# Explicitly Model Mathematics Concepts/Skills & Problem Solving Strategies

## What is the purpose of Explicitly Modeling Mathematics Concepts/Skills and Problem Solving Strategies?

The purpose of explicitly modeling mathematics concepts/skills and problem solving strategies is twofold. First, explicit modeling of a target mathematics concept/skill provides students a clear and accessible format for initially acquiring an understanding of the mathematics concept/skill. Explicit modeling by you provides students with a clear, accurate, multi-sensory model of the skill or concept. Students must first be able to access the attributes of a concept/skill before they can be expected to understand it and be able to use it in meaningful ways. Explicit teacher modeling does just that. Second, by explicitly modeling effective strategies for approaching particular problem solving situations, you provide students a process for becoming independent learners and problem solvers. While peers can sometimes be effective models for students, students with special needs require a well qualified teacher to provide such modeling, at least in the initial phases of instruction.

## What is Explicit Modeling?

Explicit modeling involves well-prepared teachers employing a variety of instructional techniques to illuminate the key attributes of any given mathematics concept/skill. In a sense, you serve as a "bridge of learning" for your student, an accessible bridge between the student and the particular mathematics concept/skill they are learning:

The level of teacher support you provide your students depends on how much of a learning bridge they need. In particular, students with learning problems need a well-established learning bridge (teacher model). They learn most effectively when their teacher provides clear and multi-sensory models of a mathematics concept/skill during math instruction.

## What are some important considerations when implementing Explicit Modeling?

The teacher purposefully sets the stage for understanding by identifying what students will learn (visually and auditorily), providing opportunities for students to link what they already know (e.g. prerequisite concepts/skills they have already mastered, prior real-life experiences they have had, areas of interest based on your students' age, culture, ethnicity, etc.), and discussing with students how what they are going to learn has relevance/meaning for their immediate lives.

• Teacher both describes and models the math skill/concept.
• Teacher breaks math concept/skill into learnable parts/steps. Think about the concept/skill and break it down into 3-4 features or parts.
• Teacher clearly describes features of the math concept or steps in performing math skill using visual examples.
• Teacher describes/models using multi-sensory techniques. Use as many "input" pathways as possible for any given concept/skill including auditory, visual, tactile, and kinesthetic means. For example, when modeling how to compare values of different fractions to determine "greater than," you might verbalize each step of the process for comparing fractions while pointing to each step written on chart paper (auditory and visual), represent each fraction using fraction circle pieces, running your finger around the perimeter of each piece, laying one fraction piece over the other one and running your finger along the space not covered up by the fraction of lesser value/area; "thinking aloud" by saying your thoughts aloud as you examine each fraction piece (visual, kinesthetic, auditory), verbalizing your answer and why you determined why one fraction was greater than the other, and having students run their fingers along the same fraction pieces and uncovered space (auditory, visual, tactile, kinesthetic).
• Teacher provides both examples and non-examples of the mathematics concept/skill. For example, in the above example, you might compare two different fractions using same process but place the fraction of greater value/area on top of the fraction of lesser value/area. Then prompt student thinking of why this is not an example of "greater than."
• Explicitly cue students to essential attributes of the mathematics concept/skill you model. For example, when associating the written fraction to the fraction pieces and their respective values, color code the numerator and denominator in ways that represent the meaning of the fraction pieces they use. Cue students to the color-coding and what each color represents. Then demonstrate how each written fraction relates to the "whole' circle:

2/4 = 2 of four equal pieces

• Teacher engages students in learning through demonstrating enthusiasm, through maintaining a lively pace, through periodically questioning students, and through checking for student understanding. Explicit modeling is not meant to be a passive learning experience for students. On the contrary, it is critical to involve students as you model.
• Scaffold your direction as students begin to demonstrate understanding through questions you ask.
• After modeling several examples and non-examples, begin to have your students demonstrate a few steps of the process.
• As students demonstrate greater understanding, ask them to complete more and more of the process.
• When students demonstrate complete understanding, have various students "teach" you by modeling the entire process.
• Play a game where you and your students try to "catch" each other making a mistake or leaving out a step in the process.

## How do I implement Explicit Modeling?

1. Select the appropriate level of understanding to model the concept/skill or problem solving strategy (concrete, representational, abstract).
2. Ensure that your students have the prerequisite skills to perform the skill or use the problem solving strategy.
3. Break down the concept/skill or problem solving strategy into logical and learnable parts (Ask yourself, "What do I do and what do I think as I perform the skill?"). The strategies you can link to from this site are already broken down into steps.
4. Provide a meaningful context for the concept/skill or problem solving strategy (e.g. word or story problem suited to the age and interests of your students. Invite parents/family members of your students or members of the community who work in an area that can be meaningfully applied to the concept/skill or strategy and ask them to show how they use the concept/skill/strategy in their work.
5. Provide visual, auditory, kinesthetic (movement), and tactile means for illustrating important aspects of the concept/skill (e.g. visually display word problem and equation, orally cue students by varying vocal intonations, point, circle, highlight computation signs or important information in story problems).
6. "Think aloud" as you illustrate each feature or step of the concept/skill/strategy (e.g. say aloud what you are thinking as you problem-solve so students can better "visualize" the metacognitive aspects of understanding or doing the concept/skill/strategy).
7. Link each step of the problem solving process (e.g. restate what you did in the previous step, what you are going to do in the next step, and why the next step is important to the previous step).
8. Periodically check student understanding with questions, remodeling steps when there is confusion.
9. Maintain a lively pace while being conscious of student information processing difficulties (e.g. need additional time to process questions).
10. Model a concept/skill at least three times.

## How does Explicitly Modeling Mathematics Concepts/Skills and Problem Solving Strategies help students who have learning problems?

• Teacher as model makes the concept/skill clear and learnable.
• High level of teacher support and direction enables student to make meaningful cognitive connections.
• Provides students who have attention problems, processing problems, memory retrieval problems, and metacognitive difficulties an accessible "learning map" to the concept/skill/strategy.
• Links between parts/steps are directly made, making confusion and misunderstanding less likely.
• Multi-sensory cueing provides students multiple modes to process and thereby learn information.
• Teaching students effective problem solving strategies provides them a means for solving problems independently and assists them to develop their metacognitive awareness.

## What Mathematics Problem Solving Strategies can I teach my students?

Mathematics problem solving strategies that have research support or that have been field tested with students can be accessed by clicking on the link below. These strategies are organized according to mathematics concept/skill area. Each strategy is described and an example of how each strategy can be used is also provided.

## What are additional resources I can use to help me implement Explicitly Modeling Mathematics Concepts/Skills and Problem Solving Strategies?

MathVIDS

MathVIDS is an interactive CD-ROM/website for teachers who are teaching math to students who are having difficulty learning mathematics. The development of MathVIDS was sponsored through funding by the Virginia Department of Education.