Posted on: April 28th, 2020 by jnovakowski No Comments
During this post-spring break phase of the school year, we are providing continuity of educational opportunities and learning experiences for our students. We are planning these opportunities through the lenses of learning priorities, equity, access and compassion. Every student’s context will be unique and we are responding with choices and options that are manageable for families at this time. A collection of resources to support the teaching and learning of mathematics and numeracy during this time have been created and curated on a page on this blog, which can be found above.
We launched our ongoing Creating Spaces for Playful Inquiry professional learning community this year with a dinner event out at IDC for 50 Richmond teachers on October 22.
After time with materials, playful inquiry mentors Briana Adams and Jess Equia shared their investigation into fibres, sewing and natural dyes with Briana’s class. After dinner together, we broke into interest groups to engage in conversation with playful inquiry mentors.
The handout for the session can be downloaded here:
Follow up Studio Series sessions have been held in The Studio at Grauer. Our first session looked at the language of wool roving and what affordances it has. Teachers considered the story of wool from sheep to sweater and considered concepts such as texture and transformation that are developed as students work with this material.
Teachers had the opportunity to touch, transform and use wool roving in different ways to help deepen their understanding of this material and how they might use it in the classroom.
Our second Studio Series session looked at the language of cotton and what possibilities it offers us as a material. We considered the story of cotton from plant to t-shirt and also discussed the social and environmental implications of the cotton industry.
Teachers used cotton in different forms – fabric, rope, embroidery floss and thread – to create with.
Thank you to all the playful inquiry mentors for continuing to grow this community in our district and a special thank you to Briana and Jess for all their contributions!
For our session together in December the grades 5&6&7 class at Quilchena examined two infographics about environmental issues and discussed how infographics use numbers and data in different ways to convey information, provoke thinking and to be persuasive. The students shared how they noticed how the infographics made some numbers large or highlighted them with colour to draw attention to them and how different types of charts or graphs can help the reader understand the information.
The students in the class have each selected a global issue that they are passionate about and have found an article in the National Geographic database to read and take notes on about their topic. They referred back to the article and their notes to find mathematical information that they could use in their own visual image that could become part of an infographic for their global issue project.
The students used apps (Pages, Paper or PicCollage) or online platforms (Canva, Piktograph) to create a visual for their project.
Through the process of creating their own visuals to share information, the teachers and I think the students will become more fluent at interpreting and analyzing infographics and other media.
Posted on: December 12th, 2019 by jnovakowski 1 Comment
For another year, we have held this three-part after school series with primary teachers to think together about the big math ideas in mathematics. For this series we focused on counting, number sense and place value understanding. Each teacher was provided with the teacher resource book Choral Counting and Counting Collections by Megan Franke, Elham Kazemi and Angela Turrou-Chan.
At our first session, we reviewed the different aspects of counting (see pedagogical content knowledge paper on counting available at the bottom of this post) and discussed ideas for both choral counting and counting collections. Teachers worked together in small groups to plan and lead a choral count that they would use with their students.
Stenhouse Publishers have an online choral counting tool you can use to plan out choral counts with your students. You can access it HERE.
Teachers were also asked to use the new SD38 Early Numeracy Assessment with some students in their classes and provide feedback on its usage and findings. This tool will be available publicly soon after this last round of trials and feedback.
At our second session, teachers shared what routines and innovations they tried with their classes and then we did counting collections together, considering ways to extend the counting collection experience by recording both the process of counting and the final count as well as recording equations that describe the count.
At our third session, we focused on place value concept development through tasks and games. Teachers had the opportunity to make some numeral materials. Its always handy to have sets of numerals available in the primary classroom for students to make connections between concrete rerpesentations of quantity with symbolic forms.
Some of the text slides from the sessions can be downloaded here:
I have been fortunate to collaborate and co-teach with some of the teachers in this series as we continue to think together about developing number sense through counting and place value tasks.
Our second session of the year was hosted at Homma Elementary on November 28 – with thanks to Sarah and Reiko for having us.
K-7 Curriculum Implementation Teacher Consultant, Jess Eguia, began our time together with a land acknowledgement and three ways to enhance land acknowledgements in our schools. She shared the beaded timeline, sharing the story of time immemorial on this land,
and the Musqueam place names map which shares the significance of key places in the territory.
Jess also shared some ideas about Indigenous ways of knowing and being that could help teachers to elaborate and extend their students’ thinking about land acknowledgements.
Using land-based materials found locally, we did some bundle dyeing.
While the bundles steamed, we headed outside for a walk along the river, sharing stories of this place over time.
We came back into the Homma library for a hot cup of tea, the unbundling and sharing what we have been trying with our students, inspired by the resource books that are inspiring us this year.
Looking forward to continuing our conversations around land-based interdisciplinary projects in the new year!
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. Each month we will zoom in and focus on one curricular content area with examples from K-12 classrooms in Richmond.
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.
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.
GEOMETRY
The foundational research that informs educators how children’s geometric thinking develops over time was developed by the van Hieles in the 1950s and published formally in the 1980s. The van Hiele hierarchical model has five broad categories (numbered 0-4).
Level 0: Visualization – analyzes component parts of figures but cannot explain interrelationships between figures and properties
Level 1: Analytic – analyzes component parts of figures and their attributes, understands necessary properties
Level 3: Formal deduction – understands and uses undefined terms and theorems meaningfully
Level 4: Rigor – advanced geometric thinking beyond the scope of the traditional secondary mathematics classroom
A more in depth explanation of the categories, along with examples, can be found 0n pages 16-17 in this excerpt of an NCTM publication linked HERE.
One of our five mathematical big ideas in our BC mathematics curriculum focuses on spatial relationships. The K-9 “meta” big idea is: We can describe, measure, and compare spatial relationships. In our curriculum spatial relationships link geometry and measurement concepts and skills together. For example, in grade 7, one of the content learning standards is: volume of rectangular prisms and cylinders.
Spatial reasoning in young children is an indicator of future overall school success, as well as more specifically, literacy and numeracy (multiple research studies across disciplines including Duncan et al, 2007 – cited in Davis, 2015). It is not a pre-determined trait but is something that is malleable and can be learned. Spatial reasoning and geometry are foundational to disciplines such as astronomy, architecture, art, geography, biology and geology and are an essential part of STEM/STEAM education and future careers.
In K-7, students learn the following mathematical content related to geometry:
An understanding of composing and decomposing both 2D and 3D shapes develops from Kindergarten through to grade 12. An understanding of what shapes make up other shapes is essential for students to apply geometric reasoning and to connect to measurement concepts such as area and volume.
Other aspects of geometry, such as positionality, perspective, dynamic movement and visualization, are embedded in the elaborations through projects, tasks and applications. These aspects are also important in the ADST curriculum, particularly in coding as well as in physical education, dance and visual arts.
In the first year of secondary school in Richmond, grade 8 students develop understanding of the big idea:
“The relationship between surface area and volume of 3D objects can be used to describe, measure, and compare spatial relationships.“
Grade 8 geometry content knowledge is focused on:
surface area and volume of regular solids, including triangular and other right prisms and cylinders
Pythagorean theorem
construction, views, and nets of 3D objects
The following are some photographs from a grade 8 math class at Hugh Boyd Secondary, where students investigate these intersecting geometry and measurement concepts in concrete, pictorial and symbolic forms.
A new Geometry 12 course was added to the choice of math courses available to our BC secondary students this fall. The five big ideas in the Geometry 12 course are:
Diagrams are fundamental to investigating, communicating, and discovering properties and relations in geometry.
Geometry involves creating, testing, and refining definitions.
The proving process begins with conjecturing, looking for counter-examples, and refining the conjecture, and the process may end with a written proof.
Geometry stories and applications vary across cultures and time.
The curricular content for the Geometry 12 course includes:
geometric constructions
parallel and perpendicular lines:
circles as tools in constructions
perpendicular bisector
circle geometry
constructing tangents
transformations of 2D shapes:
isometries
non-isometric transformations
non-Euclidean geometries
Much of the content in the Grade 12 course builds on and further develops content knowledge that is included in the K-8 mathematics curriculum.
An instructional routine that is used across K-12 is Which One Doesn’t Belong? otherwise known as WODB. In this routine, students are presented with four related items (in this case, shapes) and are asked to describe and compare their attributes and then share their thinking and reasoning to explain if they had to choose one of the shapes to not belong, which one would it be and why. This routine develops many mathematics curricular competencies as students develop and synthesize content knowledge.
A WODB poster in English and French can be found on this site HERE.
Mary Bourassa has curated a collection of WODBs on THIS SITE.
As we think about how geometry concepts develop over time, we might consider the following questions:
What would you identify as core content around geometry at the grade level/s you teach?
What curricular competencies are connected to the curricular content of geometry?
How do we support students’ development of geometric reasoning, paying attention to the different concepts and skills involved and being mindful of van Hiele’s hierarchy?What assessment techniques will give use the information we need?
What opportunities are there for your students to make math to math connections, connecting their understanding of geometry to other mathematical content areas and to other disciplines?
~Janice
References
Taking Shape: Activities to Support Geometric and Spatial Thinking K-2 by Joan Moss et al (2016)
Which One Doesn’t Belong: A Shapes Book, A Teacher’s Guide by Christopher Danielson (2016)
Spatial Reasoning in the Early Years: Principles, Assertions, and Speculations by Brent Davis and the Spatial Reasoning Study Group, 2015
Paying Attention to Spatial Reasoning: K-12 Support Document for Paying Attention to Mathematics Education, Ontario Ministry of Education, 2014 (available as a pdf online)
Understanding Geometry by Kathy Richardson (1999)
Learning and Teaching Early Math: The Learning Trajectories Approach by Douglas Clements & Julie Sarama (2009, 2014)
Open Questions for the Three-Part Lesson: Geometry and Spatial Sense K-3, 4-8 by Marian Small & Ryan Tackaberry (2018)
On November 27, I visited the grades 5&6&7 class at Quilchena to continue our focus on numeracy and for this session together I selected a numeracy task from Dr Peter Liljedahl’s website. The task continues the thinking we have been doing about water issues and and moves to thinking about agency around water conservation. We took some time together to go through what the task was asking of the students, what assumptions they needed to make, what calculations might be necessary and how they could share their recommendations.
Teachers Jen Yager and Sam Davis personalized the task by changing the names to teachers’ names from their staff. This made for some interesting comments about dental hygiene habits!
We needed to pause after the students read through and shared their understanding of the task with each other. Based on the experience we had with the last numeracy task we did, we had agreed to provide some supports to ensure students were able to get started with the task successfully. We talked through what the task was asking, what information they might need to research, what assumptions they needed to make and asked them about different ways they might approach the task.
When some of the students weren’t clear on what the differences between no flow, low flow and high flow of water was, a student quickly demonstrated for them at the sink.
The students researched the Canadian Dental Association’s recommendations for teeth brushing and did calculations for water usage. Based on their findings, they made recommendations to the teachers on ways they could conserve water while maintaining good dental hygiene. Some students wrote this up as a “report” while one student wrote a letter to her teacher with specific recommendations, backed up with her evidence.
Numeracy tasks such as these, organized by grade ranges, can be found on Dr. Peter Liljedahl’s website HERE.
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. Each month we will zoom in and focus on one curricular content area with examples from K-12 classrooms in Richmond.
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.
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.
COUNTING
“Understanding what counting is for is the starting point of an outburst of numerical inventions. Counting is the Swiss Army knife of arithmetic, the tool that children spontaneously put to all sorts of uses. With the help of counting, most children find ways of adding and subtracting numbers without requiring any explicit teaching.” (Dehaene, 1997, p.122)
Counting is considered a number concept and is connected to understanding of our number system, place value, multiples and other relationships between numbers. Within the learning trajectories research from Clements and Sarama (2014), and the critical learning phases work of Kathy Richardson (2012), the following stages are considered in the development of counting:
RIchardson
Counting Objects (one-to-one, stability, checks by recounting, cardinality, estimates, counts out a particular quantity)
One More/One Less (knows one more/one less without counting, recognizes when a number sequence is out of order)
Counting Object by Groups (counts by moving in groups, knows quantity stays same even when counting by different groups)
Clements & Sarama
Chanter
Reciter
Corresponder
Counter
Producer
Counter and Producer
Counter Backward from 10
Counter from N
Skipcounter by 10s
Counter to 100
Counter On Using Patterns
Skipcounter
Counter On Keeping Track
Counter of Quantitative Units/Place Value
Counter to 200+
Number Conserver
Counter Forward and Back
(the names of these stages are descriptive of the counting occurring, for more information visit the learning trajectories website)
Although these stages focus on whole number counting and number understanding, similar stages of development can be seen in parallel tasks when counting by fractions, decimal numbers or integers.
The skills and concepts involved in counting are developed over time and through multiple experiences:
correct sequence of number names
one-to-one correspondence: saying one number name for each object counted
cardinality: the last number said is the quantity counted
stability: the quantity of a group does not change if the objects are rearranged (also related to conservation of quantity)
relative size: more than/less than
make connections between number names, quantities and symbols
counting forwards, backwards and from any starting point
base-ten structure: how can I count or organize by tens and ones to find out how many?
There are many instructional routines that support the development of counting across the grades.
Counting Around the Circle
Counting around the circle is essentially having students count in sequence, taking turns to say the next number in the sequence, one student at a time. The starting number can be changed, the direction of count and the type of count can also be determined to practice specific skills and concepts. Norms can be put in place so that students feel supported by asking a neighbour, or having time to count ahead so they don’t feel “on the spot” when it is their turn if they are unsure of the number they need to say. An example of counting around the circle would be to begin with the number 81 and count backwards by 2s. The count could be recorded on a chart/whiteboard while the students count so the count can be discussed after the circle.
A math game related to this routine is “Buzz” where students sit in a circle and a number of the day is chosen, for example “4”. Every time a multiple of four should be said, a student says “buzz” instead. For example, 1, 2, 3, buzz, 5, 6, 7, buzz, 9, 10…
Choral Counting
Choral counting is a routine that involves having students count in unison to a preplanned counting sequence. As students count together, the teacher records the count in rows and columns providing a visual and symbolic connection to the oral counting. After counting together, the students look at the recording of the count to notice patterns and relationships.
Teacher Kristi Luk at Brighouse Elementary uses the Stenhouse Choral Counting planner to do choral counting sessions with her grades 6&7 class.
Stenhouse Publishers have an online choral counting tool to plan choral count, including counts with fractions and decimal numbers. It can be accessed HERE.
Counting Collections
Counting collections is a routine that emerged out of the research done with CGI (Cognitively Guided Instruction). In essence, students (usually in pairs) choose a collection and count it in multiple ways and record their count (quantity and process) in a way that makes sense to them. Students may begin counting collections by 1s but then continue to develop their understanding of counting by counting in multiples such as 2s, 5s, etc. Grouping tools such as cups, plates and ten frames are often used as part of the counting process. Intermediate students can count items that are already grouped like a box of eight crayons or think about counting with decimal numbers as they count dimes or quarters.
The following are some blog posts on our district blog about counting collections:
As we think about how counting and number concepts develop over time, we might consider the following questions:
What would you identify as core content around counting and understanding numbers at the grade level/s you teach?
What curricular competencies are connected to the curricular content of counting and number concepts?
How do we support students’ development of counting, paying attention to the different concepts and skills involved with counting?What assessment techniques will give use the information we need?
What opportunities are there for your students to apply/transfer their understanding of counting to authentic contexts and problems?
~Janice
References
Learning and Teaching Early Math: The Learning Trajectories Approach by Douglas Clements and Julie Sarama (2009, 2014)
How Children Learn Number Concepts: A Guide to the Critical Learning Phases by Kathy Richardson (2012)
Choral Counting and Counting Collections by Megan Franke, Elham Kazemi and Angela Chan Turrou (2018)
The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene (1997, 2011 – revised and updated edition)
Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 by Jessica Shumway
Number Sense Routines: Building Mathematical Understanding Every Day in Grades 3-5 by Jessica Shumway
I visited the grades 5&6&7 class at Quilchena for the second time on October 30. Inspired by the students’ interest in youth agents of change around climate change and by a mathematical modelling task created by Dr. Julia Aguirre about the Flint Water Crisis in the USA, we invited the students to think about the water crisis on many of our First Nations reserves in Canada.
We began by showing the students a video of water projector and advocate Autumn Peltier speaking to the United Nations 2019 Local Landscapes Forum about the water crisis in her community.
The students took notes, made connections, and recorded their wonders while they were viewing/listening to the video.
We shared three infographics about water issues in Canada and asked students to discuss the following questions:
We also asked students to consider the sources of the information in the infographics as we nurture the development of critical consumers of information.
More information and the infographics can be found HERE and HERE and HERE.
Much of the information was new to students and lots of questions came up. We discussed different types of water advisories and possible reasons why this was happening.
Students were then presented with a numeracy task. They were asked to consider how much water was needed for children for a year in a First Nations community. The purpose of the task was for students to consider the amount of water we use, issues around access to safe water and to think about an action plan for their “agents of change” thinking about how this problem could be resolved.
In hindsight, we made some assumptions that students would be able to think about all the types of information they would need to respond to this task, and know how to access this information using online sources. This was not the case, and a lot of support was needed to help students consider where they could find the information they needed. We talked about validity of sources, such as using Statistics Canada data rather than someone’s opinion on a blog post. The teachers and I realized that the students needed some mini-lessons on how to use Google as a search engine. I think we made assumptions about the students that they knew how to use technology, and they are savvy with many aspects of tech, but their fluency with accessing information was something we needed to develop. When we were able to find information, many students needed support in how to read the data tables. It became clear as we began the numeracy task, that this was much more complex of a task for the students than we had anticipated but we all persevered and made meaning at various levels and stages. For some students, support was needed with the mathematics and calculations involved.
Over the two hours we had together, students thought through various stages of the task. Some students got to the point of considering recommendations for how to reconcile the water crisis in some of our communities but not formalizing their action plans. Some students wondering what was happening to solve this issue.
We briefly looked at the Canadian government’s current plan. More information can be found HERE. This will be an ongoing conversation as we think about different ways that students can see themselves and act as agents of change.
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:
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
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:
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.
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.
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
and also more fraction investigations with a grades 4&5 class at Steves.
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.
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?
