© Witzel, 2008

A Few Math Ideas

Brad Witzel, PhD

Winthrop University

© Witzel, 2008

The number codeWrite the numeral next to its appropriate symbol. Use the key as a guide.

3 = ## 5 = ### 6 = <>###

The answers may be 1, 2, 4, 7, 8, 9, or 10a. <>#### _____b. <> _____c. ##### _____d. # _____e. <><> _____f. <>## _____g. <>##### _____h. <># _____i. #### _____

© Witzel, 2008

© Witzel, 2008

Struggles in math per grade

• When entering your class for the first time, what do students not know that they should have learned in previous grades?

• K• 1st

• 2nd

• 3rd

• 4th

• 5th

© Witzel, 2008

How could someone struggle in math with such easy concepts?

• Scalar Multiplication: You can multiply any matrix by a constant called a scalar. This is called scalar multiplication. When scalar multiplication is performed, each element is multiplied by the scalar and a new matrix is formed.

• Cumulative frequency table: To determine a percentile, a cumulative frequency table can be used. In a cumulative frequency table, the frequencies are accumulated for each item.

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© Witzel, 2008

Get rid of the tricks and teach what is happening

• Why is it that when you divide fractions, the answer is larger? Also, why do you invert and multiply?

• 2/3 divided by 1/4 = 2/3 (4/1) = 8/3

2/3 (4/1) 8/38/3

8/3

1/4 (4/1) 4/41/1

© Witzel, 2008

Give examples and explain (adapted from Miller & Mercer)

7 cars 6 groups of- 3 cars x 3 apples___ cars ___ apples

• After seeing this pattern, leave some blanks for students to fill in. Then list needed information to solve, followed by extraneous info. Once students show mastery, have them write their own word problems.

© Witzel, 2008

Build students’ problem solving skills by rewriting word problems

• Reorganize sentences to eliminate extraneous information at first

• Make obvious the variables, key words or phrases

• Build vocabulary into the organization as students progress

© Witzel, 2008

Operations

• Before teaching through concrete manipulation, focus on the eventual outcome.

• Let’s try some– Addition, Subtraction, Multiplication, Division

© Witzel, 2008

Exercises in counting

• Difficulty with math language

• Counting on and counting backward

• Transition into number operations

© Witzel, 2008

Division with fractional answers(adapted from Witzel, 2004)

Say, “How many sets of three go into 8?”

8 sticks

3 cups

Distribute sticks into cups evenly and ask, “Are there an equal number of sticks per cup? How many sticks per cup?”

2 sticks per cup with 2 more sticks that need to be divided into 3 cups.

2 sticks + 2 more sticks

cup 3 cups

© Witzel, 2008

Expanding on the same principles

÷16 ÷ 3 =

= ÷ =

1/3

© Witzel, 2008

When numbers get bigger, the process does not

355 ÷ 3 = 3

118 1/3

355 3 0 0 5 0 3 0 2 5 2 4

1 2 3 R.

© Witzel, 2008

What instruction benefits students who struggle in math?

• Teacher-directed work with guided discovery – Exogenous Constructivism (Moshman, 1982)

– Verbal and Concrete Prompts used to scaffold (Kraayenoord & Elkins, 2004)

• Procedural Instruction (Bryant, Hartman, & Kim, 2003)

– Explicit instruction (Mercer et al, 1994)

– Strategy Instruction (Maccini & Hughes, 2000)

• Teaching concepts through procedures (NCTM, 2007; Wu, 1999)

• Interactions with Math Content (Hatfield et al., 2003)

© Witzel, 2008

Some Tips for Concrete Instruction

• Concrete objects must be demonstrated• Students should be allowed to explore math

principles after knowing how to use them• Concrete does not replace teaching, it requires

more preparation• Use exploration and student language before

teaching formal math vocabulary and stepwise procedures

• Concrete instruction is not sufficient for relevance• Use concrete instruction until students are fluent• Find a way to bridge concrete understanding to

abstract computation

© Witzel, 2008

CRA approach

• CRA is the Concrete to Representational to Abstract sequence of instruction.

• Three stages of learning

• C = Learning through concrete hands-on manipulative objects

• R = Learning through pictorial forms of the math skill

• A = Learning through work with abstract (Arabic) notation

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Findings: Student Think-AloudsFindings: Student Think-Alouds

• Encouraging students to verbalize their Encouraging students to verbalize their thinking and talk about the steps they used thinking and talk about the steps they used in solving a problem – was consistently in solving a problem – was consistently effective effective

• Verbalizing steps in problem solving was Verbalizing steps in problem solving was an important ingredient in addressing an important ingredient in addressing students’ impulsivity directly students’ impulsivity directly (Schunk & Cox, 1986)(Schunk & Cox, 1986)

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Findings: Peer assisted-learningFindings: Peer assisted-learning

• Results have been consistently positive if:Results have been consistently positive if:– Student’s work in pairs and the activities have Student’s work in pairs and the activities have

a clear structure.a clear structure.– The pairs include students at differing ability The pairs include students at differing ability

levels. levels. – Both students play the role of tutor for some Both students play the role of tutor for some

of the time. of the time. – Students are trained in the procedures Students are trained in the procedures

necessary to assume the role of tutor.necessary to assume the role of tutor.

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Findings: Peer assisted-learningFindings: Peer assisted-learning

• Peer assisted-learning appears to benefit Peer assisted-learning appears to benefit both lower- and higher-performing learners both lower- and higher-performing learners because:because:– When serving as tutors, less proficient When serving as tutors, less proficient

students attended to details of problems and students attended to details of problems and the approaches their partner used to problem the approaches their partner used to problem solvesolve

– More proficient students solidified their More proficient students solidified their conceptual understanding of mathematics by conceptual understanding of mathematics by having to explain their problem solving to their having to explain their problem solving to their peerspeers

© Witzel, 2008

Don’t teach harder

• It is our job to prepare students to be successful in math at the next level. Build understanding to prepare for the next grade, next task, or next concept.

• If we can be of any assistance, please do not hesitate to ask.