1. © Witzel, 2008 A Few Math Ideas Brad Witzel, PhD Winthrop University

2. © Witzel, 2008 The number code Write 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 10 a.<>####_____ b.<>_____ c.#####_____ d.#_____ e.<><>_____ f.<>##_____ g.<>#####_____ h.<>#_____ i.####_____

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4. 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 1 st 2 nd 3 rd 4 th 5 th

5. © 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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7. 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 / 3 8 / 3 8 / 3 1 / 4 ( 4 / 1 ) 4 / 4 1 / 1

8. © 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.

9. © 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

10. © Witzel, 2008 Operations Before teaching through concrete manipulation, focus on the eventual outcome. Let’s try some –Addition, Subtraction, Multiplication, Division

11. © Witzel, 2008 Exercises in counting Difficulty with math language Counting on and counting backward Transition into number operations

12. © 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

13. © Witzel, 2008 Expanding on the same principles ÷ 16 ÷ 3 = = ÷ = 1/31/3

14. © 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 123R.

15. © 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)

16. © 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

17. © 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

18. © Witzel, 2008 18 Findings: Student Think-Alouds Encouraging students to verbalize their thinking and talk about the steps they used in solving a problem – was consistently effectiveEncouraging students to verbalize their thinking and talk about the steps they used in solving a problem – was consistently effective Verbalizing steps in problem solving was an important ingredient in addressing students’ impulsivity directlyVerbalizing steps in problem solving was an important ingredient in addressing students’ impulsivity directly (Schunk & Cox, 1986)

19. © Witzel, 2008 19 Findings: Peer assisted-learning Results have been consistently positive if:Results have been consistently positive if: –Student’s work in pairs and the activities have a clear structure. –The pairs include students at differing ability levels. –Both students play the role of tutor for some of the time. –Students are trained in the procedures necessary to assume the role of tutor.

20. © Witzel, 2008 20 Findings: Peer assisted-learning Peer assisted-learning appears to benefit both lower- and higher-performing learners because:Peer assisted-learning appears to benefit both lower- and higher-performing learners because: –When serving as tutors, less proficient students attended to details of problems and the approaches their partner used to problem solve –More proficient students solidified their conceptual understanding of mathematics by having to explain their problem solving to their peers

21. © 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.