Living the Gradual Release of Responsibility

When planning for my newest workshop on the Gradual Release of Responsibility, I had a HUGE epiphany… when thinking about what the barriers to implementing this classroom structure, I realized that while research shows us that the Gradual Release of Responsibility Framework supports learners:

Fisher and Frey Framework for Gradual Release of Responsibility

But too often, professional development follows this framework:

Fisher and Frey an Unhelpful Framework

And we know that this framework is not helpful to learners. Too often, we as designers of professional learning share our thinking and then expect teacher learners to implement on their own. This realization hit me like a brick. So how to use the Gradual Release of Responsibility framework as the design framework for professional learning?

Planning to Plan for Instruction

  • I Do: Modelling Planning
  • We Do: Shared Planning
  • You Do: Collaborative Planning
  • You Do: Independent Planning

I Do: Modelling Planning

Graphic organizers are useful tools for student and adult thinking. I began by designing a Gradual Release of Responsibility planning template that goes through the key steps in planning for instruction. Of course, it is important to recognize that the order of I Do – We Do – You Do is not the only order for instruction!

Starting with an example from the Grade 1 Science curriculum, I decided that the outcome and indicators focusing on seasonal changes would be a good place to insert student writing. From here, I started searching for a mentor text that I could use for modelling descriptive words and showing seasonal changes. I decided to use “Goodbye Winter, Hello Spring” by Kenard Pak. A helpful hint is to search a book title on YouTube to see if it fits what you are hoping for. When selecting a mentor text, it is important to identify the key questions that you might have students think about while you and they are reading.

Once I have the curricular connections and mentor text in place, I can then identify what I will include in my modelling/think aloud.

The “I Do” in this instructional sequence might include:

  • Features of the mentor text
  • Modelling writing
  • Mini lessons on writing or science related to this writing project

The “We Do” in the instructional sequence might include

  • Shared writing using the mentor text example

The “You Do: Together” in the instructional sequence might include

  • Collaborative rewriting of different passages of the book

The “You Do: Independent” in the instructional sequence might include

  • Drawings and description of one plant or animal that changes between the winter and spring seasons.

You can see a completed (DRAFT) graphic organizer for this sequence on Seasonal Changes here.

We Do: Shared Planning

In shared planning, workshop participant and I will co-construct a second instructional sequence, choosing one outcome from a pre-selected list of outcomes that all lend themselves to student writing. For example:

  • If the outcome chosen is Grade 5 Social Studies: Government Structure, we can use “Canada Votes” as our mentor text. In many communities, it would be helpful for students to expand the description of government in this book to include First Nations governance and elections.
  • If the outcome chosen is Grade 3 Mathematics: Passage of Time, we can use “A Second is a Hiccup” as our mentor text. This would allow children to explore what activities take different measures of time.

You Do: Collaborative Planning

  • At this stage, teachers will work with grade-similar peers and co-create a series of lessons that they are interested in.
  • It is important here to co-construct criteria for what makes a good plan.

You Do: Independent Planning

  • Finally, teachers will use the planning template to create a series of lessons for their own curriculum. These ideas can then be shared through a Speed Dating structure so that peers can hear creative ideas that might apply to their own classrooms.

By using the Gradual Release of Responsibility framework to learn about the Gradual Release of Responsibility Framework, the hope is that teachers will be able to experience as learners the power of this model. When we experience something as learners, we have increased understanding and confidence to use it.

Writing about Teaching Writing

I had the chance to work with the wonderful staff at Rossignol Elementary in Ile-a-la-Crosse this winter. They, like many of us, have been wondering how to support their students to be more engaged writers. They wondered:

How do we engage student writers? How do we make our students feel that they ARE writers and authors in our classrooms? There are so many blogs and ideas written on this topic, but what might work for OUR students?

There are many blogs and resources that are useful when seeking out new and innovative ideas to try. Some good ones include (but are certainly not limited to!):

Each of these articles is full of lists of creative strategies. But what works for YOUR students? This is where professional conversations and thinking about what students you have in your context can help.

We often ponder the question “Why don’t our students write more?”… something to consider is how much writing we do ourselves as adults. I have to admit that prior to creating this workshop blog, I might have gone weeks without writing outside of emails or filling in forms. To give insight into some of the barriers that our students face when writing, it is important for us to consider our own writing habits (and maybe fears!). How do we encourage ourselves as writers? My epiphany when planning for this workshop was that perhaps writing professional development needs to following the same framework that we might use for our students.

Adrienne Gear (2014) suggests the following lesson framework for each nonfiction form:

  1. An introduction to the features of the nonfiction form.
    1. This can be done by analyzing published examples of a nonfiction form.
  2. Independent write and Whole-class write can be woven together in a We DO – You DO cycle.

With this framework in mind, the teachers of Rossignol Elementary worked in collaborative groups to write the following ideas for engaging their student writers:

Shared Writing About Writing

Engaging Student Writers is about the things that we can do as teachers to encourage students to write across all curricula. There are many different strategies that fit different grade levels and different content areas. It is important that we use our professional judgement to combine our existing professional knowledge, our knowledge of our students, and the new information we learn from our colleagues and research.

Drawing and Talking to Encourage Writing in Young Children

To encourage young children to write, have them group together to talk about a common experience (sliding, wiener roast, building a snowman). As children share information, the teacher can capture the vocabulary they are using.

Teachers can

  • Record on a chart or individual cards/word strips
  • Include a picture
  • Display in the classroom

Children can

  • take those word strips/cards and draw their own picture
  • describe/talk about their picture with an adult or older student
  • label the picture (either by the child or the adult/older student)

Differentiating Writing

How might we differentiate next steps? The sequence will depend on the age and ability of young writers.

Younger writers might have an older student or adult scribe a sentence for them. They can leave enough space underneath for the child to copy the words below.Older writers might use the labels on their picture to write a sentence or sentences about their picture.

If students’ oral language skills are low, they can communicate meaning through the use of point pictures or flash cards. Key ideas related to the pictures can be created in advance by the teacher.

Extra Time for Encouraging Elementary Writers

There are many ways that we can encourage our elementary-aged writers in our classrooms.

  • When we give more time to write, we encourage writers to write more often. Allowing more time to organize their thoughts and ideas, using graphic organizers, modelling writing and brainstorming together can all contribute to student confidence.
  • Deadlines and expectations need to be communicated clearly so that students understand what needs to be produced and when it needs to be produced by.
  • Use technology like voice typing for those who can’t write as fast as they think can reduce frustration and get ideas out.
  • For those students who may be shy, strategies like passing notes, chatting 1:1 with peers about the topic, and think-pair-share can help to build confidence.

With extra time and strategies to maximize the time, students are allowed to process their thoughts and make meaning. This can help to show them that they ARE good writers.

Comic Book Writing for Engaging Writers

Comic book writing is when students write the dialogue into a blank comic template. There are various templates that you can download from sites such as this one from Scholastic.

Where to start? You might start with a “We Do” comic strip, then move to “You Do” by having students write dialogue into a given template with pictures already provided. They can then move to creating independently by choosing their template, and eventually creating their own pictures, characters and words either by using clipart or drawing their own original comic.

Mentor texts can include Manga, Amulet, Archie Comics or Marvel Comics. The use of mentor texts is key to introduce and discuss examples of dialogue and how words and pictures interact.

Comic book writing can encourage all types of writers, as it is a unique combination of visual/writing skills to tell a story.

What We Learned About Teaching Writing

I am thankful to the teachers of Rossignol Elementary for agreeing to their writing going out to an authentic audience on my Workshop Blog. By experiencing shared writing as adult learners, we discovered what some of the underlying anxieties and fears might be for our students. Worry about being wrong, worry about not being good enough, experiencing how daunting a blank piece of paper is all contribute to deepening our understanding of what to do for our students.

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As adults, we experienced the power of writing to deepen our understanding of writing to a much deeper level than just reading about writing might have done. New knowledge, combined with our professional knowledge and knowledge of our students can help us to encourage our students to BE writing, not just engage in writing.

Gear, A. (2014). Nonfiction Writing Power. Markham: Pembroke Publishers.

Johanson, T., & Broughton, D. (2014). Exploring Comprehension in Physics. Saskatoon: McDowell Foundation.

Teaching Writing Across Curricula

Reading and writing are things we expect our students to be able to do in every subject. Our curricula are full of phrases like “Describe…, Explain…, Compare…”, which all require students to organize their thinking in a certain way. The non-fiction text in our math, science, social studies, and PAA courses require our students to break down complicated information. So how do we teach our students who are struggling with taking information in and/or communicating their understanding? 

What? I Have to Teach Writing, Too??

Reading and writing are learning tools that exist across curricula. We often have our students read technical information in our science, social studies and mathematics courses and then ask them to write about their understanding during assignments and tests. Sometimes, our students come to us knowing how to do both. Sometimes, we are surprised and disappointed that they don’t seem to know how to apply their reading and writing strategies in our content area. As a math and science specialist, I want my students to use reading and writing as ways to deepen and broaden their understanding of the subjects that I am teaching them.

I would think that an ELA teacher explicitly teaches reading and writing strategies so that their students become stronger readers and writers. My intent as a science teacher is related but different. I want to use literacy instruction so that students understand science better.

I had many experiences early in my career where I assumed my students were able to research, write reports, write conclusions, or complete short answer questions on my exams. Those assumptions led to frustration and feelings of failure for both myself and my students. Sometimes, it took students failing to point out what explicit teaching I needed to do with my classes. Ultimately, if I have students who are struggling with reading and writing in my non-ELA class, I need to teach them those skills.

Reading, Writing and Comprehension

We sometimes view reading and writing as separate ideas. If we view them instead as ways that students take in and output knowledge, we can see that comprehension, or meaning making, is the bridge between the two. Reading is ONE way of taking information in, and writing is ONE way of sharing our knowledge. Our English Language Arts program recognizes that there are other, equally important modes of inputs and outputs.

Comprehension Strategies – Making Meaning

When we take information in, our mind uses different strategies to make meaning of new information. During a conversation, this occurs fluidly where we listen, make sense of information, then speak. The same is true of reading and writing. We read information, make meaning, then may write about what our new understanding is. Many literacy experts have grouped comprehension strategies into anywhere from six to thirteen strategies. Following the work of Ellin Keene, the following are SEVEN comprehension strategies that strong readers use.

For more detailed information about these seven comprehension strategies, see my earlier blog post, Comprehension Across Subjects.

Nonfiction Writing Lesson Framework

Adrienne Gear (2014) suggests the following lesson framework for each nonfiction form:

  1. An introduction to the features of the nonfiction form.
    • This can be done by analyzing published examples of a nonfiction form.
  2. Independent write and Whole-class write can be woven together in a We DO – You DO cycle.
Whole-class write:Independent write:
The teacher and class write a passage together, going through the organizing steps together. This can be done on chart paper or projected on a screen.Writing activities can vary in length. There should be multiple opportunities for each type of writing introduced in a year.
Plan and organize thinkingDraft piece of writingFocus on a writing techniqueRevision and editing techniquesPlan and organize thinkingDraft piece of writingFocus on a writing techniqueRevision and editing techniques

Nonfiction Text Features

There are several text features that are useful within nonfiction to help readers understand the information being presented. These include:

Nonfiction FeaturePurpose
MapTo show location: e.g. habitat of animals
WebList of any kind: for example, a list of food an animal eats or its enemies
Diagram, labelsDescription
Fact BoxInteresting, additional facts
Flow ChartTo show how things work together: e.g., life cycle
ChartTo sort details: e.g. facts about different species
Labels, CaptionsTo explain a diagram or picture
TimelineSequential events or dates
Diagram with LabelsComparisons

Helpful Nonfiction Mini Lessons

Mini lessons allow you to guide student writing skills without taking up a lot of time. Here are topics that Adrienne Gear suggests for helping improve writing quality. You can see some of her mini lesson resources here. Including ideas for:

  • Adding Text Features
  • Interesting Details
  • Triple-Scoop Words
  • Comparison using Similes and Metaphors
  • Voice
  • Introductions to Hook Your Reader

Sharing Our Understanding: Non-Fiction Forms of Writing

Part of introducing nonfiction forms is for students to recognize key features of each form. This can be done by analyzing published works, both in print and online. Books from your library can be used to help students visualize the type of writing you are expecting them to do. There are many different forms of non-fiction writing that students both read and are expected to write themselves. Each of these forms has their own purpose and form. (Gear, 2014)

Different forms of writing can be used within the same topic and subject area. For example:

Type of WritingWriter’s IntentExampleApplication: Weather
DescriptionTo provide reader with facts and information about a topic. Related subtopics tell us specific details about the main idea. Writers give details related to our five senses.Descriptive reports on countries, animals, plants, insects, etc.Classroom blogs.Book or movie reviews.Describe the weather in Saskatchewan in January.
InstructionTo provide reader with instructions on how to achieve a goal, do something, make something, get somewhere.How something works: e.g. manuals, how to use something, survival guides.How to do or make something: e.g. recipes, rules for games, science experiments, crafts, instructions on starting a blog page.Give the instructions for how to make a winter survival kit.
PersuasionTo share an opinion with the reader or attempt to convince the reader to take an action of some kind.Opinion piece: e.g. favourite book, movie, pet, season.Persuasive piece: e.g. you should eat a healthy diet; no school uniforms; best chocolate bar to buy; oru school is the best.Classroom blogs or online reviews.Which form of weather is deadliest to humans?
ComparisonTo share with the reader the similarities and differences between two topics or ideas.Compare (similarities) and contrast (differences): e.g. rabbits and hares; Canada and Japan; cars then and now.Compare a winter blizzard and summer hail in Saskatchewan.
ExplanationTo provide reader with facts explaining how or why something happens.Scientific explanations: e.g. how a spider spins a web, why some things float and others sink.Explain how blizzards form.
Nonfiction NarrativeTo provide reader with sequential description of events in a person’s life, a current or historical event.Biography of a famous or non-famous person.AutobiographyCurrent event/newspaperPast eventBlogs or tweetsGive a report on the Newfoundland blizzard of January, 2020.

A useful analysis is to look at our curriculum and identify where it would be most useful for students to incorporate each type of writing to deepen their understanding. This can be done in a simple chart such as the one found here.

Pre-Thinking for Writing

Writing can help students understand subject content if we have them do pre-thinking before they write. This pre-thinking has them use comprehension strategies to deepen their understanding so that they can write.

A barrier for students might be that they do not understand either the content that they are having to write about OR they do not understand the structure of what you are asking them to write.

When we have students organize their thinking before they write, they will not only understand their courses better, but they will have their thinking organized in a way that helps them write.

You can take a closer look at different forms of writing, including assessment criteria in the following summaries, as well as view helpful pre-thinking tools for each type of writing:

Ultimately, teaching meaning making and how to express understanding can help our students know the subjects we are teaching them and help them to connect school content with their lives.

Eaton, S. E. (2010, September 26). Reading Strategy: The difference between summarizing and synthesizing. Retrieved from http://www.drsaraheaton.wordpress.com

Gear, A. (2014). Nonfiction Writing Power. Markham: Pembroke Publishers.

Johanson, T., & Broughton, D. (2014). Exploring Comprehension in Physics. Saskatoon: McDowell Foundation.

Keene, E., & Zimmermann, S. (1997). Mosaic of Thought: Teaching Comprehension in a Readers Workshop. Portsmouth: Heinemann.

Public Education & Business Coalition. (n.d.). Thinking Strategies for Learners: A guide to PEBC’s professional development in reading, writing, mathematics and information literacy. Retrieved December 15, 2018, from Public Education & Business Coalition: https://www.pebc.org/wp-content/uploads/publications/thinking-strategies.pdf

Saskatchewan Ministry of Education. (2008). English Language Arts 6. Retrieved from Saskatchewan Curriculum: https://www.curriculum.gov.sk.ca/bbcswebdav/library/curricula/English/English_Language_Arts/English_Language_Arts_6_2008.pdf

Designing PD That Teachers Deserve

I have had the pleasure of working with two different organizations recently, helping their staff to understand some basic principles of designing professional learning experiences for teachers. Designing professional learning deserves as much care and attention as the planning that we expect classroom teachers to give to their classroom instruction. We do not accept undifferentiated teacher lecture as the only pedagogy in classrooms, so it is important that we design professional learning that is

  • NOT a prescriptive module that does not change, regardless of what learners need
  • NOT solely lecture-style presentation where we tell them information and leave the meaning making and application to teachers after an event.

In my decade of designing and facilitating professional learning and teaching others to design professional learning, I have been seeking out and creating ways to approach workshop design. My goal is to ensure that I provide rich, authentic, practical and differentiated adult learning to teachers and related professionals. I firmly believe that a day of teacher workshop must be as or more important than a day that teacher would have had with their students. And a day with their students is SO important.

In my learning journey, I have discovered a few key things that are the foundation for every workshop that I create and facilitate.

Expect to Learn from Participants: Partnership Principles

One of my first learning opportunities about designing professional learning was Jim Knight’s Partnership Principles. His philosophy is very simple – that the people who come to professional learning are equal in every way to the facilitators of that learning. There is no hierarchy in learning, we are colleagues and partners.

Jim Knight’s Partnership Principles identify that if we have a mindset of equality, where our learners have choice and voice with professional learning, we will create interactions that encourage dialogue, reflection where we both can learn. An ultimately, the goal is for praxis, or application and transfer of learning into teacher contexts and classrooms.

Vision Our Impact: What Change Are We Hoping For?

It is important to see what changes we are hoping for in teacher behavior, resulting in an impact on student learning. Following Thomas Guskey’s Five Levels of Evaluating Professional Development backwards, it is possible to pose questions that can be pre-thinking before we begin designing learning. This process helps us to identify WHY we are providing this professional learning. As Simon Sinek has identified in his talk on The Golden Circle, we often think about the WHAT and the HOW, but it is the WHY that inspires us. When we, as facilitators, know why, we can share that passion and enthusiasm with our adult learners. A helpful tool is to use a Thinking Map, along with the following questions:

Ask Teachers What They Need

Ideally, we can engage teacher learners before our learning event to find out what learners need. This might be informed by:

  • Observations of student behaviours – what changes are needed?
  • Observations of student learning – what gaps or areas do students need a greater focus on?
  • Observations of teacher knowledge – what would learners like to know more about or change in their own understanding?

If it is not possible to have this conversation before learning, there are different facilitation processes that can be done that can inform our facilitation. These include:

  1. Snowball – ask participants
    • What do you know about this topic?
    • What do you wonder about this topic?
  2. Notice and Wonder – provide some type of visual or media experience and then ask
    • What do you notice about this?
    • What do you wonder about this?
  3. Touch Each Page and then Professional Question Generation
    • The Touch Each Page strategy will create a focus for professional inquiry through the day.
    • Generating Questions:
      • Group Generating and Monitoring Questions – participants work in small groups to identify questions that they would like to answer.
      • Personal Inquiry – participants identify a question that they most want to answer through the day. This is put onto a sticky note that they revisit and discuss at the end of the day with a colleague.

Have an Assessment Plan: Guskey’s 5 Levels

Thomas Guskey has identified five levels of evaluation to consider when understanding the efficacy of any professional learning experience in his article “Does it Make a Difference? Evaluating Professional Development”. Too often, we take in data around student outcomes and teacher satisfaction but neglect to identify the levels in between that allow us as facilitators to draw conclusions and connections between the professional learning that we are engaging in and the impact on students.

As Guskey has stated, “Good evaluations don’t have to be complicated. They simply require thoughtful planning, the ability to ask good questions, and a basic understanding of how to find valid answers” (Guskey, 2002). It is important as we evaluate our professional learning experiences that we are looking for evidence around its efficacy, not proof that it is making a difference. Most often, there are many different professional experiences that are impacting student learning, and to identify the exact scope and impact of any one initiative is nearly impossible. Instead, it is useful to gather evidence, both qualitative and quantitative, that identifies shifts and possible contributors to changes in student learning.

Guskey’s Five Levels of Evaluating Professional Development allow us to consider the types of questions that we might ask participants during or after professional learning. Facilitators who can connect with teachers after professional learning can gather data regarding impact on student outcomes and participant use of knowledge and skills.

If you are only in contact with educators during a session, it is possible to assess levels one through three, and measure teacher intent to implement using an Agenda Assessment. An agenda assessment is an innovation that combines an agenda with an assessment of learning and can be completed throughout a professional learning experience. This information can give insight into the effectiveness of a workshop or other learning experience.

Plan for Flexibility: Have a Plan that Allows for Change

A useful planning structure is a Facilitation Guide. Like a lesson plan that a teacher might use in a classroom, a facilitation guide identifies content, process, assessment, timing and materials. This simple structure helps facilitators see how content is chunked during the day, and the sequence of instructional strategies.

Content: This is the sequence of main ideas that flows through the day. By chunking content, it is relatively easy for a facilitator to skip or skim over particular ideas. This might occur if

  1. Teachers have already identified that they know a specific piece of information; or
  2. Time does not allow for all of the concepts in the day to be covered.

Process: This column identifies the instructional strategies and key questions that facilitators might pose to encourage thinking.

Assessment: This column allows facilitators to predict what they think participants might do or say during a specific part of the workshop. It is helpful to identify

  1. what people might say if they have a misconception; or
  2. what we are looking for in participant responses that indicates that they understand.  

Timing: Just as it states, this column allows facilitators to predict the length of time that a specific process will take. This helps to know whether the workshop is at, ahead or behind timelines outlined.

Materials: This column has us list materials or resources that are used in that chunk of a workshop.

Incorporate Meaning Making Strategies: Differentiate Learning

It is important that we choose processes for learning that fit the content and amount of time provided. Considering Dylan Wiliam’s Formative Assessment Strategies, instructional strategies in professional learning are particularly powerful when they:

So, where do we find these strategies? There are many useful resources. Some of them include:

It may seem like there are many layers and lots of time needed to plan effective professional learning, but our teachers and ultimately student learning deserve our investment.

Teaching Number Operations

Addition, subtraction, multiplication and division are foundational skills that are applied to many mathematical concepts. Often, when we are hoping for student automaticity and fluency in numbers, number operations are what we are talking about.

Mathematical Models

Models are the way we are representing numbers so that we can do number operations. There are a number of different models that are helpful to students understanding number operations.

Models that Emphasize 10

Models that Emphasize Place Value

Models that Emphasize Patterns

Models that Emphasize Partitioning Number

The Importance of Partitioning Numbers

Regardless of what number operation we are talking about, it is important that children are able to break numbers into parts.

Friendly Numbers – children are often able to understand number operations with ‘friendly’ numbers like 2, 5, and 10. Breaking a 7 into a  and a 2 allows us to use number facts that are more familiar.

Place Value Partitioning – when we are working with multi-digit numbers, it is helpful for us to break numbers up into the values of their digits – for example, 327 is 300 + 20 + 7.

Number Operation Strategies

There are many different strategies that children use to perform number operations. A misconception is that all children need to know and use all strategies. It is important for us to expose children to different strategies through classroom discussion and routines such as number talks and number strings. When combined with Margaret Smith’s ideas around Orchestrating Classroom Discussion, we can set a task for students and

  1. Predict what strategies they might use. Order these from least to most complex.
  2. Observe students doing mathematical tasks – using white boards allows us to see their thinking. We can then identify different strategies being used.
  3. Have students share their thinking in an order from least to most complex. This should not include every child sharing for every task. A small handful of children sharing in a logical order can help students understand the next more complex solution. In this way, children are being exposed to other strategies, will be able to understand those that are close to their own, and increase the sophistication of their thinking.
Strategies Connection to Addition Connection to Multiplication
Counting: This is a common strategy when one of the numbers is small. Addition by counting or counting on from one number. Ex: 25 + 7 = 25, 26, 27, 28, 29, 30, 31, 32.   Skip counting by one of the numbers being multiplied. 9 x 5 = 9, 18, 27, 36, 45  
Decomposing Numbers: breaking numbers apart. Adding friendly numbers. Ex: when you need to add 12, breaking it into +10 and then +2 more.  

Making 10. Ex: when adding 5 + 7, recognizing that 5 + 5 = 10, and so it is 10 + 2 more = 12.  

Breaking one or both numbers into place value. Ex: 23 + 47 is 20 + 40; 3 + 7
Multiplying friendly numbers. Ex: when you need to multiply by 6, break it into x 5 and 1 more.          




Partial Products: Breaking one or both numbers into place value. Ex: 23 x 47 is (20 + 3) x (40 + 7)
Compensation: this is very common when a number is close to 10. Rounding one of the numbers to a friendly number, then compensating the answer at the end for the difference. Ex: 36 + 9 is close to 36 + 10, subtract 1. Ex: 36 + 11 is close to 36 + 10, add 1. Rounding one of the numbers to a friendly number, then compensating the answer at the end for the difference. Ex: 99 x 5 is close to 100 x 5, subtract 5 Ex: 101 x 5 is close to 100 x , add 5
Double/Half Recognizing that 4 + 4 is double 4, or 8. Recognizing that 4 + 3 is almost double 4, subtract 1. Recognizing that 5 x a number is the same as ½ of 10 x a number. Ex: 9 x 5 is half of 9 x 10 = 45
Standard Algorithm Traditional algorithm, symbolic regrouping. Traditional algorithm, symbolic regrouping.

A Bridge between Addition and Multiplication: Doubles

  • Doubles are one way to think about adding a number to itself, as well as the start to multiplicative thinking.
  • Doubles are an important bridge between adding and multiplication.
  • You can read more about teaching doubles here.

Addition and Subtraction

Addition is the bringing together of two or more numbers, or quantities to make a new total.

Sometimes, when we add numbers, the total in a given place value is more than 10. This means that we need to regroup, or carry, a digit to the next place. There is a great explanation of regrouping for addition and subtraction on Study.com.

Subtraction is the opposite operation to addition. For each set of three numbers, there are two subtraction and one addition number facts. These are called fact families. For example:

For the numbers 7, 3, 10:

                7 + 3 = 10

                10 – 3 = 7

                10 – 7 = 3

Fact families can be practiced using Number bonds or Missing Part cards.

As we move from single digit to multi-digit addition and subtraction, it is important that we maintain place value, and continue to move through the concrete to abstract continuum.

A helpful progression for teaching addition and subtraction can be found on the Math Smarts site.

Multiplication and Division

Conceptual Structures for Multiplication

Repeated Addition

  • This is the first structure that we introduce children to.
  • It builds on the understanding of addition but in the context of equal sized groups.

Rectangular Array/Area Model

  • This is often the second representation of multiplication introduced It is useful to show the commutative property that 3 x 4 = 4 x 3 = 12

Number Line

  • A number line can represent skip counting visually.

Scaling

  • Scaling is the most abstract structure, as it cannot be understood through counting.
  • Scaling is frequently used in everyday life when comparing quantities or measuring.

Single Digit Multiplication Facts

Multiplication facts should be introduced and mastered by relating to existing knowledge. If students are stuck in a ‘counting’ stage – either by ones or skip-counting to know their single-digit multiplication facts, it is important that they understand strategies beyond counting before they practice. Counting is a dangerous stage for students, as they can get stuck in this inefficient and often inaccurate stage. Students should not move to multi-digit multiplication before they understand multiplication strategies for single-digit multiplication.

  • It is important that students understand the commutative property 2 x 4 = 8 and 4 x 2 = 8.
  • 2 x 4 should be related to the addition fact 4 + 4 = 8, or double 4.
  • Using a multiplication table as a visual structure is helpful to see patterns in multiplication facts.

Mental Strategies Continuum

  • Same as (1 facts)
  • Doubles. (2 facts)
  • Doubles and 1 more (3 facts)
  • Double Doubles (4 facts)
  • Tens and fives (10, 5 facts)
  • Relating to tens (9 facts)
  • Remaining facts (6, 7, 8 facts)

Conceptual Structures for Division

Equal Grouping

In an equal grouping (quotition) question, the total number are known, and the size of each group is known.

  • The unknown is how many groups there are.

Equal Sharing

In an equal sharing (partition) question, the total number are known, and the number of groups is known.

  • The unknown is how many are in each group.

Number Line

Ratio

This is a comparison of the scale of two quantities and is often referred to as scale factor. This is a difficult concept as you can’t subtract to find the ratio.

Division Facts

Relate division facts back to multiplication facts families:

Ex)          6 x 8 = 48

                                8 x 6 = 48

                                48 ÷ 6 = 8

                                48 ÷ 8 = 6

Once students have understanding and fluency with single digit multiplication and division fact families they are ready to move on to multi-digit fact families.

So What do Students DO with Number Operations?

Simple computation is not enough for children to experience. They need to have opportunities to explore and wonder about numbers and how they work together. Regardless of the routine or task, children should be encouraged to use different concrete and pictorial models to show their thinking.

Some examples of rich interactions include:

Number Talks

  • Number talks promote classroom discussion. Combining number talks with visual or concrete models can help us see what students are thinking.

Number Strings

  • Number strings can help children see the pattern in number operations. They are helpful for children to see the pattern in number operations, which is the foundation for algebraic thinking.
  • You can see the structure for building number strings here.

SPLAT

  • SPLAT encourages both additive thinking and subitizing. More complex SPLAT lessons are also great for encouraging algebraic thinking with unknowns.

Problem Stories

  • Building problem stories are powerful for children to understand contexts of mathematics in their every day life.
  • Using real objects or pictures encourages children to see math in their environments.

Invitations

Games and Puzzles

  • There are so many games and puzzles that can have children play with number operations.

Open Middle Problems

  • Open middle problems allow for flexible thinking and exploration. You can see a sample here.

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