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)

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