Archive for the ‘curriculum’ Category

world tessellation day 2019

Posted on: July 4th, 2019 by jnovakowski No Comments

World Tessellation Day is celebrated on June 17 in honour of M.C. Escher’s birthday and was created by author Emily Grosvenor of Oregon. More information can be found HERE. Emily Grosvenor wrote a book about a little girl named Tessa who saw shapes, patterns and designs in the world around here. The book Tessalation was launched on Kickstarter and I was happy to support it. In celebration of the book launch, the author asked some of the supporters to do blog posts about the book and be part of a blog tour. The blog post I did can be found HERE.

World Tessellation Day was promoted in our district with connections made to our BC mathematics curriculum and it was great to watch social media posts appear highlighting tessellations! You can find some of these posts by following the hashtag #WorldTessellationDay – I have shared some posts from Richmond teachers below.

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This year, the Thursday before World Tessellation Day, I invited students to explore tessellations through materials in The Studio at Grauer after we looked at some of the pages of the book Tessalation. The investigation prompt was What can you find out about tessellations? Noticing the materials I put out, I need to remember to use some non-regular polygons as well.

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On June 17, our district’s Math Play Space featured materials that tessellate and we set up tables at the Brighouse Public Library during after school hours. Many children and families visited to investigate with the materials.

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I’m looking forward to making World Tessellation an annual public event not just in our schools but also in other places in our community!

~Janice

 

outdoor classroom day: November 1 2018 and May 23 2019

Posted on: July 3rd, 2019 by jnovakowski No Comments

Outdoor Classroom Day is an international initiative to promote the importance of children experiencing the outdoors. The event began in London, England in 2012, grew internationally in 2015 and became a global project in 2016 with the support of educators, environmentalists, play experts, NGOs, Project Dirt and Unilever. The organization supports and shares research around the benefits of playing and learning outdoors. More information can be found on the website HERE.

We participated in our first Outdoor Classroom Day on November 1 with three classes at Grauer. Each class began with some time in The Studio where we shared ideas about where and how we might see and experience mathematics outdoors. I shared a different book of images with each class to inspire their mathematical thinking.

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It was pouring rain most of the day, but being the day after Hallowe’en it was actually a good day to be outside and moving.

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Each class took on different tasks with the grades 1&2 classes looking for and thinking about estimating, counting, size (measuring) and shape.

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The grades 3&4 class worked in small groups with a task card prompting them to search for things outside involving mathematical ideas such as symmetry, fractions and measurement.IMG_4370 IMG_4377

As we were coming inside, one of the grade 2 students said, “This was the best math day ever!”

The next Outdoor Classroom Day for this school year was May 23. The grades 3 and older classes at Grauer were at the district track meet, so the three K-2 classes spent the day outside together. We took our new Outdoor Studio Wagon with us outside and it was filled with materials, tools and resources to support and inspire our mathematical thinking.

IMG_2784 We began our morning by finding out own counting collections outdoors and then recorded our counts in different ways. Some students gathered leaves, twigs or pieces of park to group and count while others found multiples in plants such as buttercups (counting the flowers by 1s and the petals by 5s) and clover (counting the leaves by groups of 3s).

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After recess, we considered different types of branding and growing patterns that we noticed in the growth of trees, plants, leaves, flowers and roots and captured these patterns with clay prints and plant prints using rubber mallets.

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Here are the contents in the Outdoor Studio Wagon:

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The next Outdoor Classroom Day is November 7 2019. You can sign up HERE.

~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.

2018-2019 primary teachers study group: session six

Posted on: June 7th, 2019 by jnovakowski No Comments

Our final session of the year was hosted at Thompson Elementary on May 16. Inspired by our core resource, Messy Maths by Juliet Robertson, we created outdoor ten frames using pieces of cotton fabric and sharpies. These ten frame can be used to count quantities of found objects to ten as well as using for grouping smaller objects like pebbles or acorns. And they are washable and re-usable and can be used in the rain which makes them ideal for outdoor learning where we live!

We also used rubber mallets on cotton cloth to create leaf and flower prints to explore the shape, size and symmetry of local plants. This is the just right time of year to do this when the cells of plants and petals are full of moisture.

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Teachers shared the different ways we have been using our focus picture book Flow, Spin, Grow by Patchen as we have found growing, swirling and branching patterns outdoors.

We also shared information about the Lost Ladybug Project – a fun way to engage students in looking closely for ladybug species, taking photographs and sharing the location of the find with the world through the website HERE

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The Thompson team toured us through their outdoor learning space and showed us their student’s mapping project.

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Thank you to Denise, Tanya and Danielle and their teacher candidates for hosting us!

We have surveyed the group and it looks like next year’s focus will be interdisciplinary learning outdoors. We will be able to connect our work around storytelling and math outdoors from the last two years as we move forward together in our professional learning.

~Janice

2018-2019 primary teachers study group: session five

Posted on: June 5th, 2019 by jnovakowski No Comments

Our fifth session of the year was hosted by Sarah Regan at Homma Elementary o April 11. Teachers shared how they had been using the book Flow Spin Grow and our French Immersion teachers were happy to have the French version now available! Teachers shared how they took photographs of the types of patterns they found outdoors and used them for inspiration in the classroom for creating patterns with materials, doing looking closely observations for science, inspiring artistic creations, etc.

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After our professional sharing, the group visited the Homma Gardens and outdoor classroom and shared ideas around how mathematics can be experienced in the garden at this time of year such as building trellises  (shape, design, symmetry, measurement) and reading seed packages (time, duration, elapsed time, measuring time, measuring depth, measuring distance apart, estimating height).

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Thank you to Homma for hosting!

~Janice

big mathematical ideas for grades 6-9 2019

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

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 No Comments

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

 

 

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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 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

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:

notice wonder poster

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