer” or “write the
answer.”
Shopping Trip Task
Joseph went to the mall with his friends to
spend the money that he had received for his
birthday. When he got home, he had $24
remaining. He had spent 3/5 of his birthday
money at the mall on video games and food.
How much money did he spend? How
much money had he received for his
birthday?
Principles to Actions pg. 51
Math Task
If the square = 1 whole, what is the
value of each piece?
What questions do
you have?
Contact Information
Kitty Rutherford
kitty.rutherford@dpi.nc.gov
Denise Schulz
denise.schulz@dpi.nc.gov
Website:
maccss.ncdpi.wikispaces.net
For all you do for our students!
For all you do for our students!
Resources Referenced
 Faulkner, V. N. (2013). Common Core.
https://www.engageny.org/sites/default/files/downloadableresources/2014/Dec/why_the_common_core_changes_math_instruction.p
df
 Jacobs, V. R., Martin, H. A., Ambrose, R. C., & Philipp, R. A. (2014).
Warning Signs!. Teaching Children Mathematics, 21(2), 107-113.
http://sites.ssis-suzhou.net/esmaths/files/2014/11/tcm2014-09-107a.pdf
 Karp, K. S., Bush, S. B., & Dougherty, B. J. (2014). 13 Rules That Expire.
Teaching Children Mathematics, 21(1), 18-25.
http://ps186.org/wp-content/uploads/13-Rules-that-Expire.pdf
 National Council of Teachers of Mathematics (NCTM)
www.nctm.org
___________________________________
Slide 1
Common
Mathematical
Misconceptions
Kitty Rutherford and Denise Schulz
NC Department of Public Instruction
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 2
___________________________________
maccss.ncdpi.wikispaces.net
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 3
___________________________________
NC EOG/EOC Percent
Solid or Superior Command (CCR)
Grade
2012-2013
2013-2014
2014-2015
3
46.8
48.2
48.8
4
47.6
47.1
48.5
5
47.7
50.3
51.3
6
38.9
39.6
41.0
7
38.5
39.0
40.0
8
34.2
34.6
35.8
Math I
42.6
46.9
48.5
___________________________________
___________________________________
___________________________________
http://www.ncpublicschools.org/accountability/reporting/
___________________________________
___________________________________
___________________________________
Slide 4
___________________________________
Why is this important?
___________________________________
4
Proficiency Rate for Grade 4 End-of-Grade Assessment in Mathematics
8
Proficiency Rate for Grade 8 End-of-Grade Assessment in Mathematics
___________________________________
2008 - 2012
2007 - 2011
2006 - 2010
___________________________________
___________________________________
___________________________________
___________________________________
Slide 5
___________________________________
Turn-and-talk…..
“Why do we see a drop in
achievement scores between
4th grade and 8th grade?”
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 6
___________________________________
“Too often, mathematics instruction gives
students the erroneous notion that learning
math is all about learning procedures, rather
than making sense of ideas.”
Marilyn Burns
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 7
___________________________________
Butterfly Method
This video clip shares an
experience of a student who
was taught how to use a "trick"
in mathematics. Needless to
say the "trick" didn't teach
mathematical understanding!
___________________________________
___________________________________
___________________________________
http://maccss.ncdpi.wikispaces.net/K-12+Resources
___________________________________
___________________________________
___________________________________
Slide 8
___________________________________
Research: 7 + 52 + 186
___________________________________
___________________________________
Young Children Reinvent Arithmetic
Constance Kamii
___________________________________
___________________________________
___________________________________
___________________________________
Slide 9
___________________________________
Math Tasks
“There is no decision that teachers make that has
a greater impact on students’ opportunities to
learn and on their perception about what
mathematics is, than the selection or creation of
the tasks with which the teacher engages students
in shaping mathematics.”
Lappan & Briars, 1995
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 10
___________________________________
Let’s look at some of these
misconceptions…..
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 11
___________________________________
“When you multiply a number
by ten, just add a zero to the
end of the number.”
___________________________________
___________________________________
7.47 x 10 ≠ 7.470
Disproven in 5.NBT.7
___________________________________
___________________________________
___________________________________
___________________________________
Slide 12
___________________________________
Addition and multiplication
make numbers bigger.
___________________________________
Works from Kindergarten onwards!
___________________________________
¼ x ¾ = 3/8
Disproven in 5.NF.4.a
0.5 x 0.2 = 0.1
Disproven in 5.NBT.7
___________________________________
___________________________________
___________________________________
___________________________________
Slide 13
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 14
___________________________________
Multiplication of Fractions
Two-fifths of the employees at a very large
company has Type A blood. If ½ of the
company’s employees donate blood what
fraction will donate type A blood.
___________________________________
___________________________________
Blue = company
1/2
1/2
___________________________________
___________________________________
___________________________________
___________________________________
Slide 15
___________________________________
Multiplication of Fractions
Two-fifths of the employees at a very large
company has Type A blood. If ½ of the
company’s employees donate blood what
fraction will donate type A blood.
Blue = company
1/5
1/5
1/5
1/5
1/5
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 16
___________________________________
Multiplication of Fractions
Two-fifths of the employees at a very large
company has Type A blood. If ½ of the
company’s employees donate blood what
fraction will donate type A blood.
___________________________________
Blue = company
Yellow = Employees with Type A blood
1/5
1/5
1/5
1/5
___________________________________
1/5
1/2
1/2
___________________________________
___________________________________
___________________________________
___________________________________
Slide 17
___________________________________
Multiplication of Fractions
Two-fifths of the employees at a very large
company has Type A blood. If ½ of the
company’s employees donate blood what
fraction will donate type A blood.
Blue = company
Yellow = Employees with Type A blood
1/5
1/5
1/5
1/5
___________________________________
___________________________________
1/5
1/2
1/2
___________________________________
___________________________________
___________________________________
___________________________________
Slide 18
___________________________________
Multiplication of Fractions
1
2
1
7
1
7
1
7
___________________________________
6
7
•
1
7
1
7
1
7
1
7
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 19
___________________________________
Multiplication of Fractions
1
2
1
7
1
7
1
7
•
___________________________________
6
7
1
7
1
7
1
7
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 20
___________________________________
Three-fourths of the class is boys. Two-thirds of the
boys are wearing tennis shoes. What fraction of the
class are boys with tennis shoes?
___________________________________
This question is asking what is 2/3 of 3/4
or what is 2/3 x 3/4.
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 21
___________________________________
Subtraction and division make
numbers smaller.
6 ÷ ½ = 12
___________________________________
___________________________________
Disproven in 5.NF.7.b
___________________________________
___________________________________
___________________________________
___________________________________
Slide 22
___________________________________
Division of Fractions
___________________________________
5÷⅓=?
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 23
___________________________________
Division of Fractions
5÷⅓=
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 24
___________________________________
Division of Fractions
___________________________________
5÷⅓=
1
2
4
3
10
11
12
5
6
7
13
14
8
9
___________________________________
15
___________________________________
___________________________________
___________________________________
___________________________________
Slide 25
___________________________________
You always divide the larger
number by the smaller number.
___________________________________
Works with 3.OA.3
___________________________________
½ ÷ 6 = 12
Disproven in 5.NF.7.a
4 ÷ 6 = 2/3
Disproven in 5.NF.3
___________________________________
___________________________________
___________________________________
___________________________________
Slide 26
___________________________________
Division of Fractions
⅓÷5=
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 27
___________________________________
Fractions are a rich part of mathematics, but
we tend to manipulate fractions by rote
rather than try to make sense of the
concepts and procedures. Researchers
have conclude that this complex topic
causes more trouble for students than any
other area of mathematics.
___________________________________
___________________________________
Bezuk and Bieck 1993
___________________________________
___________________________________
___________________________________
___________________________________
Slide 28
___________________________________
Use keywords to solve word problems.
• Key words are misleading.
• Many problems have no key words.
• The key word strategy sends a terribly
wrong message about doing mathematics.
___________________________________
___________________________________
A sense making strategy will always work.
Van de Walle & Lovin (2006)
___________________________________
___________________________________
___________________________________
___________________________________
Slide 29
___________________________________
A rule that expires:
Use keywords to solve problems.
• Keywords encourage students to strip
numbers from the problem and use them
to perform a computation outside of the
problem context.
• Many keywords are common English
words that can be used in many different
ways.
___________________________________
___________________________________
___________________________________
Karp, Bush, & Dougherty, 2014
___________________________________
___________________________________
___________________________________
Slide 30
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 31
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 32
___________________________________
Key Word Strategies
Keywords become particularly
troublesome when students begin to
explore multistep word problems, because
they must decide which keywords work
with which component of the problem.
___________________________________
___________________________________
___________________________________
Karp, Bush, & Dougherty, 2014
___________________________________
___________________________________
___________________________________
Slide 33
___________________________________
Student’s math reasoning…
___________________________________
___________________________________
http://edublog.scholastic.com/post/be-wary-using-key-words-your-math-instruction
___________________________________
___________________________________
___________________________________
___________________________________
Slide 34
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 35
___________________________________
Key Words
“Math is not about decoding
clues but about reasoning and
making sense of situations.”
“Flexibility in thinking about
operations is essential.”
(Mathematical lies we tell students, Graybeal).
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 36
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 37
___________________________________
Math Problem Types
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 38
___________________________________
___________________________________
___________________________________
How do you think students would respond to these
questions if they’ve been taught a key word strategy?
___________________________________
___________________________________
___________________________________
___________________________________
Slide 39
___________________________________
Key words don’t work…
___________________________________
___________________________________
A. 27%
B. 8 %
C. 6%
D. 60%
___________________________________
___________________________________
___________________________________
___________________________________
Slide 40
___________________________________
Key words don’t work…
___________________________________
___________________________________
A. 46%
B. 9%
C. 29%
D. 17%
___________________________________
___________________________________
___________________________________
___________________________________
Slide 41
___________________________________
When students are taught the underlying
structure of a word problem, they not only
have greater success in problem solving
but can also gain insight into the deeper
mathematical ideas in word problems.
___________________________________
___________________________________
___________________________________
Peterson, Fennema, & Carpenter, 1998)
___________________________________
___________________________________
___________________________________
Slide 42
___________________________________
Teaching students to distinguish
superficial from substantive information
in problem also leads to marginally or
statistically significant positive effects on
measure of word problem solving.
___________________________________
___________________________________
___________________________________
Fuchs et al., (2003)
___________________________________
___________________________________
___________________________________
Slide 43
___________________________________
The equal sign means “find the
answer” or “write the answer.”
Problem Types:
• Result Unknown (2 + 3 = 5)
• Change Unknown (2 + ? = 5)
• Start Unknown (? + 3 = 5)
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 44
___________________________________
___________________________________
8+4=[ ]+5
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 45
___________________________________
8+4=[ ]+5
Percent Responding with Answers
Grade
7
12
17
1st - 2nd
5
58
13
8
3rd - 4th
9
49
25
10
5th - 6th
2
76
21
2
___________________________________
12 & 17
Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School.
Carpenter, Franke, & Levi
Heinemann, 2003
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 46
___________________________________
3th Grade – 3.OA.3
___________________________________
O
O
O
O
O\
___________________________________
A.
B.
C.
D.
70%
8%
3%
18%
___________________________________
___________________________________
___________________________________
___________________________________
Slide 47
___________________________________
Causing Misconception
• When you multiply a number by ten, just add a zero to
the end of the number.”
• Addition and multiplication make numbers bigger.
• Subtraction and division make numbers smaller.
• Use keywords to solve word problems.
• The equal sign means “find the answer” or “write the
answer.”
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 48
___________________________________
Shopping Trip Task
Joseph went to the mall with his friends to
spend the money that he had received for his
birthday. When he got home, he had $24
remaining. He had spent 3/5 of his birthday
money at the mall on video games and food.
How much money did he spend? How
much money had he received for his
birthday?
___________________________________
___________________________________
___________________________________
Principles to Actions pg. 51
___________________________________
___________________________________
___________________________________
Slide 49
___________________________________
Math Task
___________________________________
If the square = 1 whole, what is the
value of each piece?
___________________________________
___________________________________
___________________________________
___________________________________
___________________________________
Slide 50
___________________________________
Contact Information
Kitty Rutherford
kitty.rutherford@dpi.nc.gov
Denise Schulz
denise.schulz@dpi.nc.gov
___________________________________
___________________________________
Website:
maccss.ncdpi.wikispaces.net
___________________________________
___________________________________
___________________________________
___________________________________
Slide 51
___________________________________
Resources Referenced
 Faulkner, V. N. (2013). Common Core.
https://www.engageny.org/sites/default/files/downloadableresources/2014/Dec/why_the_common_core_changes_math_instruction.p
df
 Jacobs, V. R., Martin, H. A., Ambrose, R. C., & Philipp, R. A. (2014).
Warning Signs!. Teaching Children Mathematics, 21(2), 107-113.
http://sites.ssis-suzhou.net/esmaths/files/2014/11/tcm2014-09-107a.pdf
___________________________________
___________________________________
 Karp, K. S., Bush, S. B., & Dougherty, B. J. (2014). 13 Rules That Expire.
Teaching Children Mathematics, 21(1), 18-25.
http://ps186.org/wp-content/uploads/13-Rules-that-Expire.pdf
 National Council of Teachers of Mathematics (NCTM)
www.nctm.org
___________________________________
___________________________________
___________________________________
___________________________________