Kitty Rutherford 2016 AIG Conference March 2, 2016 North Carolina Department of Public Instruction

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. Session description:

Welcome! “Who’s in the Room?”

maccss.ncdpi.wikispaces.net

NC EOG/EOC Percent Solid or Superior Command (CCR) Grade Math I

Why is this important? Proficiency Rate for Grade 4 End-of-Grade Assessment in Mathematics Proficiency Rate for Grade 8 End-of-Grade Assessment in Mathematics

Turn-and-talk….. “Why do we see a drop in achievement scores between 4 th grade and 8 th 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!

Research: 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

Let’s look at some of these misconceptions…..

Karp. Bush, Dougherty, 2014 & Briars, x 10 ≠ Disproven in 5.NBT.7

Karp. Bush, Dougherty, 2014 & Briars, 1995 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

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

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 1/2 1/5

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 Multiplication of Fractions 1/5 1/2

Multiplication of Fractions  1/5

Multiplication of Fractions  1/5 1/3

Multiplication of Fractions  1/3 1/5

Multiplication of Fractions  1/3 1/5

Multiplication of Fractions

Multiplication of Fractions

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.

Karp. Bush, Dougherty, 2014 & Briars, ÷ ½ = 12 Disproven in 5.NF.7.b

Division of Fractions 5 ÷ ⅓ = ?

Division of Fractions 5 ÷ ⅓ =

Division of Fraction s 5 ÷ ⅓ =

Karp. Bush, Dougherty, 2014 & Briars, 1995 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

Provide engaging Math Tasks If the square = 1 whole, what is the value of each piece?

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?

Work with someone beside you to create a problem where the typical “keyword” does not used the operation noted by the keyword strategy! Turn and Talk

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…

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.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.22% B.2% C.73% D.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

Karp, Bush, & Dougherty, 2014 and Briars, 1995 Problem Types: Result Unknown (2 + 3 = 5) Change Unknown (2 + ? = 5) Start Unknown (? + 3 = 5)

Math Problem Types

8 + 4 = [ ] + 5

Percent Responding with Answers Grade & 17 1 st - 2 nd 3 rd - 4 th 5 th - 6 th Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School Carpenter, Franke, & Levi Heinemann, 2003

8 + 4 = [ ] + 5 Percent Responding with Answers Grade & 17 1 st - 2 nd rd - 4 th 5 th - 6 th Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School Carpenter, Franke, & Levi Heinemann, 2003

8 + 4 = [ ] + 5 Percent Responding with Answers Grade & 17 1 st - 2 nd rd - 4 th th - 6 th Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School Carpenter, Franke, & Levi Heinemann, 2003

8 + 4 = [ ] + 5 Percent Responding with Answers Grade & 17 1 st - 2 nd rd - 4 th th - 6 th Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School Carpenter, Franke, & Levi Heinemann, 2003

3 th Grade – 3.OA.3 O O\ A.70% B.8% C.3% D.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.”

What questions do you have?

2014 PAEMST Math State Finalists Elementary Kayonna Pitchford Heather Landreth Meredith Stanley

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Resources Referenced  Faulkner, V. N. (2013). Common Core. resources/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),  Karp, K. S., Bush, S. B., & Dougherty, B. J. (2014). 13 Rules That Expire. Teaching Children Mathematics, 21(1),  National Council of Teachers of Mathematics (NCTM)