Principles of Evidence –Based Instruction in Mathematics
From the common core website: “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).”
We draw from recent literature reviews and research summaries:
- National Mathematics Advisory Panel Report
- RTI practice guide–Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools (Gersten, Beckmann, Clarke, Foegen, Marsh, Star, & Witzel, 2009): http://ies.ed.gov/ncee/wwc/pdf/practice_guides/rti_math_pg_042109.pdf
- Improving Mathematical Problem Solving in Grades 4-8, Practice Guide http://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=16
- Effective Mathematics Instruction (Natl Dissemination Center for Children with Disabilities) http://nichcy.org/research/ee/math
- National Math Panel fact sheet
- A Summary of Nine Key Studies: Multi-Tier Intervention and Response to Interventions For Students Struggling in Mathematics (Newman-Gonchar, Clarke, & Gersten, 2009)
- Gersten, Chard, Jayanthi, Baker, Morphy, & Flojo (2008), Mathematics Instruction for Students with Learning Disabilities or Difficulty Learning Mathematics: A Synthesis of the Research http://www.centeroninstruction.org/topic.cfm?s=1&k=ST&c=2
- Best Evidence Encyclopedia
- What Works Clearinghouse
Over-arching principles of best practice:
- Prepare for algebra and teach critical foundations in a sequential order
- Practice on basic facts fluency, including properties, and fluency with standard algorithms is important.
- Instruction that is completely teacher or student-centered is not supported by research.
- Explicit instruction on both computation and word problems
- Instruction during the intervention should be explicit and systematic.
- Interventions should include instruction on solving word problems that is based on common underlying structures.
- Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.
- Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.
Framework for Presenting Interventions
We adopt a two-tier method of presenting mathematics interventions in order to assist teachers and other educational professionals in selecting the evidence based intervention that is most likely to be effective for the child in need. In Tier 1, interventions are sorted by the Categories of Mathematics Intervention that are aligned to the Common Core State Standards. In Tier two, interventions are sorted into which stage of the intervention hierarchy they are targeted to address.
Tier 1: Categories of Mathematics Intervention (from the Common Core State Standards in Mathematics)
- Counting & Cardinality
- Operations & Algebraic Thinking
- Number & Operations in Base Ten
- Number & Operations—Fractions
- Measurement & Data
- Ratios & Proportional Relationships
- The Number System
- Expressions & Equations
- Statistics & Probability
Tier 2 – Intervention Hierarchy
After interventions are sorted into the category (or categories) each is further analyzed to considered the primacy focus in terms of the instructional hierarchy. The instructional hierarchy is a model of the stages of learning proposed by Haring and Eaton proposes that all skills are learned in common sequence. First a child acquires the new skill, followed by increasing the speed and accuracy of the skill through practice with feedback. Finally, the skill can be used in other settings or for other uses. For mathematics intervention we use an adapted model of the common reasons why students fail academically proposed by Daly and Martens (1997). This model provides a simple and quite comprehensive approach to quickly selecting functional explanations for academic issues based on the instructional hierarchy. Those interested in an in depth explanation of this framework are directed to read the original article (A model for conducting a functional analysis of academic performance problems (School Psychology Review, 26(4), 554-575). Specifically, the model we use is as follows;
- Acquisition Intervention – The academic activity is too hard and some form of direction instruction is necessary
- Proficiency Interventions – Students need more practice with feedback in order to increase accuracy and speed of the mathematic skill.
- Generalization – The student has demonstrated the skill before, but are having difficulty applying the skill in a new manner