Struggling students may miss the important step of connecting the physical model to the more general, abstract concept. Some students may also have difficulty translating from the problem to the model and back again to provide an answer. Implementing the strategies below can help you differentiate instruction.

## Best Practices with Technology

### Step 1: Provide Clear Explanations

- As students become comfortable with a particular model, move to more abstract thinking. Ask them to explain how the model helped them to know that their answer was correct. Encourage them to answer a problem by visualizing the model, rather than using the physical manipulative (see UDL Checkpoint 3.3: Guide information processing, visualization, and manipulation). Alternatively, ask them to generalize a pattern after they have completed a few similar tasks.
- Whenever students find a solution, make sure they connect the model to the original problem. If students have been using the language from the original problem, this should be a natural connection to make. If students refer to the model itself, rather than what it represents, ask them to explain their answer in terms of the original problem.
- Help students to choose their own models. Point out that models remove unnecessary information, which will help them to focus on the mathematics of the problem. Guide them to choose a model that has features that fit the problem they are working on so that they can focus on the mathematics.
What to consider when choosing a model
- Models can take many different forms—they can be physical objects, drawings, or virtual manipulatives. Students may find that each option has different advantages and disadvantages.
- For a problem involving whole numbers, take steps to make fraction pieces less confusing. Students could treat fraction pieces as “wholes” temporarily—e.g., four pieces could represent the number 4—but different sizes and shapes make it easy for students to forget that each piece should then represent the same quantity (one whole).
- Removing the option of choosing different sizes and shapes requires students to treat each whole in the same way. This will help them to remain focused on the problem at hand, rather than trying to remember what each piece means. Physical manipulatives such as base ten unit cubes or counters are both appropriate.

**IES Recommendations**

#### Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

#### Interventions should include instruction on solving word problems that is based on common underlying structures

#### Teach number and operations using a developmental progression.

### Step 2: Give Students Strategies and Models

- Help your students to understand when it is appropriate to use a model by articulating your own thinking. For example, you might say, “
*If I were having difficulty seeing how this works, I would try using 10 colored discs to make a model of the situation. The colored discs would be the 10 colored cars from the problem.*”Different types of models- Drawings
- Manipulatives
- Number lines
- Bar models
- Virtual versions of physical models

- Help your students to understand how different models can be used in different ways to illustrate a concept or solve a problem. Support them in understanding the relationship between different models, and how each model can be used to solve the same problem.
- Facilitate multiple means of expression by introducing students to varied online tools for constructing models. For example, virtual manipulatives can be particularly valuable for students with poor fine motor skills. Determine which students would benefit from being able to touch and pick up objects, and which need computer-based manipulatives.

**IES Recommendations**

#### 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.

#### Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.

Source: IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten Through 8th Grade

#### Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

#### Help students understand why procedures for computations with fractions make sense.

Source: IES Practice Guide: Developing Effective Fractions Instruction for Kindergarten Through 8th Grade

### Step 3: Provide Ongoing Formative Assessment

- Ask students to explain why they chose particular models.
Why student explanation is valuable
- Provides evidence of the student’s understanding of the modeling process and highlights any incorrect assumptions.
- Encourages students to "stop and think" about their decisions and become more strategic in decision making.
- Gives other students an opportunity to ask questions respectfully if a student selects the wrong model. This helps all students to refine their reasoning abilities and helps foster a collaborative learning environment (see UDL Checkpoint 8.3: Foster collaboration and community).

- Consider each student’s needs and learning styles when you decide what actions to take to move students closer to learning goals. Whatever actions you take, give students time to ask you questions, share their thinking, and respond to the feedback you provide.
- If a student selects an inappropriate model, ask them to explain why they chose that model. If needed, follow up by asking about the need to simplify or elaborate features, or the possibility of using other models presented by the teacher or peers.