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.

Outcomes-Based Assessment (OBA) has been on our educator radar for years. I have the pleasure of working with groups of teachers throughout Saskatchewan to dig into what we know, what we wonder about and examine logistical barriers or problems to solve in order to move forward.

# What do teachers know? What do teachers wonder about?

Professional development needs to surface teacher knowledge, including any misconceptions that might exist. Too often, professional learning facilitators assume that educators do not know anything so begin from the beginning… or assume that educators know everything and are choosing to resist change. I would argue teachers know a lot… and they, as a collective, want to do best for students and learning. Just like in a classroom, misconstruction of knowledge can occur. It is our job as learning facilitators to use our formative assessment skills to expose understanding and misunderstanding so that we know what to do next.

When teachers are asked, What do you know about Outcomes-Based Assessment? Their answers might be similar to those generated in NLSD:

It is important when broad statements are made that they are clarified by the group.

• Clarification may be needed on the term ‘learning behaviours’. These include things like attendance, behaviour, neatness, compliance with assignment expectations. Schools or systems may have other ways to communicate these ‘Hidden Curriculum’ expectations to students and parents outside of their academic achievement scores.
• Clarification may be needed around the idea that assessment is based on “where they are at right now… can change over time”. An example where a student shows competency later in the year after that unit of study has been completed. This may raise some logistical questions around how this would work within a student information system or what impact this idea has on reporting. Once specific questions or logistical barriers surface, it is possible for a school or system to determine procedures so that they can have consistency.

As Tomas Guskey states, there is NO best practice in grading. There are ‘better’ practices that we want to embrace, but there is no universal, standardized and mechanical way to generate a grade for our students.  This was an empowering point with teachers to know that their professional judgment, based on an understanding of curricular outcomes and observable student behaviours, is the most important assessment practice.

Along with what educators know, it is vital that we surface what they wonder about. Questions can frame teachers’ professional inquiry for a day of learning, as well as indicate what they need to be emphasized within the agenda. Typical questions around this topic may be:

• How do I translate an outcomes-based assessment rubric into a %?
• How do we gather, translate and score observations and conversations so that they ‘count’ like products?
• What might a teacher daybook/unit plan look like using outcomes-based assessment?
• Is all assessment outcome-based assessment?
• What do we do if an assignment is late or not handed in?
• What is the minimum/maximum number of indicators that we need to assess in order to maintain the integrity of the outcome?
• How do we use outcome-based assessment in cross-curricular teaching?

It is important that participants choose which question(s) they are most invested in to solve, and provided time within a professional learning experience to discuss possible solutions with colleagues.

Assessment practices are founded on both beliefs and knowledge. A Talking Points Strategy can help to have small groups explore and surface their beliefs about assessment.

# Starting with Curriculum

Learning targets are based on curricular outcomes. There are a number of different unit and lesson planning templates used in education. One useful process is to use a thinking map. This graphic organizer allows us to see the connections amongst curricular outcomes, instructional activities and assessment criteria.

Starting in the centre, teachers can identify the connections between the nouns (concepts) and verbs (observable behaviours) of the curriculum with the activities that allow students to show those behaviours. The assessment criteria should be related to the curriculum rather than the activity.

For example, in Saskatchewan Science 10, one part of the SCI10-CD1 Outcome: Assess the implications of human actions on the local and global climate and the sustainability of ecosystems. Some of the indicators related to this outcome might be addressed in the following progression:

By unpacking into a circular thinking map, it is possible to see how the concepts and observable behaviours work together. This will lead to a holistic view of curriculum that eradicates the question of how many indicators are important to address.

# Principles of Assessment

Rick Stiggins has developed a set of key ideas related to classroom assessment:

(Chapuis, Commodore, Stiggins, 2016)

## From Criteria to Rubrics

There are a variety of assessment tools, including checklists, portfolios, and rubrics. They all rely on clear learning targets or criteria for student success. What does success look like? What are we looking for?

Expanding on clear learning targets, Sue Brookhart shares some of her ideas on building high-quality rubrics.

Sue Brookhart’s ideas have been incorporated into this simple editable Rubric Worksheet.

# Formative and Summative Assessment

Too often, formative assessment is defined as ‘things that are not marked’, while summative assessment is defined as “things that are graded at the end of a unit”. This implies that learners can only show understanding that ‘counts’ at the end of a unit of study. So what happens to all of their thinking, work and brilliance along the way? Is it possible that a learning and assessment experience might be both or either for different students? Is it possible that formative and summative assessment are interconnected?

## Definitions

One definition for assessment is the ways in which instructors gather data about their teaching and students’ learning (Northern Illinois University, Faculty Development and Instructional Design Center). This definition implies that assessment’s purpose is multi-faceted – to inform students and teachers regarding student understanding as well as to inform teachers about their practice in teaching. Assessment, whether it is formative or summative, is a snap-shot in time that changes with instruction and understanding.

## Formative Assessment

In his book, Embedding Formative Assessment, Dylan Wiliam defines Formative Assessment as:

“An assessment functions formatively to the extent that evidence about student achievement is elicited, interpreted, and used by teachers, learners, or their peers to make decisions about the next steps in instruction that are likely to be better, or better founded, than the decisions they would have made in the absence of that evidence” (Wiliam, 2011).

This definition implies:

• Formative describes the function of the assessment rather than the form.
• Teachers, students and peers might be involved in deciding how to respond to assessment information.
• There must be a responsive action based on the data in order for the assessment to be formative. Responsive actions are instructional in nature.

If formative assessments are designed with no clear decision/action implied, then the assessment is not useful. The five key strategies for improving student achievement through formative assessment are:

 Who Where the learner is going Where the learner is now How to get there Teacher 1. Clarifying, sharing and understanding learning intentions and criteria for success. 2. Engineering effective classroom discussions, activities, and tasks that elicit evidence of learning. 3. Providing feedback that moves learning forward. Peer 4. Activating learners as instructional resources for each other. Learner 5. Activating learners as owners of their own learning.

(Wiliam, 2011, p. 46)

## Summative Assessment

Summative assessment is often described as providing information about or evaluating the attainment of understanding or achievement compared to a standard. Katie White (Softening the Edges, 2017) has created a holistic view of summative assessment as part of a larger assessment cycle.

“We engage in formative assessment, feedback and self-assessment regularly. Only after all this do we verify proficiency with summative assessment. It is at this point that we make professional judgments about whether to re-enter the learning cycle because proficiency has not yet been reached or to transition into enrichment or the next learning goal… Viewing summative assessment as part of a larger continuous cycle frees us to make decisions that are right for our learners and right for ourselves” (White, 2017, p. 139).

(The Learning and Assessment Experience at UNSW)

The goal of summative assessment is to evaluate student learning. When viewed as part of a cycle, we can see that an assessment intended to be summative may, in fact, become formative. Similarly, there may be times that an assessment intended to be formative might become summative if a learner is able to show proficiency during that experience.

If we view the terms formative and summative as how the assessment is used rather than the tool or the intent for use, it can help us to see all experiences as part of a larger assessment plan.

Brookhart, S. (2013). How to Create and Use Rubrics for Formative Assessment and Grading. Alexandria: ASCD.

Chappuis, S. J., Commodore, D. C., & Stiggins, R. J. (2016). Balanced Assessment Systems: Leadership, Quality and the Role of Classroom Assessment. Thousand Oaks: Corwin.

Guskey, T. R. (2019, February 28). Let’s Give Up The Search for ‘Best Practices’ in Grading. Retrieved from Thomas R. Guskey & Associates: http://tguskey.com/lets-give-up-the-search-for-best-practices-in-grading/

UNSW Sydney. (n.d.). Guide to Assessment. Retrieved March 12, 2019, from UNSW Student Home: https://student.unsw.edu.au/assessments

White, K. (2017). Softening the Edges. Bloomington: Solution Tree.

Wiliam, D. (2011). Embedded formative assessment. Bloomington, Indiana, United States of America: Solution Tree Press.

This blog post is a work in progress! Be sure to come back and visit in a few weeks, as I will be adding to it over time…

It can sometimes feel overwhelming when we look at all of the individual and group needs of our mathematics learners. Building readiness to learn, along with ensuring that we meet the individual needs of students might give us the impression that we need to create an individual lesson plan for each and every person in our classrooms. That sounds exhausting…

But what if we can create structures and use a variety of math instructional strategies within those structures? What if we can create diverse learning experiences that encourage mathematical thinking and growth over key concepts? This is an idea worth investigating!

Our elementary and middle years math curricula in Saskatchewan cover a number of topics, from number to patterns to shape and space and statistics. Ironically, when you look at the skills needed for students to be READY to engage in these grade-level concepts, there are only a handful of pre-skills. These pre-skills are the math concepts that are applied and used in new learning.

• multiplication and division, multiples and factors
• what a fraction is, finding equivalent fractions, improper fractions

So, how do we teach and reteach each of these key concepts in our classrooms? You can find a large number of curated resources in this Google Drive which contains folders of resources

These folders are designed to help you to teach through these continuums as well as:

• What is an instructional sequence and strategies for teaching each concept?
• What are some common misconceptions and how might we address them?
• How might we infuse technology into our mathematics instruction?
• What are some fun ways to engage in mathematics?
• How might we use math with a purpose to gain a deeper understanding of social issues?

## Concept Continuums

When we look at our Saskatchewan curriculum, we can see how concepts grow over time in these math Curricular Through Lines:

We can also pull out specific concepts and see how they grow. The following concept trajectories were created by a province-wide math leadership group a number of years ago, and show the language, strategies and concepts over time. Each continuum has four instructional strategies listed.