Archive for December, 2018

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:

NCTM_quickimages_tcm2016-12-320a

 

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.

 

Desmos in an online graphing calculator that allows for students to predict, Screen Shot 2018-12-11 at 2.52.41 PM

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

 

November thinking together: develop mental math strategies

Posted on: December 2nd, 2018 by jnovakowski No Comments

Develop, demonstrate and apply mental math strategies Screen Shot 2018-12-01 at 8.40.08 PM

is the focus of one of the Reasoning and Analyzing curricular competencies from grades K-9. For K-5 the focus is on developing mental math strategies as a means to developing fluent and flexible thinking with numbers. In grades 6-9 the focus is on demonstrating and applying these whole number strategies to new number contexts.

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As is the case with all the mathematics curricular competencies, the learning standard is the same for K-5 and then continues for grades 6-9. Grades K-5 focus on the development of mental math strategies while grades 6-9 focus on the application of mental math strategies. The grade level-ness is enacted when the curricular content and curricular competency are connected.

So for example, in grades 6-9, the competency using mental math strategies with whole numbers is applied to decimals in grade 6, integers in grade 7, fractions in grade 8 and rational numbers in grade 9 and it is this intersection of curricular content and competency that is assessed.

The suggested mental math strategies are listed in the elaborations for the curricular content for each grade. Elaborations are suggestions and support for instructional decision-making and are not meant to be used as a requirement or an assessment checklist. Mental math strategies are strategies that are intended for students to be able to do mentally. Computational fluency involves  flexible strategy use – both mentally and recorded with paper/pencil, whiteboards, etc. The strategies are transferable to working with larger numbers or to different types of numbers such as fractions and integers. Students may be introduced to the strategies by their peers during a number talk or during an instructional task. When strategies are introduced, specific mathematical language and visual scaffolds such as ten frames or number lines are often used. Some strategies may need to be practiced in different ways before students are able to use them mentally in flexible ways, and choose strategies that make sense for the numbers they are working with. As an example, the following are the computational strategies suggested in the content elaborations for grade 2:

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In our 2007  curriculum (WNCP) there was a clear definition of what Mental Math is (in the 2007 iteration of our curriculum, it was named as a mathematical process). “Mental mathematics is a combination of cognitive strategies that enhance flexible thinking and number sense. Mental mathematics enables students to determine answers without paper and pencil. It improves computational fluency by developing efficiency, accuracy, and flexibility.”

Goals of developing fluency with mental mathematics include:

  • developing confidence in doing mathematics
  • being liberated from calculator dependence
  • becoming more flexible thinkers
  • be more able to use multiple approaches when problem solving

(Rubenstein, 2001)

Strategies develop over time and complement each other. Examples of mental math strategies drawn from the elaborations in our BC curriculum framework include:

  • counting on
  • making ten
  • decomposing (to make tens/hundreds, by place value)
  • double and related doubles (doubles plus one, etc)
  • bridging over tens (transferable to hundreds, thousands etc)
  • compensating
  • adding to find the difference
  • commutative, associative and distributive principles
  • annexing zeroes
  • halving and doubling

Different visual tools can be used to support students’ development of mental math strategies such as ten frames, hundred grids (numbered and blank) and open number lines.

Teachers in Richmond have been developing their own understanding of mental math strategies Screen Shot 2018-12-01 at 9.31.56 PMthrough Number Talks over the last several years and I believe it is the most used instructional routine in our K-7 classrooms.

The following is a record of some of the number talk experience in Richmond classrooms.

SD38 Number Talks panel 2016

 

 

 

Carrie Bourne and I have started creating a math video series for Richmond educators that our available on our district portal. The series is called Doing the Math Together and the videos are intended support teachers with their understanding of mental math strategies and how they might record students’ strategies during number talks. For example, there are videos in both English and French on the importance of the “making ten” strategy.

They are located in the district portal on the Mathematics page, under the blue tile titled Doing the Math Together Video Series.

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Professional resources to support the development of mental math strategies through the development of number talks include:

Screen Shot 2018-12-01 at 9.53.51 PMScreen Shot 2018-12-01 at 9.54.48 PM                      Screen Shot 2018-12-01 at 9.54.12 PMScreen Shot 2018-12-01 at 9.54.30 PM

 

 

 

 

 

So what does it mean to be proficient with mental mathematics?

As we begin to work with the new proficiency scale across BC, we need to consider what it means to be proficient with developing, demonstrating or applying mental mathematics 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 2 student at the end of the year would be considered proficient in adding two-digit numbers mentally if they were able to be efficient, accurate and flexible when using two or more different mental math strategies such as decomposing or compensating.

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Some questions to consider as you plan for learning opportunities to develop the competency of using mental math strategies:

What strategies or knowledge do students already have about mental mathematics? What opportunities do students have to show and share what they know?

How can we encourage students to be metacognitive when using mental mathematics rather than just applying procedures? How can we develop the concept of efficiency and support students in choosing strategies that are a good fit for the numbers provided?

How can we help students understand the purpose and usefulness of developing mental math strategies?

What opportunities are we creating for students to use mental mathematics across other disciplines such as science or ADST?

~Janice

 

References

Do the Math in Your Head! (2005) by Cathy Seeley

Mental Mathematics beyond the Middle School (2001) by Rheta N. Rubenstein

Five Keys for Teaching Mental Math (2015) by James R. Olsen