Circle geometry, proofs and dynamic geometry explorations with Desmos/ Geogebra

 Here are resources for four guided lessons using free, dynamic geometry software online -- either Desmos or Geogebra. These circle geometry theorems are currently included in the BC Geometry 12 curriculum, though they were in the Math 11 precalculus curriculum for many years. (If your school is not offering Geometry 12, these are some very interesting geometric ideas that could be introduced as an enrichment topic in the senior year classes!)

Overview and lesson one (retyped)
Lesson 1 again (scanned version)
Lesson 2 (scanned)
Lesson 3 (scanned)
Lesson 4 (scanned)
Older textbook deductive proofs (scanned, somewhat blurry)

In the olden days, students used to doing geometry with pencil, compass, straightedge and paper, but from the 1990s on,  many online dynamic (as opposed to static) geometry tools became available. Geometer's Sketchpad, Cabri, Cinderella and many more are listed on a Wikipedia page here.

By far the most popular of these is now Geogebra, as it is free, open-source, and contains many lesson ideas developed by and for teachers. Take some time to look at Geogebra and experiment a bit with it.

Desmos is also taking on many similar functions to Geogebra, and is also free and available to students and teachers. You will likely use Desmos and/or Geogebra with your classes


What are your thoughts on these guided geometry experiences? How do you feel they relate to more formal deductive proofs? Do you feel that these 'prove' the theorems?

In thinking this through, please refer to John Mason's three levels of proofs --
1) Prove it to yourself,
2) Prove it to a friend or ally,
3) Prove it to a sceptic.

Here is an excellent short article by John Mason on this topic, for your reference!

For Taha and Jacob's curricular microteaching Mon Dec 2

Hi all! Here are the criteria for your peer/ self evaluations for Jacob and Taha's curricular microteaching lesson for tomorrow. (Thanks for offering clear and constructive comments as well about what worked and what might be improved -- and thanks to Mark for writing these so beautifully on the whiteboard!)

 

On 'popsicle stick bombs', longsword dancing, and (stellated) polygons!

 Here is a link to slides on this fascinating topic that we'll explore today!

And here are the two short films we saw on the geometry of longsword locks:

Steel Phoenix 

George Hart's Simons Foundation film

And here are a few photos of our class making coffee stir stick longsword locks. (If you have stills or video of us doing the longsword dance, please do post them on the class blog too!)

Planning our upcoming EDCP 342 classes to the end of term

 

On Monday Nov 25, after Jacob and Taha's curricular microteaching, we will work with popsicle stick and longsword polyhedra!

On Monday Dec. 2, we'll explore classical labyrinths and their connections with magic squares

On Monday Dec. 9, we'll engage with a math drama.

Looking forward to our workshops, guest speaker and math party in the coming weeks!

Homework reading on textbooks and how they position their readers: please post blog response

   

 

Here is an interesting critical-thinking take on the ways that math textbooks may 'speak to' their 
readers (assuming that people actually do read and listen to math textbooks...)


Wagner & Herbel-Eisenmann on math textbooks and the ways they may position their readers

Please read this thought-provoking article and blog on:

•How you respond to the examples given here -- as a teacher and as a former student
•What are your thoughts about the reasons for using/ not using textbooks, and the changing role of math textbooks in schools?

Textbooks, their history and their future

 Today we'll look at a number of textbooks, new and old, and assess them in the terms listed below. 

As you begin working as a member of a math department at a secondary school, you will be called upon to help assess textbooks that the department and school is considering purchasing. You will also be asked to make professional judgements on various alternatives to textbooks. How might you assess textbooks and their alternatives? That is what we will address in class today.

Some things to start noticing in terms of textual analysis of math textbooks, old and new. (Each observation should be considered in terms of its potential effect on mathematics teaching and learning, and on the ways students might begin to think about mathematics.) Notice:

• The qualities of the physical object. Is it heavy or light, durable or fragile? How does the paper feel and the ink smell? Does it come with a teacher guide? student workbook? CD (seems a bit old fashioned now)? web links? Will it fit in a student's backpack?

• How much does it cost? What would the investment be for a school to take on a new math textbook?

• What is the writing like? Who does it address, and what does that tell you about its view of society and mathematics learning? Does it include definitions, explanations, examples, counterexamples, word problems, instructions, stories, extensions?

• Who are the authors? What is their expertise? Who put the project together, and why? Try reading the introductory material (which hardly anyone ever reads...)

• Is it easy to navigate? Table of contents, index, glossary, answer key, appendices, diagrams, tables, different kinds of pages colour coded, ...?

• What are the graphics like? Photos, diagrams, graphs, boxed information, use of colour or not,...

• How is it designed? Use of white space, sidebars, font size and style, page layout,...

• Any examples of alternatives to paper textbooks, from publishers, or teachers, or organizations, or ...


Interestingly, even a math textbook is very much a reflection of the time, place and society it comes from! No matter that the mathematical content is slow to change -- the social and educational contexts (and printing technologies) change much quicker.
Are there any classes in high schools that don't use textbooks?

Could you imagine teaching a math class without textbooks? What is the future of the lucrative business of textbook publishing? Do you think we will soon be creating teacher-written, collaborative shared materials online that will replace current (retro) textbooks -- or not??

Here is a blog from a workshop on textbooks at CMESG (the Canadian Math Education Study Group). It has fascinating links to many of the earliest math textbooks in English (from the 16th and 17th centuries) and other varied textbooks. We will use several of these today: 

1. Sample pages from Robert Recorde's The Grounde of Artes (1632), one of the first mathematics 'textbooks' in English..and Robert Recorde, The Whetstone of Witte (full text)
   
   
2. Social Arithmetic selections part 1 (US, 1926)
Social Arithmetic selections part 2 (US, 1926)


3. Modern Algebra and Trigonometry part 2 (Dolciani et al, US,1963)
   
   
4.  Mathematics: A Human Endeavor (Harold R. Jacobs, US, 1970) part 1 and Part 2
   
   
5.   Mathquest 8 (Addison-Wesley publishers, Canada, 1988)
   
   
6.  Precalculus 11 (McGraw Hill Ryerson publishers, Canada, 2011)
   
   
7.  Math that matters (Stocker, 2006, Toronto)
   
   
8. Khan Academy Math as textbook?
   
   
9.  Excerpts from Stewart's Calculus

Blog reflection on flow in the secondary math class for Monday Nov 18, 9 AM

 

Hi all! As a follow-up to our exploration of 'flow' , please write a short blog post reflecting on the film we saw in class on Monday: Mihalyi Czikszentmihalyi's TED talk on 'flow' and happiness.

In your post, please reflect on what you take away from this talk and the idea of 'flow':

Have you experienced a state of flow through certain experiences? What prompts it? Is it sometimes connected with mathematical experiences?

Is it possible to achieve a state of flow in secondary math classes? Can we, as teachers, help create the conditions for a flow state for our students as they learn math? If so, how -- and if not, why not?

Looking forward to reading your blog posts, and to following up with more on flow states and ideas about The Thinking Classroom!