Workshop Blog

Assess-Respond-Instruct: Building Math Readiness

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?”

For example, in Saskatchewan curriculum Grade 6 Saskatchewan Patterns and Relations

  • 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.

Some helpful organizational hints include:

  • 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.

If you are interested in learning more about the Assess-Respond-Instruct Framework for building readiness, or would like to bring professional development to your staff in this area, please contact Terry@johansonconsulting.ca

Teaching Patterns in Early Years

Patterns are everywhere. Exploring and identifying patterns can help children understand our number system, operations, spatial understanding and the foundations of algebra. Mathematics is the study of patterns and exploring them through play can begin mathematical and algebraic thinking in early years. Click here for a downloadable version of this post.

There are several big ideas related to patterns:

  1. Patterns exist and occur regularly in the natural and man-made world.
  2. Patterns can be recognized, extended and generalized using words and symbols.
  3. The same pattern can be found in many different forms – physical objects, sounds, movements and symbols.

The progression of patterns through Saskatchewan Curricula:

When viewing patterns, it is useful to know the following terms:

  • Element – an action, object, sound or symbol that is part of a sequence.
  • Core – the shortest string of elements that repeats.
  • Pattern – a sequence of elements that has a repeating core.

Children will develop their ability to recognize and manipulate patterns differently. Some children will move through the following progression:

Exploring patterns also gives children practice and exposure to other mathematical ideas, including:

  • Counting and cardinality – counting the number of items in the unit of a repeating pattern, or how many items are added in an increasing pattern.
  • Adding and subtracting – generalizing about an increasing or decreasing pattern – how many more or less.
  • Position and spatial properties – which element comes next, which element is between two others, reversing order of elements.

How might you teach patterns?

As with many mathematical ideas in early years, it is important to connect ideas. Learning is not linear! It is important that children use physical materials from their environment to build and explore patterns rather than relying on drawing and colouring patterns. Buttons, toys, linking cubes and natural materials can all be used to create patterns.

The Measured Mom has a list of fun ways to engage young children in exploring patterns. It is fun to take children outside. Megan Zeni describes how you might have children explore Patterns Outside and in Nature.  

Repeating Patterns

Repeating patterns can be introduced using concrete objects, sounds, body movements or symbols. Exploring with a variety of materials can help children identify what is creating a pattern.

Pattern Strips can be made using any shape or object. Students can work independently or in groups to copy the pattern on a strip using real objects. These patterns can then be extended. Watch whether they are copying each element separately or if they have identified the core of the pattern and are able to place all of the elements of the core at the same time. This might look like:

  • If the pattern is red/blue/red/blue – children will place the red and blue at the same time.

A significant step in understanding patterns is when children are able to identify that the same pattern exists even when the materials are different. Using some type of symbol, children are able to code a pattern and compare it to other patterns. If they choose to code the pattern using the alphabet, they might describe it as A-B-A-B or A-A-B-A-A-B. An extension with pattern strips is to create the same pattern with different materials.

Pattern Match can happen in many forms.

  • You can give each group a set of different pattern strips, and they find which strips are showing the same pattern.
  • Children can work in groups, one child is the pattern caller. They choose 3-4 pattern strips and lay them face up on their table. They then ‘secretly’ choose one of the strips and calls out the pattern code. Their group members try to identify which strip is being read.

Growing Patterns

In Saskatchewan, children begin to explore increasing patterns in grade 2, and decreasing patterns starting in grade 3.  The beginning of understanding growing patterns is for children to experience building them with concrete objects.

It is important for children to record their observations. A table can help students record the number for each step in the pattern. Using a table, students can predict how many items are needed to create a certain step in the pattern.

Patterns with Numbers

Number patterns are woven throughout our number system, how we perform operations and the ways we represent numbers. John Van de Walle and LouAnn H. Lovin (Teaching Student-Centered Mathematics K-3, 2006) have created a number pattern activity that has students identify how a number string continues by identifying the pattern present.

Rectangle: Rounded Corners: What’s Next and Why?
Show students five or six numbers from a number pattern. The task for students is to extend the pattern for several more numbers and to explain the rule for generating the pattern. The difficulty of the task depends on the number pattern and the familiarity of students with searching for patterns. Here is a short list of patterns, some easy enough for Kindergarten.
1, 2, 1, 2, 1, 2, …	a simple alternating number scheme
1, 2, 2, 3, 3, 3, …	each digit repeats according to its value
5, 1, 5, 2, 5, 3, …	the courting sequence is interspersed with 5s
2, 4, 6, 8, 10, …	even numbers, skip counting by 2s
1, 2, 4, 5, 7, 8, 10, …	two counts, then skip one
2, 5, 11, 23, ….	double the previous number and add 1
1, 2, 4, 7, 11, 16, ….	the number being added increases by 1
2, 12, 22,32, …	add 10
Most of the preceding examples also have variations you can try. Make your own! (Van de Walle & Lovin, 2006)

Skip Counting

Skip counting is an excellent source of patterns. We often limit skip counting to small numbers like 2, 3 or 5. We also often start skip counting at 0. Children can explore the patterns that are created when we skip count by larger numbers, changing the start number. It is a great idea to use a calculator, so students don’t get bogged down in computation!

  • If we start at 7 and skip count by 5’s, what pattern do we see?
    • 7, 12, 17, 22, 27, 32…
  • If we start at 7 and skip count by 55’s, what pattern do we see?
    • 7, 62, 117, 172, 227, 282, 337, …
  • What do we notice about these two patterns?

Patterns in the Hundreds Chart

Start and Jump on the 100s Chart

Using a hundreds chart, have students colour in the pattern created by one of the Start and Jump Numbers sequences given. If different students represent different patterns, what do they notice?

  • How do patterns change when
    • the start number changes?
    • The jump number changes?
  • Which skip count numbers create columns?

Let’s Build the 100’s Chart

  • Using a pocket 100s chart or interactive 100’s chart. Place the following number cards in the pockets:
    • 4, 10, 17, 32, 48
  • Gather students so they can see the 100’s chart easily.
  • Hold up a number related to one of those in the chart, such as 18. Ask “who would like to place this number?”. Explain how you know where to put it.
  • Choose numbers to place based on the number concepts you are working on:
    • If you are working on adding 10, choose numbers that emphasize that concept.
    • If you are working on skip counting, choose numbers that emphasize what you are counting by.

Game: Arrow Clues

  • Clues can be created on cards or written large enough for all players to play the same clues.
  • Arrow clues can look like:
  • undefined
  • Differentiation:
    • Students can play with or without a 100s chart to refer to.
  • Have students describe the impact of each of the types of arrows on the VALUE of the number.

Missing Number Puzzles

Using the patterns in the 100’s chart, children can figure out the missing numbers when only a part of the 100s chart is provided.

Teaching Place Value

Place value and number sense are foundational concepts on which others build over the years in mathematics. Some of the big ideas within place value include:

Concept Progression Over Time

In Saskatchewan, our curriculum identifies the following ideas:

  • In Kindergarten, children learn that counting tells us how many. The whole numbers are in a particular order and there are patterns in the way we say them that help us remember their order.
  • In Grade 1, children understand place value in individual numbers – they look at 17 as a quantity. We can compare and order numbers.
  • In Grade 2, children understand that the value of the digit depends on its location or place.
  • In Grade 3, children consolidate their understanding that the place determines a number’s value.

Ideas for Teaching Place Value

Rekenreks, 5 and 10 Frames

Number sense is a foundation of place value. Relating numbers to ‘friendly’ 5 and 10 are key ideas that can move children past counting.

Try This – Use a rekenrek to show the following:

  1. Representing numbers – how might children use these tools to represent 7? 3? How do they know?
  2. Quick flash – flash a number of beads on a rekenrek and have children tell you what the number is. How do they know this is the number? Are they counting? Or comparing to the ‘friendly’ 5 or 10?
  3. Model numbers in a number string – showing 4, then 5, then 6. Some children will see the pattern of 1 less than 5, 5, and 1 more than 5. You can then repeat with 3, 5, and 7.

Now try to think through these activities using 5 and 10 Frames and linking cubes to show numbers. How is this the same and different than using a rekenrek? There are a number of games and activities involving dot cards and 10 frames that can emphasize 10.

You can find out more on the Building Math Minds Rekenrek activities site.

Subitizing

Subitizing is a foundational skill and occurs when children know that a number of objects is present without counting. Subitizing can occur with random displays of objects or dots, or patterned dots like you would see on a dice, dominoes or ten frames.

Try ThisBuilding Math Minds has a great site for subitizing games. You can find some ideas in this Evergreen Games Overview.

100s Chart

The hundreds chart is an important tool for children to see patterns in our number system. There are a number of games and activities that you can try to emphasize different math ideas.

Try This – There are a number of blogs and vlogs that teachers have created to highlight the 100’s chart. Buggy and Buddy does a good job curating ideas from a number of sources. You can also have children try to find the missing numbers on a 100’s chart to emphasize the patterns in our number system.

Base 10 Blocks

Based 10 blocks are a foundational manipulative to help children understand our number system.

Try This – Go to Hand2Mind website and scroll down to view the lessons provided. These are organized by grade band so that you can find what might fit your students best. Use the base 10 blocks provided to try to work through some of these lessons

Place Value Misconceptions

Misconceptions can be created by a mis-applied pattern, or incomplete understanding of number concepts. The following are some place value misconceptions that occur in Early Years, and some possible instructional strategies to address them.

Misconception: A number is a number, and does not represent a bundle of 10, 100, 1000 etc. objects regardless of its position in a number.

Example: 1 means one, so when it is placed in a number 17, it still represents one rather than 10.

What to do about it? Use the concrete to abstract continuum to represent 17:

  1. Place value blocks or other counters, such as coffee stir sticks.
  2. Arrow Cards
  3. Find the digit on the 100’s chart

Misconception: Students represent numbers after 100 as they sound.

Example: Students think that the number after 100 is 1001, then 1002, 1003, etc.

What to do about it: Use a chart that goes beyond 100, have children fill in the next numbers after 100.

Misconception: The student orders numbers based on the value of the digits, instead of place value.

Example: 67>103 because 6 and 7 are bigger than 1 and 2.

What to do about it: Have students represent numbers using base 10 blocks and then write out expressions using > and < when comparing.

What to do about it: Have students show numbers on a number line to see which numbers are further from zero to the right.

Misconception: The student struggles with the teen numbers, as they are different from the pattern in other decades.

Example: Students may say “eleventeen” or may not understand that 16 is ten and six. They may also think that sixteen is 61 because we say the number six first.

What to do about it: Christina, The Recovering Traditionalist, has curated a number of games and ideas for addressing how to teach the teens.

Having Fun with Math

Mathematics should be playful, and there are a number of games that can build fluency in mathematics.

Combo-10

This game allows students to see how numbers fit together to make 10 using domino-like game pieces. It is for groups of 2 – 4 players.

Try This – Play with at least two people or groups. Each group needs 1 set of dominoes. Lay them face down. Each person/group draws 7. The rest are the draw pile.

  • The player with the highest double (or most dots if there are no doubles drawn) plays first. A piece can be played if the number of dots on one side of the domino adds to 10 with a domino on the table. Doubles can be laid sideways, allowing more arms to grow.
  • A wild card is a domino whose dots add to 10. If you play a wild card, you can play twice.

Snap

Snap is a game played with linking cubes. Each pair receives 10 linking cubes. Players may want to start with the cubes in a stack, alternating colours:

Try This – One player has a stack of 10 cubes behind their back. ‘Snap off’ part of the stack and show the part that is remaining to your partner.

The partner tries to guess how many were snapped off and hidden from view. The unknown part is revealed.

Variations:

  • Using more or fewer blocks in the stack.
  • Breaking the 10 cubes apart and hiding some of them underneath an opaque glass or container.

Race to 100

The goal of this game is to get to 100 first without going over.

Try This – Play the Game

Each player starts at 1. The first player uses a spinner or dice to generate a number. They can move up the 100s chart by their number of tens or ones until one player gets to 100 without going first.

Variations:

  • Each player gets 6 turns. The closest to 100 without going over wins.
  • Continue playing until a player lands exactly on 100. If the roll takes them over 100, they lose that turn.

Math Swat

Adapted from https://kidsactivitiesblog.com/

cool math game

Flyswatter math combines the fun of moving and slapping with the chance to learn number recognition and solving math problems.

Creating the game board: The game board can be as small or as large as you would like and include the number range and type of numbers that you are working with in your classroom.

Try This – Play a Game with two lines of players. Each line has their own swatter.

  • Counting: swat the numbers in order – in either direction.
  • Number recognition: say the number and have learners swat the correct symbol.
  • Counting and 1:1 correspondence: give a number of counters, blocks, etc – they count and then swat the number.
  • Addition or subtraction facts: give the fact, swat the correct sum.
  • Addition facts: give the sum and one addend, swat the missing part.
  • Skip counting: swat the numbers as they count by 2s, 5s, etc.

Using Technology in Mathematics

Technology can be used to enhance mathematics in a number of different ways:

Place Value Online Games

As you know, not all online games are created equally! Sometimes, they are just online worksheet with little engagement. Sheppard Software is a site that encourage practice through play, including flexible thinking about place value.

Try This – Try playing one of the place value games, Underline Digit Value, on Sheppard Software.

Interactive Whiteboards

These whiteboards all allow you and students to share thinking. They can include audio, pictures, and mark ups. Some apps are free, while others require a subscription.

Try This – Log into one of the interactive whiteboards below that you have not used before. Use the username and password provided on the sticky note!

Interactive Manipulatives  – ICT Math

These interactive manipulatives can be used to explore math ideas. These tools are web-based and do not require a log in or download.

Try This – Go to the Arrow Cards tool in the “Teaching Tools” at ICT Math. You can show the value of numbers using arrow cards along with either rek-n-reks or base 10 blocks simultaneously. Show the value for 3299. What happens when you add one more ones digit?

QR Code Scavenger Hunt

This teaching idea comes from Kristin Kennedy and is available free on Teachers Pay Teachers. It would be relatively easy to create your own based on this idea.

Planning for Outcomes-Based Assessment

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?

Used to Know I ThinkProfessional 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:

Know Complete

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. 

question mark

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.

Unpacking Outcomes

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:

Outcome Unpacking

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:

Stiggins Principles

(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?

Criteria Statements

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

Description Statements

Rubric Pitfalls

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

Used to Know I Think 3

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.

Used to Know I Think 1

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).

Formative Summative Cycle

(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.

Used to Know I Think 2

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.

 

 

 

Math Instruction for ALL Students

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!

I am still learning

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.

For example, when we are learning about adding and subtracting fractions, we need to know about:

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

When you analyze grade-level outcomes in mathematics, you will often see combinations of the following pr-skills:

  1. Number Sense and Place Value
  2. Addition and Subtraction
  3. Multiplication and Division
  4. Parts of a Whole – Fractions, Decimals and Percent
  5. Algebraic Thinking

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: https://bit.ly/2zrNfqd which contains folders of resources 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.

Blog at WordPress.com.

Up ↑