Our conversations in mathematics teaching are often centered around the gaps that we observe in student understanding, or how students are not ready to learn the grade level mathematics that we are trying to teach them. When we look at our teaching practices in other subjects, we know that it is important to Activate and Connect Prior Knowledge, and to provide responsive instruction if there are skills that our students are missing. The same holds true in mathematics, but how do we do this in a structured, systematic and efficient way in our classrooms? The Assess-Respond-Instruct framework, developed with and implemented by teachers from across Saskatchewan, does exactly that.

Foundationally, the Assess-Respond-Instruct framework provides opportunities for

• teachers and students to know whether students have an understanding and fluency with prior knowledge, and
• filling gaps in knowledge and build fluency, and
• engage in grade level mathematics.

In order to embark on this way of teaching, some key questions that drive our planning are:

## Differentiation vs Modification

A key idea within mathematics is the difference and similarity between differentiation and modification. Working with a school last week, we brainstormed the following key ideas:

Sometimes, a student needs a modified curriculum because they are unable to grasp mathematical concepts. This determination is made with much consultation with the education team, parents, and students. Communication is key between home and school to ensure that parents understand that their child is not working towards grade level outcomes. Rather, they are on a modified curriculum with modified assessment expectations. When a child is working towards a modified curriculum, it should still be differentiated. Students need to experience a variety of modes and strategies to help them achieve their unique learning goals.

The difficulty is when a student or class is inadvertently experiencing a modified curriculum without the pre-thinking and opportunities to engage in grade level mathematics. This might look like a child being identified as ‘not being able’ to add and subtract in grade 4, so they only work on addition and subtraction when their classmates are working towards multiplication and division. In this example, the child is not given an opportunity to engage in grade level outcomes, so the gap in their learning is even larger the following year.

So what is a possible solution? The Assess-Respond-Instruct Framework!

## Designing Pre-Assessments

Pre-assessments should focus on mathematics knowledge that students need in order to be ready to engage in new, grade-level instruction.

### Content to Pre-assess

We can identify the pre-skills necessary for a new unit of study by mapping curriculum and asking ourselves “What might students know before this grade to help them understand the content at our grade level?”

• P6.1 – Extend understanding of patterns and relations in tables of values and graphs.
• P6.2 – Extend understanding of preservation of equality concretely, pictorially, physically, and symbolically.
• P6.3 – Extend understanding of patterns and relationships by using expressions and equations involving variables.

In this Grade 6 Saskatchewan example, the blue concepts are grade-level, while the yellow concepts are mathematical ideas that appear in curriculum before Grade 6. By mapping curriculum, you can see that new, grade-level instruction is only one small step beyond what students have experienced in the past.

Analyzing the pre-skills from the example above, we can see that they can be clustered in the following way:

It is important to identify the extent to which students might need to understand a concept. In this Grade 6 example, students need to understand and be fluent in addition with single digit numbers, and subtraction of double digit minus single digit numbers. We would not usually expect students to work with larger numbers when we are solving algebraic equations at this grade level. Even though curriculum has students learn and practice addition and subtraction to 10,000 in Grade 5, we do not need students to use these large numbers in THIS unit of study, so we would not pre-assess or respond to those large numbers.

The content of a pre-assessment for this example unit of study would include:

• Addition of single digit numbers and subtraction of numbers no larger than 100.
• Multiplication of single digit numbers and division of numbers no larger than 100.
• Representing relations, including tables of values and graphs.
• Solving one step equations, including balance scale representations and missing value equations.

### Forms of Pre-assessments

You might have assessment data that you have gathered through school-system pre-assessments, through tools like Pearson’s Numeracy Nets, or you can develop your own simple Pre-assessments.

## How might we respond to gaps in pre-skills?

We need to consider both the content and structure that we are using to respond to gaps in understanding. Many teaching innovations focus on one or the other. I would suggest that we need to consider both the content of intervention as well as the process, or structure, that we use to have students interact with that content.

### What is Responsive Content? Differentiating Mathematics Intervention

Too often, our mathematics intervention in upper grades involves symbolic practice of a topic that a student is unsure of. Rather than only focussing on symbolic practice, we need to differentiate our intervention – additional practice worksheets are not enough if a student does not understand.

What does differentiation look like in mathematics? If we consider NCTM’s ways of representing algebraic ideas, and the Theory of Multiple Intelligences, a simple way to look at differentiation for every math concept might be:

Whether we are looking at responsive instruction or new instruction, it is important that students are given opportunities to learn new concepts:

• Concretely and visually
• Video – this can help auditory learners watch and listen to math concepts
• Written explanations – a simple and concise description of that mathematical idea
• Game – a way to interact with peers and have mathematical conversations
• Practice – to build fluency with foundational math ideas

A planning organizer is helpful in identifying the components that you will have ready for students who need intervention in each topic. The concepts that we focus on for responsive instruction are those identified in our pre-skill analysis of our next unit of instruction. The modes of responsive instruction need to be thought out for each concept, or skill. This provides a robust framework for intervention.

## A Classroom Structure – Responsive Stations

There are many ways to structure your classroom to ensure that your students are receiving the instruction that they need. These might include classroom routines that focus on readiness skills, or rotational stations like Daily 3 Math. One innovative structure is Responsive Stations.

### Implementation

Once you have the pre-assessment data, students go to those stations that their data on their pre-assessment indicates that they need.

• Colour coding your boards helps students know where they are heading.
• Use a tracking sheet to monitor which pre-skills each student needs to address.
• Use stickers as rewards to track what stations have been done.
• Use a short post-assessment to determine that a student understands the content.
• Use bins of materials at each station to help keep organized.
• Include an enrichment station for those students who have pre-skills in place. This enrichment station can include games, additional math topics, and ideas such as creating new videos or games based on math concepts.

Once you have provided opportunities for students to be ready for your grade-level instruction, you can then teach new concepts using rich instructional practices that we know help students understand. Through the year, your class will revisit the same pre-skills over time, as many topics repeat as pre-skills throughout curriculum.

## Sample Responsive Station Planners

These planners are living documents, created and revised by teachers.