Math Instructional Strategies

1. Math Instructional Strategies What is the latest research telling us?

2. Research states the following interventions are found to be effective…. • Reinforcement and corrective feedback for fluency • Concrete-Representational-Abstract Instruction • Direct/Explicit Instruction • Demonstration Plus Permanent Model • Verbalization while problem solving • Big Ideas • Metacognitive strategies: Self-monitoring, Self-Instruction • Computer-Assisted Instruction • Monitoring Student Progress • Teaching Skills to Mastery (Sharon D. Hardy, Feb. 2005)

3. Concrete-Representational-Abstract Instruction What is it? • Each math concept is modeled with concrete materials • Multiple opportunities to practice and demonstrate mastery using concrete materials • Concept/Skill is next modeled at the representational (semi-concrete) level which involves pictures that represent the concrete objects. • Multiple opportunities to practice and demonstrate mastery by drawing solutions • Concept/Skill is finally modeled at the abstract level • Multiple opportunities to practice and demonstrate mastery at the abstract level and then moving to a new math concept/skill • As the teacher moves through this process, the numbers/symbols should be used in conjunction with the concrete materials and representational drawings

4. Building Mathematical Concepts Concrete Manipulatives Pictorial Representation Abstract Symbols 4 + 4 = 8 2 x 4 = 8 I I I I I I I I *Significant time must be spent working with concrete materials and constructing pictorial representations in order for abstract symbol and operational understanding to occur

5. C-R-A Demonstration

6. Direct & Explicit Teaching What are the critical components? • Concept/skill is broken down into critical features/elements • Teacher clearly describes concept/skill • Teacher clearly models concept/skill • Multi-sensory instruction • Teacher thinks aloud as modeling occurs • Teacher models examples and non-examples • Cueing • High levels of teacher-student interaction

7. Direct/Explicit Instruction How do I implement it? • Students should have prerequisite skills to perform the new skill • Break down the skill into logical learnable parts • Provide meaning contexts (story problem with age and interests of students ) • Provide multi-sensory inputs • Think aloud with each step • Link each step of the problem solving process (e.g. restate what you did in the previous step, what and why the next step is important to the previous step). • Check student understanding with questions and remodeling when there is confusion • Maintain a lively pace • Model a concept/skill at least three times before beginning to scaffold your instruction

8. Explicit Teacher Modeling http://coe.jmu.edu/mathvids2/strategies/em.html -this site will show videos of explicit teaching in the concrete, representational, and abstract formats using a math story.

9. Sarah’s Bag Read and Understand Sarah has a bag full of blocks. She has red, blue, and purple blocks. Sarah made a bar graph of the blocks in her bag. The graph looks like this. How many blocks are in Sarah’s bag? Underline the question in the story. Plan Use this problem solving strategy to help you. Solve Draw blocks in the bag to show how many blocks Sarah has. Write the number of blocks in her bag. ________

10. Additional Math Resources http://www.carrollk12.org/instruction/elemcurric/math/tkproblemsolving.htm -This site will direct you to not only templates for creating your own math stories but additional resources as well, such as operational posters/maps and brief constructed response formats.

11. Metacognitive Strategies What are they? • A memorable “plan of action” that provides students an easy to follow procedure for solving a math problem • Taught using explicit teaching methods • Includes student’s thinking as well as their physical actions • Must accurately represent the task

12. DRAW Strategy • Discover the sign • Read the problem • Answer or Draw a conceptual representation of the problem using lines, tallies, and/or checks • Write the answer and check

13. STAR Strategy (for older students) • Search the word problem -read the problem carefully -ask yourself “What facts do I know?” “ What do I need to find out?” • Translate the words into an equation form -choose a variable -identify the operation (s) -represent in the concrete form -draw a picture of the representation (semi-concrete form) -write an algebraic equation (abstract application) • Answer the problem • Review the solution -reread the problem -ask “Does the answer make sense? Why? -check answer

14. Using the STAR Strategy Problem: In State College, PA, the temperature was -2F. The temperature rose by 9 degrees F by the afternoon. What was the temperature in the afternoon? Concrete Stage: • Search problem and write down facts. • Translate equation into a picture form (Student put two tiles in the negative area of their mat and nine tiles in the positive area to represent -2 and +9.) Then, cancel opposites, -2 and +2 • Answer the problem by counting the remaining tiles +7 • Review by rereading the problem and checking the reasonableness of the answer. *Need 80% mastery on two probes before going to semi-concrete.

15. STAR Strategy Use the previous problem, to demonstrate this strategy in the representational and the abstract phase. Representational- Instead of manipulative, students represent word problems using drawings of the algebra tiles. Abstract- Represent and solve math problems using numerical symbols, answer the problem using a rule, and review the situation: -2F + (+9F) = x Apply the rule for adding integers, solve the problem x = +7

16. Scaffolding Instruction What is it? • Provides students with the crucial learning support they need to move from initial acquisition concept/skill toward independent performance of the math concept/skill. • Relies both on your observation and on your decision making skills • Occurs after you have initially described and modeled the instructional concept/skill a multiple number of times • Systematic fading of teacher modeling as students demonstrate they have initially acquired the concept/skill. • Immediate and specific feedback, both corrective and positive reinforcement is provided with student responses • Process continues until students demonstrate complete understanding of math concept/skill and demonstrate they are independent

17. Scaffolding Instruction http://coe.jmu.edu/mathvids2/videos/strategies/si/si_clip2_med.mov Demonstrates the implementation of scaffolding: • Models skills at least 3 times • Performs skills while asking questions and answering them aloud • Provide immediate and specific feedback • When answers are incorrect, praise for the effort and risk-taking behaviors while describing and modeling the correct response • When increased competence is shown, fade your direction, prompting students to do more of the problem-solving process. • When you are confident they understand the problem-solving process, invite them to problem solve with other students ( the students direct the problem-solving process with questions and responses) • Let the students responses and nonverbal behaviors guide you to when to fade or continue with prompts

18. What would you do? Objective: The student will understand how to perform the order of operations using the mnemonic “Many Dogs Are Smelly” Sample problem: 7 + 3 x 2

19. Teaching Skills to Mastery • Practice is a means to improve fluency (it does not help them learn a new skill). • When the skills are accurate, provide student practice in a variety of ways: - Instructional games • Self-correcting materials • Learning groups • Peer tutoring • Planned discovery activities “Real competence only comes with extensive practice” (Anderson, Reder, & Simon (1995))

20. Online Resources • The Learning Toolbox http://coe.jmu.edu/learningtoolbox • Math Vids http://coe.jmu/mathvidsr • The Learning Stream http://ferdig.coe.ufl.edu/video/ • Carroll County Schools http://www.carrollk12.org/