Transcript Common Mathematical Misconceptions

Kitty Rutherford
Collaborative Regional Education Workshop (CREW)
October 6, 2016
North Carolina Department of Public Instruction
Session description:
Come explore common mathematical misconceptions that
hinder student’s conceptual understanding. What does the
research say about quick fixes and short cuts? Participants
will experience engaging tasks with effective strategies that
build a solid foundation to develop your AIG learners into
strong mathematical thinkers.
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?
Proficiency Rate for Grade 48End-of-Grade Assessment in Mathematics
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 by 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
What Kind of tasks was Kyle provided?
Make my
appointment for
3 o’clock - that’s
when we have
math
Nobody likes
math!
We have it at the
end if the day so
if we don’t get to
it,
it is okay!
Teachers’ belief influence the decisions that they make
about the manner in which they teach mathematics.
Students’ beliefs influence their perception of what it
means to learn mathematics and their dispositions towards
the subject.
Let’s look at some of these
misconceptions…..
¼ x ¾ = 3/8
Disproven in 5.NF.4.a
0.5 x 0.2 = 0.1
Disproven in 5.NBT.7
Karp. Bush, Dougherty, 2014 & Briars, 1995
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.
7.47 x 10 ≠ 7.470
Disproven in 5.NBT.7
Karp. Bush, Dougherty, 2014 & Briars, 1995
6 ÷ ½ = 12
Disproven in 5.NF.7.b
Karp. Bush, Dougherty, 2014 & Briars, 1995
Division of Fractions
5÷⅓=?
Division of Fractions
5÷⅓=
Division of Fractions
5÷⅓=
1
2
4
3
1
0
1
1
1
2
5
6
7
1
3
1
4
1
5
8
9
½ ÷ 6 = 12
Disproven in 5.NF.7.a
4 ÷ 6 = 2/3
Disproven in 5.NF.3
Karp. Bush, Dougherty, 2014 & Briars, 1995
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
If the square = 1 whole,
what is the value of each piece?
Provide engaging
Math Tasks
• 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
How many teachers in your
school have a keyword poster
hanging in their room?
Turn and Talk
Work with someone beside you to create a
problem where the typical “keyword” does
not used the operation noted by the
keyword strategy!
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…
https://www.youtube.com/watch?v=zIJ6ybuDrhc
Key Words
“Math is not about decoding clues but
about reasoning and making sense of
situations.”
“Flexibility in thinking about operations is
essential.”
Graybeal, 2014
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.
B.
C.
D.
27%
8%
6%
60%
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
Math Problem Types
Problem Types:
Result Unknown (2 + 3 = 5)
Change Unknown (2 + ? = 5)
Start Unknown (? + 3 = 5)
Karp, Bush, & Dougherty, 2014 and Briars, 1995
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
A.
B.
C.
D.
70%
8%
3%
18%
Causing Misconception
 Addition and multiplication make numbers bigger.
 When you multiply a number by ten, just add a zero
to the end of the number.”
 Subtraction and division make numbers smaller.
 Use keywords to solve word problems.
 The equal sign means “find the answer” or “write
the answer.”
What questions do
you have?
Announced September 2016...
2014 PAEMST
Elementary Math Awardee
Kayonna Pitchford
Cumberland County
https://www.paemst.org/
Announced September 2016...
2015 PAEMST
Secondary Math Awardee
Lauren Baucom
Union County
https://www.paemst.org/
2016 PAEMST State Finalists
for Elementary Mathematics
Candace Crothers
Chapel Hill-Carrboro
Heather Landreth
Pitt County
Claudia Fann
Guilford County
https://www.paemst.org/
Contact Information
Kitty Rutherford
[email protected]
Website:
maccss.ncdpi.wikispaces.net
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NC Mathematics
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@ncmathematics
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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.pdf
 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