Archive for the ‘secondary’ Category

December thinking together: visualize to explore mathematical concepts

Posted on: December 11th, 2018 by jnovakowski No Comments

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

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

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

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

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

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

 

Instructional Resources

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

 

 

 

 

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

 

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

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

 

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visualize and graph linear relationships and functions and relations.

 

 

So what does it mean to be proficient with visualizing?

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

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

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

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

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

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

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

~Janice

 

References

Thinking Through and By Visualizing (NRICH)

The Power of Visualization in Math by Jeremiah Ruesch

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

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

 

September thinking together: mathematics curricular competencies

Posted on: September 28th, 2018 by jnovakowski No Comments

For the 2018-19 school year, the “thinking together” series of blog posts will focus on the curricular competencies 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 competenecies and then each month we will zoom in and focus on one curricular competency and examine connections to K-12 curricular content, possible learning experiences and assessment.

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

The curricular competencies are intended to reflect the discipline of mathematics and highlight the practices, processes and competencies of mathematicians such as justifying, estimating, visualizing and explaining

The curricular competencies are connected the the Core Competencies of Communication, Thinking  and Personal & Social. More information about the Core Competencies can be found HERE.

 

Screen Shot 2018-09-28 at 9.45.26 PMThe curricular competencies along with the curricular content comprise the legally mandated part of the curriculum, now called learning standards. This means these competencies are required to be taught, assessed and learning achievement for these competencies is communicated to students and parents.

Something unique about the mathematics curricular competencies is that they are essentially the same from K-12. K-5 competencies are exactly the same with some slight additions in grades 6-9 and then building on what was created in K-9 for the grades 10-12 courses. Because they are the same at each grade level, to be assessed at “grade level” they need to be connected to curricular content. For example, one of the curricular competencies is “estimate reasonably” – for Kindergarten that will mean with quantities to 10, for grade 4 that could mean for quantities to 10 000 or for the measurement of perimeter using standard units and for grade 8 estimating reasonably could be practiced when operating with fractions or considering best buys when learning about financial literacy.

The new classroom assessment framework developed by BC teachers and the Ministry of Education focuses on assessing curricular competencies and can be found HERE.  A document outlining criteria categories, criteria and sample applications specific to K-9 Mathematics can be found HERE. The new four-point proficiency scale provides language to support teachers and students as they engage in classroom assessment.

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As we are 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 it means to be a mathematician and what mathematicians do?
  • What opportunities can be created over the school year for students to name, be aware of, practice, develop and reflect on the core and curricular competencies in mathematics?
  • How can we make the core competencies 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 competencies and content? What curricular content areas complement and are linking to specific curricular competencies?

~Janice

May thinking together: How can we weave Indigenous content and perspectives into the teaching and learning of mathematics?

Posted on: June 12th, 2018 by jnovakowski 1 Comment

Screen Shot 2018-06-12 at 11.25.11 PMThe First Peoples Principles of Learning is a foundational document in the redesign of BC’s curriculum frameworks. The Principles were developed by FNESC (First Nations Education Steering Committee) and the poster in English can be found HERE and in French can be found HERE. As Jo Chrona would say, the FPPL are much more than the poster – they are principles that are inclusive of all children in BC while honouring Indigenous ways of being and knowing. FNESC has developed teaching resources such as the In Our Own Words resources for K-3 and the Math First Peoples resource for Grades 8&9 (currently being updated) but much of the information and ideas in the resource can be adapted for all grade levels.

 

On May 17, Leanne McColl, Lynn Wainwright and myself attended the 8th annual K-12 Aboriginal Math Symposium. Educators from across BC attend this symposium. Information about the symposium can be found HERE and there is a tab on the website that links to archived resources.

I have attended this symposium for years and was fortunate to share a project from The Studio at Grauer at this year’s event. Some of the slides from my presentation can be found HERE , under May 2018.

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A focus of my presentation was on three of BC’s mathematics curricular competencies. These competencies are part of the learning standards for the K-9 mathematics curriculum and are aligned with the First Peoples  Principles of Learning and the Core Competencies.

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The BC Numeracy Network has archived different types of resources to support the redesigned curriculum. Under the Connections tab, there is a page dedicated to resources that support the weaving of the First Peoples Principles of Learning into mathematics teaching and learning.

Link to BCNN page here

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In the Richmond school district, two of the four goals of our Aboriginal Education Enhancement Agreement (AEEA) are focused on all learners (not just those with Indigenous ancestry) developing an understanding about the First Peoples Principles of Learning, our local First Nations community and Indigenous worldviews and perspectives as part of engaging in the process of reconciliation through education.

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Teachers often ask me about where to start in this area and are concerned about not doing things properly or that they do not have enough knowledge themselves. I suggest that teachers contact someone in their district about local protocols and then try something in collaboration, maybe inspired by one of the above suggested resources. Look for authentic connections within your community and across disciplines in the curriculum..  Some of the things that I have done to continue to learn more in this area are: read articles and books recommended to me, seek out opportunities to learn from elders and Indigenous community members and colleagues, get involved with district or university-based collaborative projects,  connect with your district’s Aboriginal Education team, attend workshops and tours offered through museums, cultural centres and local Indigenous organizations. There are lots of opportunities to learn and see connections to mathematics…we need to go forward together with an open mind and an open heart.

To consider…

How can the First Peoples Principles of Learning be embedded in our mathematics teaching and learning? How do BC’s mathematics curricular competencies reflect these principles?

One of the principles is that “learning takes patience and time” – how does this principle bump up against some ideas around the teaching and learning of mathematics?

How might we work towards the goals of our Aboriginal Education Enhancement Agreement within our mathematics classrooms? What role could mathematics play in the process of reconciliation?

What does it mean to use authentic resources, stories and elements of culture in our mathematics teaching? How is this affected by the land and the story of the place where we live and teach? Who can help us think about these ideas? Where can I learn more and find resources?

What opportunities do your students of Indigenous ancestry have to see their community, family and culture represented in the mathematics they are learning at school? Within our diverse community, how do all students see themselves reflected in their mathematics experience? What is the relationship between our students’ mathematical identities and their personal and cultural identities?

What interdisciplinary projects might connect mathematics with Indigenous knowledge and worldviews?

~Janice

Talk With Our Kids About Money 2018

Posted on: May 12th, 2018 by jnovakowski

As part of a national financial literacy month every April, the Richmond School District participates in Talk With Our Kids About Money Day (TWOKAM) the third Wednesday in April. Financial literacy is a new part of BC’s redesigned mathematics curriculum with a content learning standard at each grade level from K-grade 9.

To raise awareness of the resources available to teacher, local CFEE (Canadian Federation for Economic Education) representative Tracy Weeks shared materials and information at our Elementary Math Focus Afternoon in January.

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In April, an assembly was held at Burnett Secondary with CFEE president Gary Rabbior talking to students about financial literacy.  Tracy Weeks (CFEE) facilitated an information session for parents at Hamilton Elementary on April 9.

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On April 18 – national TWOKAM day – a finale event was held for parents and students at Brighouse Elementary. Student projects from Burnett Secondary were on display and guest speaker Paul Lermitte shared ideas with parents for developing financial literacy with their children at home. Thank you to Brighouse for hosting this well-attended event!

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We hope to continue to grow the idea of “Money Fairs” (think financial literacy fairs like science fairs) in our district as we continue to teach and learn about financial literacy in our classrooms.

TWOKAM video

TWOKAM – CFEE website link

~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

graduation numeracy assessment – January 2018

Posted on: May 11th, 2018 by jnovakowski

On January 26, I spent part of my morning at Richmond Secondary working with the whole staff to examine the Graduation Numeracy Assessment – how numeracy is defined, the numeracy processes, example questions and ways to embed numeracy tasks in all courses. Educators worked in cross-disciplinary groups to choose one of the sample questions to work through together, being mindful of how their students might engage with these questions.

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The overview slides from the morning can be found here: GNAoverview_Richmond_January_2018

Detailed information about the BC Graduation Numeracy Assessment can be found through the BC curriculum website. There is a design specifications package, pre-assessment tasks that students/classes can do before the assessment to learn about the numeracy processes, a collaborative learning guide, videos, sample questions, scoring guide and student exemplars as well as information on the background and development of the assessment and information for parents.

link to Graduation Numeracy Assessment information 

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In Richmond, two of our secondary schools (SLSS and Burnett) participated in the gradual roll-out of the writing of the Graduation Numeracy Assessment. Two or three classes from each school participated in the assessment and will receive their results in April. Both schools collected student feedback and the Vice-Principals shared this feedback along with their logistical recommendations at a secondary vice-principals meeting in April.

~Janice

February thinking together: How can we rehumanize mathematics?

Posted on: March 1st, 2018 by jnovakowski

For February, let’s consider how can we rehumanize mathematics?

I was able to attend and present at the NCTM Regional Conference in Chicago this past November/December.

Words that came up over and over again throughout the conference were: EQUITY, ACCESS, EMPOWERMENT, POWER, AUTHORITY, PRIVILEGE, IDENTITY and AGENCY.

These may not be the words you may initially think of when you think about major themes at a mathematics education conference but there is a definite shift in how the role of mathematics within a society is viewed, and who is being included and what voices are being heard.

The conference opened with a panel of speakers addressing issues of access, equity and empowerment.

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The link to the archived Facebook live video form the opening session of the NCTM Regional Conference in Chicago on November 29 2017 is HERE

Kassia Omohundro Wedekind shared her thoughts on “hands down, speak out” practices, disrupting the traditional practice of having students putting their hands up to respond to questions, which in classrooms often reflects the inequities in society at large. She encourages teachers to create conversation maps of the mathematical discourse in their classrooms and to encourage all students to see themselves as mathematicians and contributors to the discourse.

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Dr. Tyrone Howard discussed the issues of inequity in classrooms with regards to race. He affirmed the importance of context and that students’ issues of identity need to be addressed.

Annie Perkins shared practices from her own secondary classroom that support all her students in seeing themselves as mathematicians. She developed “The Mathematicians Project” with a focus on introducing students to mathematicians that are “not just white dudes”.

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Dr. Rochelle Gutierrez is a well recognized researcher and author in the area of equity in mathematics. Over the last few years, her papers and presentations have focused on the idea of re-humanizing mathematics. Photos of some of her slides provide a beginning glimpse into her work. What stands out for you? What are curious about learning more about?

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Another presentation from Dr. Rochelle Gutierrez that might be of interest and would be great to share with during a pro-d day to open up discussion around equity, access and culturally responsive pedagogies for all students:

NCTM 2016 ShadowCon: Stand up for Students 

Dina Williams closed the opening panel with stories from her classroom and her version of a powerful song –  A Change is Gonna Come.

Change has been a long time coming.

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So to think about…

Do all of your students see themselves as mathematicians?

What mathematicians do you talk about in your class? Do you provide “mirrors and windows” for your students to see themselves in the curriculum and in others?

What types of mathematics do we privilege?

What is culturally responsive pedagogy in mathematics education?

How is mathematics necessary for engaging with ideas of social justice?

How can we broaden our understanding of math to de-center what is typically viewed as “school mathematics”?

How do we create mathematics experiences in our schools that are inclusive for all learners?

~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

 

November thinking together: What is number sense?

Posted on: November 29th, 2017 by jnovakowski

What is number sense?

How are numeracy and number sense the same and how are they different?

What routines nurture the development of number sense?

 

~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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