Archive for December, 2019

November thinking together: geometry across the grades

Posted on: December 11th, 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. 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 2: Informal deduction – understands abstract relationships among figures, follows informal proofs
  • 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:

  • single attributes of 2D shapes and 3D objects
  • comparison of 2D shapes and 3D objects
  • multiple attributes of 2D shapes and 3D objects
  • construction of 3D objects
  • regular and irregular polygons
  • line symmetry
  • classification of prisms and pyramids
  • single transformations
  • triangles
  • combinations of transformations

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. 
  • Finding invariance amidst variation drives geometric investigation. 
  • 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)

intermediate numeracy project: water conservation task

Posted on: December 10th, 2019 by jnovakowski No Comments

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.

~Janice

October thinking together: counting across the grades

Posted on: December 2nd, 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. Each month we will zoom in and focus on one curricular content area with examples from K-12 classrooms in Richmond.

KDU_knowdounderstand

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

  1. Counting Objects (one-to-one, stability, checks by recounting, cardinality, estimates, counts out a particular quantity)
  2. One More/One Less (knows one more/one less without counting, recognizes when a number sequence is out of order)
  3. Counting Object by Groups (counts by moving in groups, knows quantity stays same even when counting by different groups)

Clements & Sarama

  1. Chanter
  2. Reciter
  3. Corresponder
  4. Counter
  5. Producer
  6. Counter and Producer
  7. Counter Backward from 10
  8. Counter from N
  9. Skipcounter by 10s
  10. Counter to 100
  11. Counter On Using Patterns
  12. Skipcounter
  13. Counter On Keeping Track
  14. Counter of Quantitative Units/Place Value
  15. Counter to 200+
  16. Number Conserver
  17. 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:

Counting Collections K-3

Introducing Counting Collections in Kindergarten

Extending Counting Collections Grades 1-4

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

Counting: Reggio-Inspired Mathematics Pedagogical Content Knowledge four-pager