Bradley S. Witzel, PhD Winthrop University witzelb@winthrop.edu Math Interventions for K-8 students...

Bradley S. Witzel, PhDWinthrop University

witzelb@winthrop.edu

Math Interventions for K-8 students

Eastern Pennsylvania Special Administration Conference

Some birds struggle in flight

Some birds don’t

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. #### _____

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

K 51 62 73 84 HS

Struggles in math per grade

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6© Witzel, 2010

Who is at fault?

The college professor said, "Such wrong in the student is a shame, Lack of preparation in high school is to blame.“Said the high school teacher, "Good heavens, that boy is a fool.The fault, of course, is with the middle school."The middle school teacher said, "From such stupidity may I be spared, They send him to me so unprepared.“The elementary teacher said, "The kindergartners are block-heads all. They call it preparation; why, it's worse than none at all.“The kindergarten teacher said, "Such lack of training never did I see, What kind of mother must that woman be.“The mother said, "Poor helpless child, he's not to blameFor you see, his father's folks are all the same." Said the father, at the end of the line,"I doubt the rascal's even mine!"

7© Witzel, 2010

If you were a struggling student in the school where you currently teach explain:

a) How would the teachers know you are struggling?b) What kind of help would you receive for reading,

math, and social difficulties?c) Who would recommend the help?d) Who would provide the help?e) For how long would you receive help?f) How would the school know when you do not need

help or you need a different kind of support?

Our purpose today

You will learn• some potential changes for helping struggling

students• research-supported instruction (math)

– CRA; think alouds; strategy instruction

• ideas for implementation• how to assess and screen students

Special Education vs. General Education

RtI- Tiered Instruction and Intervention

Tier 1: BenchmarkSchool-wide (not Title I) research-supported instruction available to ALL students including standards-aligned concepts and competencies, and instruction.

Tier 2: StrategicAcademic and behavioral strategies, methodologies and practices designed for students not making expected progress in the standards-aligned system. These students are at risk for academic failure.

Tier 3: Intensive InterventionsAcademic and behavioral strategies, methodologies and practices designed for students significantly lagging behind established grade-level benchmarks in the standards aligned system. © Witzel, 2010 11

*******Your Turn*******

• What are your tier 1 instructional ideas?– Name the instruction– What evidence supports your instruction?

• What are your tier 2 interventions?– Name the intervention– What evidence supports your intervention?

What will we do to prepare for RtI in Math, where RtI was not initially

intended?1. Possible interventions2. Research synthesis and meta-analyses3. Progress Monitoring

Assess your implementation (based on IES Practice Guide, Gersten, et al, 2009)

Category Recommendation Yes No How

Overall plan Assessment

Screening all students to identify those who need interventions

Intervention content

Interventions that focus on whole numbers (K-5) and rational numbers (4-8)

Intervention instruction

Interventions are taught explicitly

Structural word problem instruction

Interventions include visual representations

Interventions include at least 10 minutes on fluent fact retrieval

Assessment Progress monitoring for those receiving interventions as well as those at-risk

Motivational strategies for those in interventions

Recommendations

Tier 1: PALS; Explicit instruction; concrete prompts; student interaction

Tier 2: CRA; Strategy instruction

Tier 3: Modified curriculum standards

Research and support provided by: The National Mathematics Panel and Gersten, Baker, and Chard with the Center on Instruction in Math

Interventions

Content and Instruction

Explicit Instruction• Explicit instruction consistently resulted in large effects

both for learning single skills as well as multiple related skills in complex problem solving.

• How do we balance teacher-led instruction with student engagement?

• These findings must be tempered by the fact that the measures on which the effect sizes were calculated were all researcher-developed. Gersten, Baker, & Chard (2008)

Explicit Instruction

The National Mathematics Advisory Panel stated that “Explicit systematic instruction typically entails teachers explaining and demonstrating specific strategies and allowing students many opportunities to ask and answer questions and to think about the decisions they make while solving problems” (p.48).

Explicit Teaching Steps Best Practices consistent in math research

literature:– Advance Organizer– Model– Guided Practice– Independent Practice– Feedback– Maintenance and Generalization

Observe these steps in http://etv.jmu.edu/mathvids/© Witzel, 2010 19

A sample lesson format

1. Warm-up (maintenance) 2. Statement of relevance3. Describe / Model4. Guided Practice5. Independent Practice (process feedback)6. Word Problem (generalization)7. ***Fluency Probe (product feedback)

Let’s try some examples

• First some modeling “I do it”

• Then well work together on some “We do it”

• Then you try some on your own “You do it”

Models of Information TransferCecil Mercer

• “I do it”• “We do it”• “You do it”

Matt McGueThe 5 steps to Apprenticeship1. “I do, you watch, we talk”2. “I do, you help, we talk”3. “You do, I help, we talk”4. “You do, I watch, we talk”5. “You do, someone else

watches”

Feedback and Generalization

• Appropriate feedback

• When should homework involve multiple independent practice problems?

• When should homework involve math problems that explain the application in their own lives?

Grouping strategies

• Station and center teaching• Small group help• Peers

• Keep students moving efficiently through groups. The younger the students, the less time per activity and the smaller the group.

Peer-assisted learning

• Peer assisted learning provides extensive opportunities for students to practice solving math problems and to interact with peers about mathematics

Peer assisted-learning

• 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.

Peer assisted-learning

• 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

Feedback to Students about their Performance

• Providing students with feedback about their performance resulted in moderately large effects.

• For students with disabilities, these effects were smaller.

Clarity in our work with students

What do we focus on?• General Education Curriculum-and-• Number Sense• Basic Facts and Operational Accuracy• Basic Facts and Operational Fluency• Fractions, Decimals, and Proportions• Word Problem Solving

Practice operational facility

• The RtI Panel (Gersten, Beckman, Clarke, Foegen, Marsh, Star, and Witzel, 2009) concluded that all students (K-8) receiving interventions should receive at least 10 minutes of practice per day in fact fluency.

• K-5 should focus on whole numbers• 4-8 should focus on rational numbers

Number Sense (Chard, 2006)

• It has been found that students who evidence math difficulties early struggle in understanding and task performance with number sense concepts, such as counting, quantification or magnitude of number, number to numeral identification, base-10 and place value, and fluent arithmetic strategies.

**New slide**

Ten FramesUsing patterns to earn

numeracy skills and numbers (subitization)

• 3+4=7

• 5+2=7

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Around the tree and around the tree,that’s the way we make a three

Down and over and down some more, that the that’s the way we make a four

Dot notation (Simon & Hanrahan, 2004)

Use different dots for different place values

Numeral and quantity

Base-10 and place recognition

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Language Experiences

d) Base ten Tens Ones

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Teach computation with the “why”

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Practice operational facility

• The RtI Panel (Gersten, Beckman, Clarke, Foegen, Marsh, Star, and Witzel, 2009) concluded that all students (K-8) receiving interventions should receive at least 10 minutes of practice per day in fact fluency.

• K-5 should focus on whole numbers• 4-8 should focus on rational numbers

Findings: Visuals and Graphic Depictions of Problems (CRA)

• Graphic representations of problems and concepts are widely used in texts both in the U.S. and in nations that perform well in international comparisons

• Teaching students to use graphic representations of the underlying concepts of a problem results in moderate effects.

Findings: Visuals and Graphic Depictions of Problems

• Effects were larger when teachers provided students with multiple opportunities to apply graphic representations to specific problems

• Effects were also enhanced when teachers taught students to select appropriate graphic representation and why a particular representation was most suitable

Findings: Visuals and Graphic Depictions of Problems

• When teachers used graphic representations to demonstrate problems only, results were much less consistent.

• Visuals were not particularly useful unless students were provided opportunities to practice using them.

• Highest effect sizes were for CRA with clear and explicit stepwise consistency (Gersten et al., 2009; Witzel, Mercer, & Miller, 2003; Witzel, 2005)

• Concrete to Representational to Abstract Sequence of Instruction (CRA)

• Concrete (expeditious use of manipulatives)• pictorial Representations• Abstract procedures

• http://www.rtitlc.org

Concrete to Representational to Abstract

48© Witzel, 2010

Let’s try some CRA from Multisensory Algebra and the Middle School

Intervention Series

• From the basic facts• To variations that often cause trouble• To crazy things like fractions• To simple equations

CRA with fractions

• After teaching her students subtraction and negative integers through the fraction computation using sticks, Mrs. Straube was able to transition into more abstract terms

2. 6 3/8 6 and 3/8

- 2 7/8 - 2 and - 7/8

4 and - 4/8 3 8/8 – 4/8 = 3 4/8

CRA with expansions on fractions using identity (Zeichner, 2006)

3 (5 + z) 3 (x – 7)

7x 12 7x 12 12 5 5 12

3 (5 + z)3 (x – 7)

Prepare instruction as a series of events

CRA with more difficult fractions

• 5/8 ÷ 2/3

Divide the numerator into 3 groups per tally mark. Then, count the groups of 2.

Lay down an equivalent fraction until you have a whole number of groups

Common Sense for CRA(From Witzel, Riccomini, & Schneider, 2008)

• Multimodal forms of math acquisition to aid memory and retrieval (Engelkamp & Zimmer, 1990; Nilsson, 2000)

• Multiple learning styles are being met to aid relevance and motivation (Oberer, 2003)

• Meaningful manipulations of materials allow students to rationalize abstract mathematics

(Demby, 1997; Noice & Noice, 2001)

• Provides an alternative to algorithm memorization of math rules

• Transportable without concrete materials (Witzel & Allsopp, 2007)

Who benefits from CRA instruction?

• Students with memory concerns• Not an algorithm stepwise approach• Pictorial step allows students to carry manipulative everywhere

• Students with low on-task performance• Students with poor math background

and• Students who lack conceptual knowledge

-or-• Students who have difficulty relating the concept to the

math skill / task© Witzel, 2010 55

What not to do.

Find another way

Trigonometric ratios

hypopp

Measurement - relates the concepts of measurement to similarity and proportionality in real-world situations.

Extend Explicit Instruction to Include Student Think-Alouds

• Encouraging 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 directly (Schunk & Cox, 1986)

Think alouds within explicit instruction

• Explicit instruction requires an instructional sequence that involves the teacher modeling strategies or procedural steps by acting out the problem and verbally describing what thought processes the students should perform. By explaining the reasoning of each step students can link accurate answers to conceptual understanding.

Clues and Undo’s for procedures

5/6X + 4 = 8,solve for X

Clues Undos

Multiply by 5/6 Divide by 5/6

Add 4 Subtract 4

Algebraic - The student describes, analyzes, and generalizes a wide variety of patterns, relations, and functions. (MA.D.1.3)Operations - selects the appropriate operation to solve problems involving addition, subtraction, multiplication, and division of rational numbers, ratios, proportions, and percents, including the appropriate application of the algebraic order of operations. (MA.A.3.3)

Extend Explicit Instruction to Include Student Think-Alouds

• Encouraging 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 directly (Schunk & Cox, 1986)

Think alouds within explicit instruction

• Explicit instruction requires an instructional sequence that involves the teacher modeling strategies or procedural steps by acting out the problem and verbally describing what thought processes the students should perform. By explaining the reasoning of each step students can link accurate answers to conceptual understanding.

Findings: Student Think-Alouds

• Verbalizing appeared to be most effective when multiple approaches to solving problems were demonstrated and students were encouraged to think-aloud as they solved multiple practice problems.

Math Language and Problem Solving

• Math is a language and should be taught explicitly.

1.Explicitly teach vocabulary terms2.Present at least one consistent word problem

approach3.Incorporate math language throughout

instructional delivery

Reading Levels of Math Texts in ESGrade Textbook

publisher Area of Textbook Directions

readability range

Word Problems readability range

3 Popular textbook program 1

2-digit addition, perpendicular lines, lines, angles, and fractions

2nd -5th 4th – 6th

6th – 8th 2nd – 6th

solution and least common multiple

8th – 9th 4th – 6th

6th – 7th 3rd – 6th

Learning Strategies• RIDE

– Read the problem correctly– Identify the relevant information– Determine the operations and unit for expressing the

answer– Enter the correct numbers and calculate then check

• RAPQ (adapted)– Read the problem– Ask what the point is and what is needed to answer– Put the ideas in your own words (paraphrase on paper)– Question your answer and the equation

• See additional learning strategies– http://coe.jmu.edu/learningtoolbox

KW(N)S

What we know

What we want to know

What we do not know

Strategy to solve the problem

Schema-based strategies (see the work of Jitendra and Montague)

• Vary Size of groups unknown: – In school, there are 12 calculators for 24 students to share. How many

students will share each calculator?

• Whole unknown: – Isabelle earned $30 for each day that she worked at the

church store. She worked for 5 days. How much money did she earn?

• Referent unknown (compared is part of referent): Laura and Isha went running. Isha ran 8 laps. She ran 1/2 as many laps as Laura. How many laps did Laura run?

Sample Word Problems

1. The map shows that you have traveled 4 out of 10 miles on your trip. Your friend tells you that you are 3/5 of the way there. Are you 3/5 there? Show why or why not using abstract notation

2. The temperature was 4 degrees below zero. Recently, the temperature rose up 5 degrees. What is the temperature as it relates to zero now? (1) Set up the equation; (2) Explain your reasoning; and (3) Solve the problem.

Change Problem (Gersten et al, p.27)

Compare problem (Gersten et al, p. 28)

Summary• Results of these research syntheses suggest that

students who are struggling with mathematics benefit from:– Verbalizing and use of visuals for problem solving;– Explicit instruction in how to use specific skills and multi-

step strategies;– Their teachers receiving feedback from formative

assessment to modify instruction;– Peer-assisted learning opportunities in which they focus

on problem details and observe models of proficient students’ problem solving

Math is different

“ Do not keep forever on the public road, going only where others have gone, and following on after the other like a flock of sheep. Leave the beaten track occasionally and dive into the woods. Every time you do so you will be certain to find something that you have never seen before” (AGB, 1914).

Be the Problem-Solver

Conclusion

Agodini, R., Harris, B., Atkins-Burnett, S., Heaviside, S., Novak, T., & Murphy, R. (2009). Achievement effects of four early elementary school math curricula: Findings from first graders in 39 schools (NCEE 2009–4052). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.

Allsopp, D. H., Kyger, M. M., & Lovin, L. H. (2007). Teaching mathematics meaningfully: Solutions for reaching struggling learners. Baltimore, MD: PaulH. Brookes Publishing Co.

Butler, F. M., Miller, S. P., Crehan, K., Babbitt, B., & Pierce, T. (2003). Fraction instruction for students with mathematics disabilities: Comparing two teaching sequences. Learning Disabilities Research and Practice, 18, 99–111.

Center on Instruction. (2007). A synopsis of a synthesis of empirical research on teaching mathematics to low-achieving students. Portsmouth, NH: RMC Research Corporation: Author.

Fuchs, L. S., Fuchs, D., Yazdian, L., Powell, S., & Karns, K. (n.d.). Peer-assisted learning strategies: Kindergarten math: Teacher manual. Available from Peer-Assisted Learning Strategies. Web site: http://www.kc.vanderbilt.edu/kennedy/pals.

Fuchs, L. S., Fuchs, D., & Prentice, K. (2004). Responsiveness to mathematical problem-solving instruction: Comparing students at risk of mathematics disability with and without risk of reading disability. Journal of Learning Disabilities, 37(4), 293–306.

Gersten, R., Beckmann, S., Clarke, B., Foegen,A.,Marsh, L., Star, J. R., &Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RTI) for elementary and middle schools (NCEE 2009–4060).Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/ practiceguides/.

Gersten, R., Clarke, B. S., & Jordan, N. C. (2007). Screening for mathematics difficulties in K–3 students. Portsmouth, NH: RMC Research Corp, Center on Instruction.

References

Maccini, P., & Gagnon, J. C. (2000). Best practices for teaching mathematics to secondary students with special needs. Focus on Exceptional Children, 32(5), 1–21.

Maccini, P., Mulcahy, C. A., &Wilson, M. G. (2007). A follow-up of mathematics interventions for secondary students with learning disabilities. Learning Disabilities Research & Practice, 22(1), 58–74.

National Mathematics Advisory Panel. (2008). Foundation for success: The final report of the National Mathematics Advisory Panel. U.S. Department of Education Washington, DC. Retrieved March 2008 from www.ed.gov/MathPanel.

Riccomini, P.J. & Witzel, B. S. (in-press). Rti and mathematics. Thousand Oaks, CA: Corwin Press. Riccomini, P. J., & Witzel, B. S. (2009). Computation of integers: Math intervention for elementary

and middle grades students. Upper Saddle River, NJ: Pearson Education, Inc. Riccomini, P. J., Witzel, B. S., & Riccomini, A. E. (in press). Maximize development in early childhood

math programs by optimizing the instructional sequence. In N. L. Gallenstein & J. Hodges (Eds.), Mathematics for all. Olney, MD: ACEI.

Witzel, B. S. (2005). Using CRA to teach algebra to students with math difficulties in inclusive settings. Learning Disabilities: A Contemporary Journal, 3(2), 49–60.

Witzel, B. S. (2009). Response to intervention in mathematics: Strategies for success. Peterborough, NH: Staff Development for Educators.

Witzel, B. S., & Riccomini, P. J. (2009). Computation of fractions: Math interventions for elementary and middle grades students. Upper Saddle River, NJ: Pearson Education, Inc.

Witzel, B. S., & Riccomini, P. J. (in-press). Solving equations: A math intervention. Upper Saddle River, NJ: Pearson Education, Inc.

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