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April thinking together: communicate mathematical thinking in many ways

Posted on: April 30th, 2019 by jnovakowski No Comments

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 No Comments

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

creating spaces for playful inquiry: thinking about the hundred languages – April 2018

Posted on: May 16th, 2018 by jnovakowski

For our final session of this year’s Creating Spaces for Playful Inquiry professional learning series, we focused on the Hundred Languages – a grounding element of the educational approach from the childcare centres in Reggio Emilia, Italy. The Hundred Languages concept is based on a poem by Loris Malaguzzi who suggests that all children have a hundred languages (or more) in which to express themselves and that are role as educators (and school systems) is to nurture these languages, not suppress them.

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As Richmond educators entered the room, they were invited to they were asked to reflect on how the hundred languages are living in their classrooms.

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The Richmond educators who visited Opal School in Portland over spring break shared their reflections on the experience through documentation panels.

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Carrie Bourne,  Jen Yager and Julie Curran shared what they learned at Opal and how they have taken some of these ideas up in their own teaching contexts.

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Marie Thom and I shared some of our experiences from our Canadian Study Tour of Reggio Emilia in March. I shared some ideas I saw about intersecting digital and analog languages through digital landscapes and Marie shared the power of the language of food and the metaphor of the table to bring people together.

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After dinner together (enacting the table metaphor) our interest groups met with playful inquiry mentors to share ideas and go deeper with their understanding about playful inquiry.

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We collected feedback from educators who have attended this three part series as we reflect on our learning from this year and think ahead to next year.

“Love the opportunity to collaborate with others and hear others share about their thinking/learning and what they are trying in their classrooms. It is thought-provoking and inspiring.”

“Playful inquiry and teaching is a learning process, always growing and changing and best in collaboration with others teachers and peers.”

“This series has kept me inspired when I’ve felt uninspired or simply tired.”

“This series completely changed the lens through which I see my role as the teacher and the roles of the students.”

There was considerable interest in creating opportunities for teachers to visit others’ classrooms to see playful inquiry in action and to be able to collaborate with colleagues from across the district.

 

Regardless of how how things unfold for professional learning opportunities in our district for next year, we know we have a strong and growing community of educators committed to teaching and learning through playful inquiry. Thanks to all of the educators involved in this series for their contributions and participation!

~Janice on behalf of the Playful Inquiry Mentors

2017-18 big mathematical ideas for grades 3-5

Posted on: May 13th, 2018 by jnovakowski 2 Comments

In its fourth year, a group of grades 3-5 teachers came together three times after school to think about the big mathematical ideas for this grade range, considering the pedagogical content knowledge needed to teach and assess student learning. Our first session of the year on October 18 focused on the number concepts big ideas in our curriculum which at gates 3-5 focus on a deep understanding of fractions.

We began with an image from fractiontalks.com – a website curated by Canadian math educator Nat Banting. We considered what students needed to understand about fractions to engage with this task and anticipated how are students might respond to the challenge of figuring out what fractional part of the large square is the shaded blue triangle.

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We considered how different materials provided different affordances for thinking about fractions, particularly thinking about different ways to represent fractions – set, area and linear. Some of the text slides from the session and a handout follow.

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Unfortunately, I had to cancel our January session due to illness.

We came together again on April 11 and based on feedback from the group, discussed computational fluency and the role of inquiry in learning mathematics. We revisited instructional routines such as Which One Doesn’t Belong? (wodb.ca) and considered how these routines incorporate questioning, wondering and nurture the curricular competencies in mathematics.

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~Janice

2017-18 primary teachers study group: session 5

Posted on: May 13th, 2018 by jnovakowski

On April 12, our study group met on the dyke of the middle arm of the Fraser River. We were joined by “Indigenous Plant Diva” and current storyteller in residence for the Vancouver Public Library, Cease Wyss. A short video about Cease can be found HERE.

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As we walked along the river, Cease pointed out different plants to us and shared knowledge and stories about the plants. Paying attention to a plant’s colour, shapes and texture can indicate part of the body or ailment it can provide medicine for. For example, red berries often support blood, muscles and organs.

Cease explained the importance of cattails to cleanse the water along the river as well as providing food and nesting materials for birds. We learned how some plants like dead nettle and chickweed can be used as salves to treat skin ailments and how other plants such as stinging nettle or salmonberry leaves can be infused in hot water to create teas to address different ailments.

We learned to identify plantain (frog’s leaf), dead nettle, chickweed, Nootka rose, sheep sorrel and horsetail, the oldest plant on the planet.

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Teachers left with so much new knowledge about local plant species. This knowledge building is an important part of our study group and was something that was requested by teachers to enhance they work they are doing with their students around storytelling outdoors. We can find ways to share this new knowledge with our students and weave this in to our storytelling experiences.

~Janice

March thinking together: What is computational fluency?

Posted on: May 12th, 2018 by jnovakowski

Computational fluency is defined as having efficient, flexible and accurate methods for computing.

-NCTM, 2000

Computational fluency develops from a strong sense of number.

(BC Math Curriculum, Big Idea, K-9, 2015)

 

In BC’s redesigned curriculum, computational fluency has been given a heightened emphasis. In mathematics, there are typically four strands of topics/content and in this iteration of our curriculum, a fifth strand – computational fluency –  has been added and this is reflected in the big ideas and curricular competencies and content.

The meta big idea around computational fluency in our BC K-9 Mathematics curriculum is:

Computational fluency develops from a strong sense of number.

There is a big idea for computational fluency at each grade level:

K: One-to-one correspondence and a sense of 5 and 10 are essential for fluency with numbers.
Grade 1: Addition and subtraction with numbers to 10 can be modelled concretely, pictorially, and symbolically to develop computational fluency.
Grade 2: Development of computational fluency in addition and subtraction with numbers to 100 requires an understanding of place value.
Grade 3: Development of computational fluency in addition, subtraction, multiplication and division of whole numbers requires flexible decomposing and composing.
Grade 4: Development of computational fluency and multiplicative thinking requires analysis of patterns and relations in multiplication and division.
Grade 5: Computational fluency and flexibility with numbers extend to operations with larger (multi-digit) numbers.
Grade 6: Computational fluency and flexibility with numbers extend to operations with whole numbers and decimals.
Grade 7: Computational fluency and flexibility with numbers extend to operations with integers and decimals.
Grade 8: Computational fluency and flexibility extend to operations with fractions.
Grade 9: Computational fluency and flexibility with numbers extend to operations with rational numbers.

As computational fluency with whole numbers is focused on in the earlier grades, it is expected that students will apply number sense and computational fluency and flexibility to their work with decimal numbers, greater numbers, integers and fractions.

For addition and subtraction and then multiplication and division, students develop computational fluency over three years – beginning with emerging fluency, then developing through proficiency and then moving on to extending fluency with increased flexibility and ability to apply strategies across contexts and content.

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For example, with addition and subtraction:

In Grade 3, the curricular content learning standard is “addition and subtraction facts to 20 (emerging computational fluency)“.

In Grade 4, it is “addition and subtraction facts to 20 (developing computational fluency)”.

And in Grade 5, it is “addition and subtraction facts to 20 (extending computational fluency)”.

It is also important to be aware of what comes before and after these three stages of development. In grades 1 and 2, students are introduced to the concepts of addition and subtraction as well as the related symbolic notation. They begin to practice mental math computational strategies building on their understanding of five and ten and decomposing numbers to work flexibly with addition and subtraction questions. In grades 6&7, students apply computational strategies that they have developed for addition and subtraction facts with greater whole numbers, decimal numbers and integers.

There is a similar progression for multiplication and division facts.

A note about memorizing…memorizing is one form of learning but is not necessarily related to students having computational fluency. Many teachers in our district report that their students have memorized their addition or multiplication facts but need support with thinking flexibly and fluently with numbers. In our BC mathematics curriculum, the expectation is that by the end of Grade 3 for addition and the end of Grade 5 for multiplication,  that most students will be able to recall their facts. In a previous curriculum, recall was defined as being able to compute within three seconds. For some students, there may be instant memory retrieval and for other students they may bring the sum or product to mind through an efficient mental computational strategy or associative retrieval process.

Number Talks are an essential instructional routine in developing strategies, mathematical discourse and creating awareness about computational fluency. Key resources include:

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Number Talks

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So some questions to think about…

How would you define computational fluency? What does it look like? sound like?

What do your students need move towards more developed computational fluency?

What do you need to understand more about regarding a continuum of learning and specific strategies related to computational fluency?

What are different ways to develop computational fluency? What instructional routines, games or tasks could we use for practice?

How can we communicate the goals of computational fluency to parents?

~Janice

creating spaces for playful inquiry: thinking about reflection – January 2018

Posted on: May 11th, 2018 by jnovakowski

On January 25, Richmond educators gathered at Grauer for our second dinner session of our Creating Spaces for Playful Inquiry professional learning series. This month our focus was on reflection and time. Educators shared their experiences engaging in playful inquiry with their students and considering the role of reflection in documentation and making both students’ and our professional learning visible.

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After dinner together, teachers participated in interest groups facilitated by our playful inquiry mentors.

A handout with curricular connections to the idea of reflection can be found here: Reflections_Provocations_Jan25_2018

~Janice

2017-18 primary teachers study group: session 4

Posted on: May 10th, 2018 by jnovakowski

On March 1, the primary teachers study group met at the Richmond Nature Park. We shared resources for learning about local living things and discussed the different services the Nature Park provides to schools and the community. The Nature Park is situated on a bog which is a very unique ecosystem.

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We visited different areas of the park, watching the birds come and go from the feeders, walking along the trails and boardwalk area.

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How does looking closely at a found object help you think about its story? What is the story of this (skeleton) leaf?

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There was still snow on the ground in some of the more shaded areas of the park and we used the snow as a story context. How could we use the snow as a background for map-making? We used found natural materials to create a map of a special place to inspire memories and story.

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The Nature House has lots of interactive displays. living things (including a functioning bee hive), and lots of information about species of plants and animals living in Richmond. Brochures are available listing local plants, birds and insects as well as brochures with self-guided tours of the park. We were all keen to continue to build our own knowledge of local species to be able to weave this knowledge into the outdoor learning experiences we are creating for our students.

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The Nature Park Society’s website can be found here: Richmond Nature Park Society

The City of Richmond’s Nature Park web page can be found here: City of Richmond – Nature Park

~Janice

January thinking together: What is numeracy?

Posted on: January 31st, 2018 by jnovakowski

This year I am going to share a monthly focus as a way for educators in our district (and beyond, of course!) to think together, collaborate and share ideas around K-12 mathematics education. On the list are number sense, estimation, reasoning, spatial awareness…it is a list in progress so suggestions are welcome.

My intentions are to begin each month with a blog post highlighting the focus area in our BC mathematics curriculum and connecting it to the broader field of mathematics education. I plan to share links to websites and resources, share books that I have found helpful and provide examples of mathematical tasks from Richmond classrooms. During each month, I will also tweet out related links, ideas, blog posts and photographs from classrooms.

For January, let’s consider what is numeracy?

This January, two of our secondary schools – Steveston-London and Burnett – had students take part in the managed implementation of the Graduation Numeracy Assessment. Other secondary schools in our district are considering both pedagogical and logistical details as they approach the first regular sitting of the assessment for students in June 2018. The Graduation Numeracy Assessment is a new graduation requirement for BC students along with a Graduation Literacy Assessment. Students currently in grades 10 and 11 will begin writing the assessment and have three opportunities to write the assessment to improve their proficiency mark if they choose to. The assessment is not linked directly to a mathematics course or grade and it is thus, the responsibility of all K-12 educators to nurture and develop numerate students. Just as literacy isn’t just about literature, numeracy is not just about numbers – numeracy is being able to apply all areas of mathematics to make sense of the world around you and solve problems relevant to you or others.

For the purposes of the assessment, the Ministry is defining numeracy as:

Numeracy is the ability, willingness, and perseverance to interpret and apply mathematical understanding to solve problems in contextualized situations, and to analyze and communicate these solutions in ways relevant to the given context. 

As students engage with numeracy tasks, they work through a sequence of five numeracy processes:  interpret, apply, solve, analyze and communicate.

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For a more detailed analysis of the concept of numeracy, Dr. Peter Liljedahl and Minnie Liu, share their ideas in an article in Vector, the BCAMT journal: Vector Summer 2013 – Numeracy, pages 34 -39

The following information about the Graduation Numeracy Assessment is available online on the Ministry’s curriculum website:

Graduation Numeracy Assessment Design Specifications 2017

Link to online Graduation Numeracy Assessment sample assessment

Graduation Numeracy Assessment – information for parents

GNA student-choice questions scoring guide and exemplars

Pre-assessment collaborative learning videos

I highly recommend that all BC educators try the sample assessment available online (linked above) to get a sense of the types of questions we can all be using with our students, regardless of grade or course. Last Friday, on a professional development day, the whole Richmond Secondary School staff worked in groups to collaborate on some of the sample assessment questions and to consider how to embed opportunities for numeracy in their courses.

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Looking forward to continuing the conversation around numeracy and what it means to be numerate.

~Janice

 

October thinking together: What is balanced numeracy?

Posted on: October 31st, 2017 by jnovakowski

This year I am going to share a monthly focus as a way for educators in our district (and beyond, of course!) to think together, collaborate and share ideas around K-12 mathematics education. On the list are number sense, estimation, reasoning, spatial awareness…it is a list in progress so suggestions are welcome.

My intentions are to begin each month with a blog post highlighting the focus area in our BC mathematics curriculum and connecting it to the broader field of mathematics education. I plan to share links to websites and resources, share books that I have found helpful and provide examples of mathematical tasks from Richmond classrooms. During each month, I will also tweet out related links, ideas, blog posts and photographs from classrooms.

For October, let’s consider what is balanced numeracy?

I have been fortunate to be a part of a Ministry initiated project drawing together educators from across BC to form the BC Numeracy Network. Our first project together was to consider balanced numeracy within the redesigned mathematics curriculum.

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The group of educators involved in this network began our professional collaborative inquiry together considering the question, What is balanced numeracy? Although we drew on the practices of balanced literacy, we looked to the current research in mathematics education to inform our thinking. In thinking about a balanced numeracy approach, we looked beyond mathematical content and foundations (which needs to remain in the foreground) to also consider the role of the learning environment, instructional and assessment approaches, the habits, dispositions and competencies of the discipline of mathematics, the importance of connection-making and the role of engagement in learning. We considered balance in terms of the day, week, month, term and year as well as what learning opportunities are provided to our students for individual practice and problem-solving, small group work and whole class routines, discourse and mathematical community building.

The BC Numeracy Network has created a website full of linked references and resources to support teachers’ professional learning: https://sites.google.com/view/bcnumeracynetwork/home

As you think about balanced numeracy, consider…

What do you consider when planning a balanced literacy program in your classroom? What aspects of this might apply to thinking about balanced numeracy?

How are numeracy experiences different from mathematics experiences and how are they connected?

What opportunities can I provide for my students to engage in small group work in mathematics/numeracy? (The North American classroom landscape suggests that most “math time” is either whole class or individual work).

How am I balancing the two components of the BC learning standards in mathematics – curricular competencies and curricular content?

How might providing balanced numeracy experiences for my students extend or enhance their thinking about mathematics?

~Janice

 

There are many resources now available to support aspects of balanced numeracy in the classroom. Consider the following professional resources:

Math Workshop by Jennifer Lempp

Guided Math in Action by Dr. Nicki Newton

Guided Math by Laney Sammons

Dr. Peter Liljedahl of SFU has many numeracy tasks available on his website

This part of our website defines numeracy in the BC context and looks at the parallels between balanced literacy and balanced numeracy.

 

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