Archive for the ‘assessment’ Category

professional learning from the summer of 2019

Posted on: September 2nd, 2019 by jnovakowski No Comments

Hi there,

It was a full and fun summer. I had lots of time to work on projects, read books, spend time with family and friends, tend to the garden, learn some new things and enjoy being outside where we live. I was also fortunate to travel a bit for work and build in some exploration time in the places I visited. All things that I like and bring my joy.

I receive and purchase a LOT of professional books. Books are a weakness for me and I often don’t have the time to read every new professional book I get. Because I often am asked to recommend books to schools, districts, etc my reading process is that I read the summary on the back cover or inside, I read through the tables of contents and then I skim through the whole book to get a sense of the flow of the book and to see how images, infographics etc are used. Finally, I choose one section or chapter of interest to read through completely. I feel okay about recommending books based on this process. Over the summer, I enjoy taking the time to read a selection of professional books cover to cover, usually about one a week in combination with my other reading for enjoyment. I received a new work iPads in June with an Apple pencil and my commitment this summer was to practice sketchnoting. The following are the sketch notes summarizing the professional books I read this summer.

Paper.Professional Reading Summer 2019.1

 

Paper.Professional Reading Summer 2019.2

 

Paper.Professional Reading Summer 2019.3

 

Paper.Professional Reading Summer 2019.4

 

Paper.Professional Reading Summer 2019.5

 

 

 

Paper.Professional Reading Summer 2019.6

IMG_4733In July I was invited to contribute to the updating of FNESC’s First Peoples Mathematics teaching resource. The existing resource was focused on grades 8&9 and can be found on the FNESC website here. The updated resource will focus on grades 5-9 and include adaptations for senior grades and K-5. It will be sent out to teachers to review this fall and will likely be ready spring 2020.

 

 

I attended a conference about early mathematics research in Portland, Oregon. The conference focused on current research and sessions were led be researchers and educators from across the USA. I learned about the DREME network from Stanford and the resources they offer and I was also fortunate to attend a session led by educators from the Boulder Journey School.

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My husband rode his bike down to Portland and met me there so we made a little holiday out of it. It was my first time attending this conference and I hope to go again next year. More info can be found here.

 

I had less then 12 hours at home from our trip to Portland before I flew off to Chicago. I was honoured to be invite to a Public Math Gathering organized by the Public Math group. More info about their initiatives can be found here. There were educators from Minnesota, Wisconsin, Washington and Chicago as well as artists and museum folk from the Chicago area. We participated in a neighbourhood event on the Friday evening and then visited the “famous” Mr. Bubble laundromat where we observed how math initiatives (form the group as well as provided by the Chelsea Clinton Foundation) were being used by families in the space. We then spent the afternoon designing and prototyping new math installations for the laundromat. We went back to the laundromat Sunday morning to observe how our installations engaged the public. It was such an inspiriting experience to work with such a diverse group of people around math.

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The summer always gifts me some sustained time to devote to writing projects. For many summers it was my academic writing but the last few summer I have been working on a book with a colleague, Misty Paterson. We finished off our edits in early July and sent things off to the printers. We held a book launch for Pop-Up Studio in Vancouver on August 28 – so great to be able to finally hold a book that took on a life of its own. More information about the book can by found on MIsty’s website here.

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I am one of the members of the BC Numeracy Network (if you aren’t familiar, the website is here) and a subgroup of us met this summer to begin work on a resource to support professional learning in mathematics teaching and learning. It is always great to be able to work with colleagues that have become good friends. Look for our project coming out this fall!

IMG_5733And my final writing project of the summer was the first issue of our BC Reggio-Inspired Mathematics Project magazine. The first issue is called Thinking About Mathematics through materials. Teachers from eight Coast Metro districts contributed to this magazine which captures our collaborative professional inquiry focus from the last school year. More information about the magazine, including ordering information, can be found on our website HERE.

 

 

For the last couple of years my curiosity has been piqued by images I have seen of beautiful geometry art shared on twitter. With some investigation I found an online Islamic Geometry course that many math teachers on twitter have taken and I signed up with a commitment to do the first introductory/basic course this summer. I learned a lot, made lots of connections (was excited to visit the Islamic Art wing in The Art Institute of Chicago) and am inspired to find ways to embed my new learning in math studio experiences.  Information about the course I took can be found here. Thank you to Samira Mian for her detailed explanations and lovely videos.

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Another highlight of a very mathy summer was visiting the Numbers in Nature  exhibit at Science World where I got some great ideas for projects and having our district’s Math Play Space at Richmond’s annual Garlic Festival. You can read more about that here.

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Wishing you all a wonderful September,

Janice

June thinking together: connect mathematical concepts to each other, other areas and to personal interests

Posted on: June 18th, 2019 by jnovakowski 2 Comments

This month’s curricular competency focus is connect mathematical concepts to each other, to other areas and to personal interests. This curricular competency is the same across grades K-12.

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This competency falls under the organizer of  ”Connecting and Reflecting” and is linked to metacognition, synthesizing concepts and ideas, reflective thinking and self-assessment. There are links with this curricular competency to the Core Competencies of Communication and Positive Personal and Social Identity.

Elaborations are suggestions for educators to consider as they plan for developing this curricular competency:

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Drawing on the literacy research of David Pearson, one framework for thinking about mathematical connections is to consider creating opportunities for students to make:

  • math to math connections
  • math to self connections 
  • math to world connections

Many teachers have seen classroom-based evidence of learning when students demonstrate an ability to make math to math connections and feel students who can connect and see relationships between concepts have strong number or spatial sense and a stronger understanding of the mathematical ideas involved. Instead of learning about fractions in grade 4 for three weeks and maybe not encountering formally again at school until grade 5, teachers weave math concepts together throughout the year to help nurture math-to-math connections. After being introduced to both concepts of fractions and decimal numbers during focused studies, students are asked questions such as “How are fractions and decimal numbers connected?” These types of questions are included in the elaborations for the Big Ideas in our BC Mathematics curriculum.

Other examples include:

“How are addition and subtraction related?”

“How are multiplication and division related?”

“What is the relationship between area and perimeter?”

“What is the connection between patterning and algebra?”

Math-to-math connections can also be considered across grades (how did learning about fractions with pattern blocks last year help you think about fractions with Cuisenaire rods this year?) or across forms (concrete, pictorial, symbolic) or across problem types.

Math-to-self and math-to-world connections enhance understanding of personal, social and cultural identity as well as an understanding of issues in the world around us. A student might make a connection to skip counting or multiples to scoring in basketball or a student might see an infographic or graph on a website and use proportional reasoning to make sense of the information. When making connections, students see how mathematics can be used as a language to both receive and express information about themselves and the world around them. We often ask students: “Where does math live here?” as a way for them to make connections to different places and contexts or areas of study.

Where does math live…

in the game of basketball?

at the beach?

in the study of biology?

at the grocery store?

in the weather?

at the playground?

in cooking and baking?

in the newspaper?

Related to the idea of connection-making is transfer and application. Students may learn facts or skills but they need to be able to transfer, apply or build on that learning in other areas. This is the essence of numeracy – to be able to apply mathematical understanding in new contexts, situations or with new problems.

 

Some questions to prompt students to make connection include:

What does this remind you of?

When have you done a problem like this before?

What do you already know about this?

What materials have you used to think about this concept?

Where else have you experienced this idea?

Where can you find or use this concept in the world around you?

 

Some questions to consider as you plan for learning opportunities to develop the competency of connecting mathematical concepts to each other, to other areas and to personal interests:

How can we plan for mathematical connections in different learning contexts such as the gym, music class, art room, library or learning outdoors or in the community?

What opportunities do we create to intentionally nurture students’ connection-making across math topics and across disciplines?

How is connection-making in reading comprehension connected to connection-making in mathematics?

How might we capture and curate mathematical connections that students make to make this learning visible?

~Janice

*Please note: This is the last in this year’s series of monthly blog posts on BC’s curricular competencies for mathematics.

big mathematical ideas for grades 6-9 2019

Posted on: May 30th, 2019 by jnovakowski

Similar to the K-2 and grades 3-5 big math ideas series, this year we offered a grades 6-9 series. For a variety of reasons, we were only able to hold one session in April. We focused on the big idea of computational fluency.

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Each teacher received the book Making Number Talks Matter by Cathy Humphreys and Ruth Parker. The focus of number talks is to develop computational fluency through practice of and discussion of mental math strategies for number operations.

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Another focus of our session was inclusive instructional routines that develop number sense, computational fluency and curricular competencies such as reasoning.

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District posters are available for routines such as Splat, Number Talks, and Which One Doesn’t Belong. They can be found in English and French on this blog on the top of the site. An example of one of these posters is:

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At grades 6-9, developing and extending computational fluency with whole numbers across all four operations is an essential, foundational component of our BC mathematics curriculum. At this grade range, students are also connecting and transferring many of these strategies to operations with decimal numbers, integers and fractions.

Looking forward to next year, it is our hope to have more opportunities for teachers to bridge teaching and learning experiences from elementary to secondary.

~Janice

May thinking together: explain and justify mathematical ideas and decisions

Posted on: May 26th, 2019 by jnovakowski

This month’s curricular competency focus is explain and justify mathematical ideas and decisions. This curricular competency is the same across grades K-12 and is included in the Grades 10-12 courses with the addition of ”in many ways“.

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This competency falls under the organizer of  ”Communicating and Representing” is also connected to the Core Competency of Communication, particularly the aspect of explaining and reflecting on experiences.

Elaborations are suggestions for educators to consider as they plan for developing this curricular competency:

  • mathematical arguments

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What is a mathematical argument?

A mathematical argument is the debate and discussion of a mathematical problem or task. This involves the explanation and justification of the reasoning, problem-solving process and the solution. As stated by Small (2017), the ability to create a sound mathematical argument is developed over time.

A common instructional routine in our district is Number Talks. During this routine, students are asked to share their mental math strategies for solving questions involving number operations. Part of this routine is defending or “proving” their solution through their strategy explanation. Other students may agree with, build on or argue with the strategies used. A focus of this routine is both building mathematical discourse structures as well as building the listeners, connectors and reflectors needed in a mathematical community. During Number Talks, students listen to each others’ explanations and justifications and then also use mathematical language to communicate their own mathematical arguments. Before orally sharing their explanations to the whole group, students are often given the opportunity to turn and talk, or think in their head to formulate and rehearse their explanations.

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In the book Teaching Mathematical Thinking, author Marian Small (2017) suggests the language that develops during mathematical argumentation and discourse may sound like this:

“I agree with ______ because _______.”

“I didn’t understand why you __________.”

“I disagree with ___________ because ____________.”

“I wonder why you _____________.”

“What if you had _____________.”

Small (2017) provides some examples of open question that nurture mathematical argumentation. For example, for grades 3-5 students:

Liz says that when you multiply two numbers, the answer is more likely to be even than odd.

Do you agree or not? Why?

And for grades 6-8:

A store employee noticed that an item’s price had been reduced by 30% and realized it was a mistake. So she added 30% back to the reduced price. Avery said the price is the same as it used to be but Zahra disagreed.

With whom do you agree? Why?

What tasks like these are we presenting to students to intentionally nurture and practice the development of explaining and justifying mathematical ideas and decision-making?

Mathematician Dan Finkel shares the importance of conjectures and counterexamples in his playful instructional approach. More information can be found on his website mathforlove.com

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In the following example from Dan, a student made a conjecture that if you multiply both factors by two, the product will stay the same. Can you think of a counterexample that disproves this?

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In their book But Why Does It Work? Mathematical Argumentation in the Elementary Classroom (2017), authors Susan Jo Russell et al share an efficient teaching model focused on mathematical argument for developing the ability of students to justify their thinking and engage with the reasoning of others. Their model supports students in:

  • noticing relationships across sets of problems, expressions or equations
  • articulating a claim about what they notice
  • investigating their claim through representations such as manipulatives, diagrams, or story contexts
  • using their representation to demonstrate and explain why their claim must be true or not
  • extending their thinking from one operation to another

In their book Teaching with Mathematical Argument (2018), authors Stylianou and Blanton suggest that a focus on justification and explanation of thinking can celebrate the diversity of thinking within our classrooms. From their book:

“How can argumentation be a goal and an expectation for all students? One strategy is to embrace students’ use of diverse strategies. This diversity can then be used to plan cognitively demanding instruction that includes argumentation and that allows all learners to build from their own thinking and access their peers’ thinking to develop their understanding of new concepts. Rich, open tasks that invite argumentation are challenging because of their open nature. However, their openness also allows access to students who struggle in mathematics. Being open implies having more than one entry point, which makes such tasks accessible to students who often struggle to follow one particular procedure.”

By honouring the diverse thinking of the learners in our classrooms, we are also nurturing the important idea that there isn’t “one right way” to do or think about mathematics. Creating entry points for all students to explain and justify mathematical ideas is part of creating a safe mathematical community for all.

Some questions to consider as you plan for learning opportunities to develop the competency of explaining and justifying mathematical ideas and decisions:

How do we support students and families in understanding that explaining and justifying your answers and processes is an important part of mathematics?

What problems and tasks are we presenting to students to intentionally nurture and practice the development of explaining and justifying mathematical ideas and decision-making?

What visual and language supports might support students as they engage in mathematical discourse and argumentation?

What opportunities do students have to notice patterns and relationships, make conjectures and generalizations across mathematical concepts? What ways could they share and explain their mathematical ideas by using materials, pictures or diagrams, stories or contexts or numbers and symbols?

How might technology provide access for students or transform the way they are able to explain and justify their mathematical ideas and decisions?

~Janice

References:

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But Why Does It Work? Mathematical Argument in the Elementary Classroom

by Susan Jo Russell, Deborah Schifter, Virginia Bastable, Traci Higgins, Reva Kasman

Heinemann Publishers, 2017

 

 

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Teaching with Mathematical Argument: Strategies for Supporting Everyday Instruction

by Despina Stylianou and Maria Blanton

Heinemann Publishers,  2018

 

 

Screen Shot 2019-05-26 at 7.58.18 PMTeaching Mathematical Thinking: Tasks and Questions to Strengthen Practices and Processes

by Marian Small

Teachers College Press/Nelson, 2017

 

 

Promoting Mathematical Argumentation by C. Ramsey and W. Langrall (2016). Teaching Children Mathematics (volume 22), number 7, pages 412-419.

April thinking together: communicate mathematical thinking in many ways

Posted on: April 30th, 2019 by jnovakowski

This month’s curricular competency focus is communicate mathematical thinking in many ways. This curricular competency is the same across grades K-9 and is included in the Grades 10-12 courses as “explain and justify mathematical ideas and decisions in many ways“.

This competency falls under the organizer of  ”Communicating and Representing” which includes the following related competencies:

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Elaborations are suggestions for educators to consider as they plan for developing this curricular competency:

  • communicate using concrete, pictorial and symbolic forms
  • use spoken or written language to express, describe, explain, justify and apply mathematical ideas
  • use technology for communication purposes such as screencasting and digital photography and videography

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There are clear connections between the Core Competency of Communication with this grouping of curricular competencies. A one-page table showing the language of both types of competencies can be downloaded here:

SD38 K-5 Math Communication_Avenir

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An important part of communicating mathematical thinking in many ways is to be able to use different forms such as concrete (materials or math manipulatives), pictorial (drawings, diagrams, tallies) or symbolic forms (numerals and symbols).

An example from primary classrooms of how students may move from concrete to symbolic notations is with the use of materials such as base ten blocks. Students may communicate their understanding of numbers by creating that number with materials and then recording the symbolic notation. The following are some examples from a grades 2&3 classroom at Cook Elementary that show how children used concrete, pictorial and symbolic forms to help them solve and communicate their solutions for mathematical problems.

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As students begin to understand a concept, such as multiplication, they usually construct a representation with materials to build understanding. These representation may then be recorded pictorially and then labels are added using symbolic notation. This fluency between forms is important and the connections between representations is essential to conceptual understanding. A student may be presented with a symbolic form (such as an equation) and asked to show a concrete form or pictorial form that “matches”. The following are examples from a grades 2&3 classroom at Tomsett Elementary.

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For our intermediate and secondary students, it is still important to be using concrete materials, especially when students are developing their understanding of a new concept such as fractions, decimals, or integers. The following are examples from a grades 4&5 classroom at Homma Elementary

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and also more fraction investigations with a grades 4&5 class at Steves.

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In our curriculum, the terms “concrete, pictorial and symbolic” are used in ways for students to think about concepts but also to communicate and represent their thinking. In some other jurisdictions around the world, the term CRA is used to reference an instructional approach to concept development, standing for Concrete, Representational and Abstract. More information can be found HERE. There is some overlap between the the CRA framework and how our curriculum focuses on concrete, pictorial and symbolic communication of mathematical thinking and understanding.

Another area of focus in our district is using iPad technology for students to communicate their thinking and learning. One of the most common uses of the devices in math is to use screen casting apps such as doceri, ShowMe, Explain Everything or 30Hands. When students screencast, they can take a photograph or video of what they are doing and then annotate with arrows, words etc and then orally describe their problem-solving process or thinking. For example, in a grade 8 class at Hugh Boyd Secondary, students took images of number balances they used to develop their understanding of equivalence in algebraic equations and then communicated their thinking by orally explaining their understanding.

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Some questions to consider as you plan for learning opportunities to develop the competency of communicating mathematical thinking in many ways:

How is the core competency of communication noticed, named and nurtured during the teaching and learning of mathematics?

What different materials are students learning to use, think through and represent with? What materials are mathematically structured and what other types of materials might we offer to students?

What opportunities are we providing for students to share their thinking in different ways? Are students provided with choices and is there a balance in the different ways students can communicate their mathematical thinking?

How might technology provide access for students or transform the way they are able to communicate their mathematical thinking?

 ~Janice

March thinking together: engage in problem-solving experiences connected with place, story and cultural practices and perspectives

Posted on: March 14th, 2019 by jnovakowski

This month’s curricular competency focus is engage in problem-solving experiences that are connected to place, story, cultural practices and perspectives relevant to local First Peoples communities, the local community, and other other cultures. This curricular competency is the same across grades K-12 and courses and falls under the organizer of “Understanding and Solving” which suggest the focus of using contextual and meaningful experiences to support mathematical understanding.

Elaborations are suggestions for educators to consider as they plan for developing this curricular competency:

  • in daily activities, local and traditional practices, the environment, popular media and news events cross-curricular integration
  • have students pose and solve problems or ask questions connected to place, stories and cultural practices

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The focus and thinking behind this curricular competency are the ideas of authenticity, meaningfulness, engagement and connectedness. Not all mathematics learning needs be contextualized or connected to “real life” but for many students who may see math as something that they do at school between 9 and 10am and don’t yet see the relevance of the math they are learning, providing tasks and problems that connect to place, community and culture may support their mathematical thinking and learning and broaden their understanding and appreciation for what math is and how it can be experienced. Experiential and holistic learning are foundational to the First Peoples Principles of Learning and these are considerations for all learners. The First Peoples Principles of Learning also remind of us of the importance of connecting learning through place and story, working with others and developing a self of self, family, community and culture. This curricular competency is aligned with the Personal and Social Core Competency – positive personal and cultural identity, personal awareness and responsibility and social responsibility.

Some resources to consider:

Messy Maths by Juliet Robertson (elementary resource for taking math learning outdoors)

Tluuwaay ‘Waadluxan Mathematical Adventures edited by Dr. Cynthia Nicol and Joanne Yovanovich (mathematical adventures from Haida Gwaii developed by community members, elders and educators)

BC Numeracy Network – Connecting Community, Culture and Place

First Peoples Mathematics 8&9 developed by FNESC - this teacher-created resource is being revised to reflect the current BC mathematics curriculum and provide more learning experiences across grades and disciplines.

 

Blog posts from this site with related information:

Place-Based Mathematics

Place-Based Mathematical Inquiry

Primary Study Group 2018-2019 – Outdoors Math

Indigenous Content and Perspectives in Math

 

Some questions to consider as you plan for learning opportunities to develop the competency of engaging in problem-solving experiences connected to place, story and cultural practices and perspectives:

How does place/land/environment inspire mathematical thinking? What potential numeracy or problem-solving tasks emerge when we think about local land-based contexts?

What problems or issues are facing the local community? How might mathematics help us to think about and understand these problems or issues? What information or data might be collected and shared? How can we use different tools to communicate mathematical information to create an opportunity for discussion and engaging in a problem-solving process?

How does Indigenous knowledge connect, intersect and support the curricular competencies and content in our mathematics curriculum? Who is a knowledge holder in your local First Nations community that you could learn from and with? 

What are authentic resources? What stories and cultural practices are public and able to be shared? What doe it mean to use authentic resources, stories, and elements of culture in our mathematics teaching? How are resources specific to a local context? Who can we go to to find out more information and learn about local protocols?

What cultural practices in your community have mathematics embedded in them? How might we use the structure of “notice, name and nurture” to expand awareness of what mathematics is and how it can be experienced?

How can stories help us think about the passage of time, relationships, connections and mathematical structures, actions and models?

~Janice

big mathematical ideas for grades 3-5 2019

Posted on: March 13th, 2019 by jnovakowski

This is the sixth year of this after school series that focuses on the big mathematical ideas encountered by teachers working with students in grades 3-5. This year this group met three times during term three.

Our first session was on January 17. Each teacher received the professional resource Number Sense Routines by Jessica Shumway.

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The focus of our first session was on multiplicative thinking and computational fluency.

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We began by working on a math problem together, from the book, and considered the different ways our students might engage with the mathematics.

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And then looked at visual routines from the book that support multiplicative thinking through spatial structuring.IMG_7304

We also considered games that provide purposeful practice for developing computational fluency and reasoning around multiplication, such as the array-based game, How Close to 100? from Mindset Mathematics.

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Our second session was on February 7 and after sharing the visual routines that we tried with our students, we discussed the big ideas around decimal numbers.

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IMG_7975Our focus from the book was using number routines such as Today’s Number as well as Number Talks with fractions and decimal numbers. We also connected using visual supports such as 10×10 grids in games to practice decimal computation and develop understanding of decimal numbers in both fractional and place value-based ways.

Some games and a recording sheet for thinking about decimal numbers from the session can be downloaded here:

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Our third session was held on March 7 during which we focused on the big idea of area, connecting this concept to both multiplication and the visual routines we had learned earlier in the series (arrays, spatial structuring, decomposing into parts).

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We also focused on the instructional routine of notice and wonder and how it can be used to have students make sense of a mathematical situation or problem as well as create an opportunity for students to ask questions that can lead into mathematical investigations.

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Annie Fetter of the Math Forum has made many math teachers aware of Notice and Wonder over the years and an overview document is available:

Intro I Notice I Wonder NCTM

For this session, a new SD38 math instructional routine poster was created and it is available in both English and French:

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notice wonder poster french

These posters are also all available on this blog, under the poster tab at the top!

Thank you to Grauer Elementary for the use of The Nest to host this series!

~Janice

 

February thinking together: develop, use and apply multiple strategies to solve problems

Posted on: February 28th, 2019 by jnovakowski

This month’s curricular competency focus is using multiple strategies to solve problems. There is a development in how strategies are used from K-12 and for what types of problems.

In K-5 the curricular competency language is “develop and use multiple strategies to engage in problem solving” with elaborations including examples of strategies involving visual, oral and symbolic forms and through play and experimentation.

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In K-5, we support students in developing a repertoire of strategies to draw upon and we encourage the practice of choosing and using these strategies in different problem solving experiences ranging  from structured word/story problems, open problems or questions or problem-based or numeracy tasks. During the development of strategies, students will notice similar strategies being shared by their classmates and these strategies might be named such as “looking for a pattern” or “acting it out” or “represent with materials”. Naming strategies such as these helps to enhance mathematical communication, discourse and community in the classroom when discussing mathematical problems.

As with many of the curricular competencies in math, there are slight variations between grade bands, showing the developing application and demonstration of these competencies.

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In grades 6-9 the curricular competency language is “apply multiple strategies to solve problems in both abstract and contextualized situations” with elaborations including examples of strategies focusing on those that are familiar, personal or from other cultures. Students in this grade range are refining and reflecting on their own use of problem solving strategies and we encourage students to listen and learn from their peers in order to consider new ways to think about a mathematics problem.

 

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In grade 10 the curricular competency language is “apply flexible and strategic approaches to solve problems” with elaborations such as deciding what tools to use to solve a problem as choosing from a list of known strategies such as guess and check, solve a simpler problem, model, use a chart, role-play or use diagrams. The numeracy processes for engaging in numeracy tasks are related to this competency at the secondary level - interpret, apply, solve, analyze and communicate.

 

Although specific strategies such as “guess and check” or “solve a simpler problem” are not named specifically in the elaborations from K-9, it is these more formally named strategies that are developed with understanding, meaning and purpose over time. Alternative or personally derived or preferred strategies may also be developed by students and shared with their solutions, supported with their reasoning and explanations to demonstrate their understanding of the problem and the mathematics involved.

Many math educators and researchers have found over decades of research and classroom experiences that students who have multiple strategies or approaches to problems are more fluent and flexible in their thinking. An important aspect of using multiple strategies is knowing when a particularly strategy is helpful or efficient. Not all strategies are suitable for all problems and this an important part of the progression of developing this competency in mathematics  One particularly effective instructional strategy is engaging students in comparing the strategies they used to solve a problem. Researchers have recently examined the cognitive process of comparison and how it supports learning in mathematics. The sharing and comparison of multiple student strategies for a problem was found to be particularly effective for developing procedural flexibility across students and to support conceptual and procedural knowledge for students with some background knowledge around one of the strategies compared. (Durkin et al, 2017 – referenced below). Based on their findings, the researchers share some significant instructional moves that will support student learning:

1) regular and frequent comparison of  alternative strategies

2) judicious selection of strategies and problems to compare

3) carefully designed visual presentation of the multiple strategies

4) small group and whole class discussions around comparison of strategies with a focus on similarities, differences, affordances and constraints

 

Examples of what the use of multiple strategies might look like in the classroom include:

Primary: The teacher reads the story The Frog in the Bog and asks the grade 1 students to figure out how many critters are in the frog’s tummy. The teacher invites the students to think about how they might solve this problem and what they will need. The students work on their own or with a partner to solve the problem through building with materials, acting it out, drawing or recording with tally marks and numbers. Some students accompany their solutions with an equation and one student records his ideas orally using iPad technology. As the students are working, the teacher pauses the students and asks them to walk around the room and see what their classmates are doing and see if they can find a new idea for their own work. After solving the problem, the students prepare to share their solutions and strategies with the class and the teacher gathers the students on the carpet and chooses some students who used different strategies to share. The teacher records the strategies on the chart and then asks the students if they have a new idea for a strategy for the next time they do a problem like this.

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Intermediate: In a grades 6&7 class, the teacher projects the first three figures of a visual pattern on the class whiteboard (examples on visual patterns.org). The teacher asks the students what they notice about the figures and records some of the students’ responses and then asks them to consider what comes next. Students are asked to consider what strategies or approaches might help them think about this. After some thinking time, the teacher asks the students to turn and talk with one or two other students and compare each others’ strategies and consider new ways of thinking about the problem. The teacher then invites the students to apply more than one strategy to solve what figure 43 will look like. The students share their solutions and strategies with the teacher recording the different strategies through different representations such as a drawing, a narrative, an expression, a table or a graph. The teacher then facilitates a discussion comparing the representations and how they are connected and support the understanding of the problem.

(with thanks to Fawn Nguyen and Marc Garneau for the inspiration)

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Secondary: Students in a grade 10 class are assigned to be in random groups of three and work on a numeracy task on a whiteboard or window around the classroom. The class has been learning about prime factorization and the teacher shares the following problem orally:

Prime numbers have exactly two factors – 1 and itself. Which numbers have exactly 3 factors? Exactly 4 factors? And so on. Given any positive integer, n, how can you tell exactly how many factors it has?

Each group of students begins talking and sharing their ideas. Students begin to record their thinking, using diagrams, charts, numbers, etc. and build on and challenge each others’ thinking about the problem and approaches to solving it. Students move around the room and watch or engage with other groups. The teacher facilitates students’ sharing of solutions and approaches to the problem and then provides a set of related problems for students to continue practicing with, either in their groups or independently.

Numeracy tasks such as this one can be found HERE and HERE and HERE.

(with thanks to Mike Pruner and Dr. Peter Liljedahl for the thinking classroom inspiration)

 

Some questions to consider as you plan for learning opportunities to develop the competency of using multiple strategies and approaches to solve problems:

What strategies and approaches do you notice your students using? Are some students “stuck” using the same strategy? How could you nudge students to try different strategies and approaches?

What different types and structures of math problems are being provided to your students? Are students flexible with their strategy choice or approach, making decisions based on the problem they are working on?

How might you and your students record their strategies and approaches to make this thinking visible?

What opportunities are we creating for students to watch and listen to others think through, choose and apply strategies and solve problems? How might this support their learning?

What tools, materials and resources do students have access to to support choice and application of different strategies and approaches when solving math problems?

~Janice

References

Elementary and Middle School Mathematics: Teaching Developmentally by John van de Walle et al

Teaching Mathematics through Problem-Solving (NCTM) edited by Frank Lester and Randall Charles

Why Is Teaching With Problem Solving Important to Student Learning (NCTM Research Brief)

Durkin, K., Star, Jon. R. & Rittle-Johnson, B. (2017) Using Comparison of Multiple Strategies in the Mathematics Classroom: Lessons Learned and Next Steps, ZDM: The International Journal on Matheamtics Education 49(4), 585-597.

 

big mathematical ideas for K-2 2018

Posted on: December 19th, 2018 by jnovakowski

This fall we hosted a three-part after school professional learning series focusing on the big mathematical ideas in Kindergarten thru Grade 2. We have been doing this series for grades 3-5 teachers for the last five years and this year have added series for K-2 and grades 6-9 teachers. The focus of the series is to look at the foundational math concepts within the grade band and consider ways to develop those concepts and related curricular competencies. Other curricular elements such as core competencies, First Peoples Principles of Learning, use of technology and assessment are woven into the series.

September 27

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We discussed three instructional routines focused on counting: choral counting, count around the circle and counting collections. The following are the professional resources that were recommended and every teacher attending was provided with a copy of Christopher Danielson’s new book How Many? and the accompanying teachers guide.

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We shared the idea of unit chats which is the essence of the book How Many? What could we count? What else could we count? How does the quantity change as we change the unit we are counting?

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We also introduced Dan Finkel’s website and his section of photographs that can be used for unit chats HERE.

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Between the first and second sessions, teachers were asked to try one of the counting routines, read parts of the How Many? teacher guide, try a unit chat with their classes and do the performance task with one of their students.

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October 25

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We spent the first part of our session together sharing with each other about a counting routine they did with their class, how their students responded to unit chats and their findings from the performance task. Teachers brought video, photos and student work to share and discuss.

We discussed the importance of research-based learning trajectories/progressions to inform our instructional and assessment practices. The BC Numeracy Network has collated several learning trajectories/progressions HERE (scroll down to the bottom of this page).

We introduced the draft of the new SD38 Early Numeracy Assessment Tool which is intended to use with students from the end of Kindergarten through grade 2 to create class learning profiles and well as help identify specific learning goals for students. It can also be used by schools to monitor student progress over time. The assessment tool focuses on key areas of number sense and the tasks are drawn from the BC Early Numeracy Project and the work from the Numerical Cognition Lab at Western University. Teachers were asked to complete the assessment with one student they were curious about learning more about.

November 22

We began our session sharing how it went with the new K-2 assessment tool. The teachers had lots of good feedback and suggested edits which will now be taken back to the district committee for final revisions.

We shared some different materials and experiences to support the development of K-2 students’ number sense, connecting the ideas of counting, subitizing, connecting quantities and symbols and ordering/sequencing. One of our favourite materials is Tiny Polka Dot, which I personally believe should be in every K-2 classroom (available in Canada through amazon.ca HERE).

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We also went over the ten frame games and tasks that can be used in K-2 classrooms for purposeful practice during math workshop or small group instructional time.

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Teachers and their students took photographs to contribute to our own digital How Many? book and it is a work in progress but the collection we have so far can be found here (best viewed via Chrome):

How Many? digital book

Look for information and  next steps for our SD38 K-2 Numeracy Assessment Tool in the new year!

~Janice

 

 

 

 

 

 

December thinking together: visualize to explore mathematical concepts

Posted on: December 11th, 2018 by jnovakowski

This month’s focus is on the curricular competency: visualize to explore mathematical concepts.

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In the 2007 WNCP mathematics curriculum, visualization is defined as involving “thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the visual-spatial world”. Concepts such as number, spatial relationships, linear relationships, measurement, and functions and relations can be explored and developed through visualization.

In the new BC grades 10-12 courses, the elaborations for this curricular competency are:

  • create and use mental images to support understanding
  • visualization can be supported using dynamic materials (e.g., graphical relationships and simulations), concrete materials, drawings, and diagrams

Visualization and spatial reasoning involve the relationship between 2D and 3D shapes as well as dynamic imagery such as different perspectives, movement, rotations and reflections. Visualizing involves an interplay between internal imagery and external representations  (Crapo cited in NRICH article below). Students need experience with concrete and visual representations/pictures/models as well as being able to visualize something in their minds, often referred to as the “mind’s eye”.

Canadian and International research has shown that there are links between strong abilities to visualize and success in mathematics. One widely used psychological assessment for visualization involves “The Paper Folding Test”  in which a paper is folded and a hole is placed through a specific location and the participant is asked to visualize what the paper will look like when it is unfolded, utilizing the ability to generate, maintain and manipulate a mental image, (Lohman, 1996 cited in Moss et al 2016). A recent study also found a link between the ability to visualize and success with solving mathematical word problems, citing the ability to mentally visualize and make sense of the problem contributed to success in diagramming and solving problems (Boonen et al 2013 cited in Moss et al 2016). The Canadian work of (Moss et al 2016 ) and their Math for Young Children research project focuses on spatial reasoning and the importance of developing students’ flexible use of visualization skills and strategies.

 

Instructional Resources

Screen Shot 2018-12-11 at 4.11.50 PMThe book Taking Shape (referenced below) provides several visualization tasks on pages 30-35 but visualization is an important component of most of the spatial reasoning tasks in the book.

 

 

 

 

Quick Images is an instructional routine that supports the visualization of quantities and shapes. Dot patterns and Screen Shot 2018-12-11 at 2.26.05 PMcomposition of shapes are often used as quick images. More information and videos can be found on the TEDD website HERE.

 

A short article from the NCTM explaining the connection between visualization and subitizing can be found here:

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Screen Shot 2018-12-11 at 2.28.51 PMFawn Nguyen has compiled a collection of visual patterns HERE. Visual patterns provide the first three steps of the pattern and then students are asked to visualize the next steps, which involves both arithmetic, algebraic and geometric thinking.

 

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visualize and graph linear relationships and functions and relations.

 

 

So what does it mean to be proficient with visualizing?

As we begin to work with the new proficiency scale across BC, we need to consider what it means to be proficient with visualizing to explore mathematical concepts in relation to the grade level curricular content. As more teachers across the provinces the the scale, we will have examples of student proficiency that demonstrates initial, partial, complete and sophisticated understanding of the concepts and competencies involved.

For example, a grade six student at the end of the year would be considered proficient with visualizing geometric transformations if they were able to follow directions to mentally translate, rotate and reflect a 2D shape and show or describe the resulting orientation/position.

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Some questions to consider as you plan for learning opportunities to develop the competency of visualizing:

How is the core competency of communication developed through the process of visualization? What different ways can students show and explain what they are visualizing – using materials, pictures or words?

How do the competencies of estimating and visualizing complement each other to support reasoning and analyzing in mathematics? How can using visual referents support estimating?

How can we help students understand the purpose and usefulness of developing visualization skills and strategies? What examples can we share of scientists and inventors that used visualization to develop theories and ideas?

What opportunities are we creating for students to practice and use visualization skills and strategies across different mathematical content areas such as geometry, measurement, number, algebra and functions?

~Janice

 

References

Thinking Through and By Visualizing (NRICH)

The Power of Visualization in Math by Jeremiah Ruesch

Spatial Reasoning in the Early Years: Principles, Assertions, and Speculations by Brent Davis and the Spatial Reasoning Study Group, 2015

Taking Shape: Activities to Develop Geometric and Spatial Thinking by Joan Moss, Catherine D. Bruce, Tara Flynn and Zachary Hawes, 2016