Archive for the ‘math’ Category

September thinking together: mathematics curricular content

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

For the 2019-20 school year, the “thinking together” series of blog posts will focus on the curricular content in the mathematics curriculum.  The “thinking together” series is meant to support professional learning and provoke discussion and thinking. This month will provide an overview of the curricular curricular content and then each month we will zoom in and focus on one curricular content area with examples from K-12 classrooms in Richmond.

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The curricular content is the “know” part of the know-do-understand (KDU) model of learning from BC’s redesigned curriculum.

The curricular content develops and builds over time. Each grade level has core curricular content knowledge and these are reflected in the big ideas for each grade level. There are five big ideas that reflect five strands of curricular content – number and number operations, computational fluency, geometry and measurement, patterning and algebraic relationships and data analysis and probability. A sixth content area in mathematics, financial literacy,  is new this curriculum.

Many areas of curricular content in mathematics are applied in other disciplines such as science, physical education and ADST.

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The curricular content along with the curricular competencies  comprise the legally mandated part of the curriculum, now called learning standards. This means that both curricular content and curricular competencies are required to be taught, assessed and proficiency/learning achievement is communicated to students and parents/guardians.

As we begin a new school year and are thinking about year plans and overviews we might consider the following questions:

  • What opportunities do students have to learn about what math is? How can we expand students’ thinking about all the areas of mathematics reflected in our curriculum? What do you think your students would say if you asked them: What is math?
  • What would you identify as core or foundational concepts or mathematical ideas at the grade level/s you tech?
  • How can we make the connections between curricular content and curricular competencies in mathematics visible in our classrooms and schools?
  • As we are planning for instruction and assessment, how are we being intentional about weaving together both curricular content and competencies What curricular content areas complement and are linked to specific curricular competencies?
  • How do you plan for opportunities for students to make math-to-math connections over the school year? For example, what connections are there between number and geometry or patterning and data analysis?

~Janice

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.

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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

The Math Play Space at the Richmond Garlic Festival

Posted on: September 1st, 2019 by jnovakowski 1 Comment

After launching our district’s Math Play Space this spring at some school and public library events, we took part in our first big community event on August 19 at Richmond’s Garlic Festival. In its eleventh year, the Garlic Fest draws visitors from all over the Lower Mainland and has an active Kids Zone Area.

 

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IMG_6273I arrived bright and early to set up our space and by 10am our tent area was filing up with families. I was thankful for the shifts volunteered by our math mentor teachers and teacher consultants. It was interesting to watch families interact with the materials – some parents and grandparents stayed back and watched their children play, others played with their children and other adults jumped in and played on their own. Parents had questions about the materials being used and some had questions about the BC math curriculum. I was able to provide them with our district brochure and hope to have it translated into multiple languages by our next community event.

Our tent was full throughout the day with the Kids Zone coordinator saying that we were the most visited area of the day!

 

 

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We did change out some of the materials over the course of the event but the two most used materials were the tiling/tessellating turtles and the magnatiles.

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I particularly loved watching the children who had made garlic crowns at a booth in the area playing with the materials!

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For more information on the Richmond School District’s Math Play Space, including upcoming events, please see HERE.

~Janice

world tessellation day 2019

Posted on: July 4th, 2019 by jnovakowski

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

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

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

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

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.