Workshop Blog

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.

Differentiating Instruction – Why, How, What?

How often have we been in a conversation with a colleague about trying to meet the needs of all of our students, and we hear the dreaded phrase “well, just differentiate”… this blanket statement can bring about visions of creating 18 different lesson plans for our 18 students. This is not sustainable, so what is differentiation REALLY? How do we meet the needs of diverse learners and keep our sanity?

Picture Feedback 2

Workshops focussing on differentiation are, ironically, often not differentiated. It is important that all professional learning, including those experiences based on the topic of differentiation, attempt to have teachers experience differentiated learning as well as reinforce the foundations of how and why we differentiate content, process, product and environment for students. 

Planning for Differentiation

It is important to understand not only specific strategies but to also know why we might differentiate. What information do we need as teachers in order to plan appropriately for our individual students as well as our whole class experiences? We need to know a combination of Learning Styles, Multiple Intelligences, content readiness, and student interests in order to Plan for Differentiation.

Planning Differentiation

This information can be compiled into a Learner Profile Card or a Whole-Class Preferences Summary Chart to allow both students and teachers to know what and how learners might learn best.

Something that is often an ‘aha’ for adults is to consider whether they are “Think to Talk or Talk to Think” learners. If someone is a think-to-talker and is forced to jump into group work without first having the chance to get their thoughts in order, they may have a feeling of being unsafe expressing their ideas. If a talk-to-thinker is forced to read quietly before they are allowed to talk, they may find that their minds wander and are unable to focus. This same sense of safety is true for student learners as well.

One of the foundational researchers in the area of differentiation is Carol Tomlinson, who describes differentiation as

  1. Being curious about our students,
  2. Having relationships between teachers and students; and
  3. Providing a variety of learning experiences to learners

Differentiating Content

Why:

Differentiating content allows you to address gaps in understanding to build readiness. We know in literacy that activating prior knowledge is essential for students to make connections to new learning. This is true in other subjects as well. Assessing prior knowledge allows gaps to be addressed before new concepts are introduced. Differentiating content allows students to ACCESS information and learning.

What:

Your curriculum drives the knowledge, concepts, skills, and understandings a student needs to know and use. While the curricular outcome cannot change for individual students, the delivery format for content such as video, readings, audio, reading level can be differentiated. Content can also be chunked, shared through visual graphic organizers, or addressed through jigsaws to reduce the volume of information each individual needs to interact with. Themes can be based on personal interest to increase interest and understanding if a specific topic is not required by the curriculum.

How:

  • Use pre-assessment to determine where students need to begin, then match students with appropriate activities. Pre-assessments may include:
    Pciture Feedback 4
    • Student/teacher discussion,
    • Begin a KWL chart – what we know/want to know/learned,
    • Journal – what you already know about,
    • Brain dump – list all of the things you know about a topic, cluster with other class members, and
    • Snowball.
  • Use texts or novels at more than one reading level.
  • Present information through both whole-to-part and part-to-whole.
  • Use a variety of reading-buddy arrangements to support and challenge students when working with different texts.
  • Re-teach students pre-skills or provide enrichment for students who already demonstrate an understanding of pre-skills.
  • Use texts, video or different media to convey information.
  • Use Bloom’s taxonomy or Webb’s depth of knowledge to encourage thinking about content at several levels.

Differentiation Process

Why:

Differentiating process is about how students make sense of new learning. What is happening in each individual brain is based on their learning preferences, multiple intelligences, and background. Learners need time to take in, reflect on and make sense of new learning before moving on. Processes help students monitor their comprehension and determine what they do and do not understand. Learning processes also allow teachers to formatively assess student progress and provide feedback in real time.

What:

There are many different words used to describe learning processes – instructional strategies, discursive strategies, comprehension strategies… all of these are ways that learners interact with and make sense of new learning. Providing more or less structured support for learning, planning for a variety of instructional strategies based on the variety of learning styles in a classroom during a unit of study, and providing opportunities for self-reflection and self-assessment, and providing individual, pair/small group and whole group learning experiences are some key ideas for differentiation process.

How:

  • Assess learning styles, multiple intelligences, learning preferences, etc. to understand individual learning profiles as well as your class profile.
  • Use tiered activities that allow all students to work on the same outcome but with different levels of support.
  • Provide different learning experiences based on interests – i.e. when exploring mixtures and solutions, some students might choose to learn concepts through cooking, while others may learn concepts through art.
  • When planning a unit of study, ensure that concepts are interacted with using a variety of modes. For example, in mathematics, a planning form for math could be based on the idea of multiple representations of mathematical ideas:
Math Example Differentiation
  • Use a variety of comprehension strategy tactics.
  • Provide choice for students for how they are going to take notes, summarize information, make connections.
  • Use reflective strategies, such as:
  • Literature Circles(which also support content and product differentiation).
  • Different classroom structures, such as stations/centers, choice boards, flexible grouping all allow for different processes to be occurring simultaneously.

Differentiating Product

Why:

Differentiating product allows for student choice and allows learners to use their strengths to represent their understanding. Product choices all align to curricular outcomes, so learning is not compromised. Student voice and choice increase learner engagement. Products are the way that students represent their thinking about a curricular outcome. Differentiating the type of product being created allows you to see what they know about the curricular topic rather than the skill they needed to package that representation.

What:

Product differentiation is often cited as the most common form of differentiation and is often in the form of choices. You as the teacher may provide those choices and students pick from a variety of formats, you may have students propose their own designs or a combination of the two. How much responsibility and autonomy you provide for your students will depend on factors such as student understanding of their own strengths, age and time. When providing choice, it is important to co-construct clear criteria for success so that all products, regardless of form, are all being assessed on curricular outcomes rather than the form of a product. A rule of thumb is that the same checklist/rubric/assessment tool should be able to be used for all products on the same outcome, whether they are a paper, video, play, board game, etc.

How:

  • Allow students to help design product choices.
  • Co-construct assessment criteria.
  • Allow for varied working arrangements – individual, pair, group
  • Provide for or encourage students accessing a variety of resources.
  • Ensure that all products are at the same level of Bloom’s Taxonomy or Webb’s Depth of Knowledge.
  • Use a common assessment tool (checklist, rubric, etc.).

Conclusion:

When teachers plan by connecting content, process, product and learning product with student readiness, interests and learning profiles, students are more engaged and are able to understand ideas with a higher level of complexity.

Works Cited

McCarthy, J. (2015, August 28). 3 Ways to Plan for Diverse Learners. Retrieved from Edutopia: https://www.edutopia.org/blog/differentiated-instruction-ways-to-plan-john-mccarthy

New South Wales Education. (2015). Decide What to Differentiate. Retrieved from Phase 4 – Differentiating Learning: http://www.ssgt.nsw.edu.au/differentiating_learning.htm

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.

Building Coherent Teams

Michael Fullan (2016) states that coherence is “a shared depth of understanding about the purpose and nature of the work in the minds and actions individually and especially collectively” and is not about specific strategies, frameworks or alignment. So, how might we build coherent teams? How do we determine the ‘right’ actions and de-emphasize actions that are distractions? How might we focus on actions that enhance our collective as well as our autonomy? There are some processes and skills that are helpful.

Positive Communication

To build coherent teams, we need to know and practice communication skills, including paraphrasing and posing questions. A general conversation flow includes:

The Art of Paraphrase

“The purposeful use of paraphrase signals our full attention. It communicates that we understand the teacher’s thoughts, concerns, questions and ideas; or that we are trying to … well-crafted paraphrases align the speaker and responder, establishing understanding and communicating regard. Questions, no matter how well-intentioned, distance by degrees, the asker from the asked.”

(Adapted from Wellman and Lipton)

Things to keep in mind when paraphrasing:

  • Attend fully.
  • Listen with the intention to understand.
  • Capture the essence of the message but in a shorter format.
  • Reflect the essence of voice, tone and gesture.
  • Paraphrase before asking a question.
  • Use the pronoun “you” instead of “I.”

Intentions of Paraphrasing

Well-crafted paraphrases with appropriate pauses trigger more thoughtful responses than questions can alone. Three types of paraphrase, shown in the chart below, widen the range of possible responses. Each type supports relationship and thinking but the paraphrase that shifts the level of abstraction is more likely to create new levels of understanding. Conversations that utilize paraphrasing often move through a pattern of acknowledging to summarizing to abstracting, but there is no right pathway for the conversations.

Types of Questions

Horn and Metler-Armijo (Toolkit for Mentor Practice, 2010) identify three types of questions that are useful for professional conversations:

  • Clarifying questions – are asked to further understanding of the questioner. These types of questions convey that the questioner is actively interested.
  • Probing questions – are asked to have the speaker think more deeply about the concerns, challenges, or actions being taken. These types of questions dig into ideas to move from generalizations to specific ideas.
  • Mediational questions – are “intentionally designed to engage and transform the other person’s thinking and perspective” (Costa and Garmston, 2002). These types of questions are designed to open up and broaden thinking.

Mediational Questions

A special comment on “Why”…

Why questions are part of our everyday language. Why are you late? Why do you not have a pencil? Why are you doing that?

When we are having a conversation that may be emotional or highly charged, a question that begins with “Why” may create a sense of defensiveness. Consider a situation where someone has made a certain decision. Compare the reaction to “Why have you done this action?” vs “What is the impact your decision has had on…?”. A question that begins “What” or “How” is often more thought provoking and has less potential to create a defensive response.

Rectangle: Rounded Corners: Communication skills are key to building coherent teams, as they allow deep conversations to occur amongst team members, provoking thought through active listening.

Liberating Structures

Liberating structures, when used regularly, allow all team members the opportunity to work together to produce solutions, ideas and feel that everyone is contributing to an organization’s next steps. It is possible for every person to generate ideas and lead change.

Integrated~Autonomy

When considering how to best meet the needs of a system and the schools within a system, it is important that we view centralization/standardization and autonomy as both achievable and desirable rather than viewing them as opposite and competing interests. The Integrated~Autonomy liberating structure can help us to:

  • Develop innovative strategies to move forward.
  • Avoid wild swings in policies, programs or structures.
  • Evaluate decisions by asking “are we boosting both Coherence and Autonomy?”.
  • Increase quality of communication between school-based and Increase quality of communication between school-based and central office leaders.

Imagine actions that work towards BOTH increased standardization/centralization and increased Autonomy.

Some Examples

  • Attendance policies and consequences for non-attendance – what policies should be set centrally and which decisions should be made locally?
  • Planning documents required for hand-in/approval – format, timing and requirements for year, unit and lesson plans?
  • Parent conferences and reporting communication – what determined (format, process, content) centrally and what locally?

Structuring the Invitation:

  • Explore the question: Will our purpose be best served by increased local autonomy, including customization and site-based decision-making OR will our purpose be best served by increased coherence, including integration, standardization and centralized decision-making?
  • How might we be more coherent AND more autonomous at the same time?

Troika

This collaborative problem-solving strategy allows for colleagues to share possible solutions in a safe, non-judgmental environment.

In groups of three, learners sit in a triangle facing one another with no table between them. One person is the ‘client’, and the other two are the ‘consultants’.

  1. The client describes their dilemma, barrier or issue for about 2 minutes. The consultants might ask clarifying questions at this time.
  2. The client turns their chair so that their back is to the two consultants. The consultants discuss possible solutions to the client’s issue without any input, affirmation or cues from the client. The client might write down those suggestions that are most helpful. This might last for 2 – 4 minutes.
  3. The client turns around and summarizes what suggestions are most helpful that they might try.

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