Vanessa's Calculus Reform Presentation

http://prezi.com/mqtlrkzqeseh/?utm_campaign=share&utm_medium=copy <-- View it here! :)

Murugan's Presentation on Technology

Please click here to download.

Keri's Presentation on Manipulatives

Click here to see my findings on manipulatives in math education

David H's Power Point on Technology

https://drive.google.com/file/d/0B1hvQaGNvqyEY3dtMnpzMjdheFE/view?usp=sharing

I hope the link works!

Dave

Terry Jones' Monte Python-esque film on the history of mathematics: The Story of One

Here is a link to Terry Jones' film!  You can find worksheets, etc. online to go with the film, if you like that sort of thing.

Some resources and starting points for the using the history of mathematics in teaching

Here are some links to a few of the many interesting resources for using math history in teaching mathematics:

1) From Berlinghoff and Gouvea, Math through the ages: A gentle history for teachers and others: sections on the history of Renaissance Italian intrigue & the solution of cubics; and a section on the Pythagorean Theorem.

2) From Olivastro, Ancient puzzles: A section on ancient Egyptian arithmetic.

3) From Classic brain puzzlers: a few examples of early puzzles from different sources.

4) From Joseph, The crest of the peacock: Non-European roots of mathematics: more on ancient Egyptian arithmetic.

5) From Berggren, Episodes in the mathematics of medieval Islam:  A section on dust board arithmetic.

6) From Byrne, The elements of Euclid: Excerpt from Oliver Byrne's amazing colour-coded 1847 edition of Euclid (though unfortunately scanned in B&W here!) Here's a link to an online pdf in full colour.


John Mason & Louis Charbonneau led a 1997 CMESG Working Group that focused on history in mathematics learning -- see p. 35 onward in this link.

Irene Percival from SFU wrote her PhD thesis on history in the mathematics classroom, and has produced lots of materials for use in the K-12 math classroom. Here are some links to some of them:

A visual/ cut-out proof of the Pythagorean Theorem
Babylonian arithmetic

2005: Linking history to school math curriculum

Here is a site that illustrates Hindu-Arabic dust board arithmetic, as described in the Berggren excerpt above.

Our last set of readings for the course -- on the history of math education, for our April 1 class

Hi everyone. Here's our last set of class readings for you to blog about -- these ones about integrating the history of mathematics in math classes:

1) Jankvist: Whys and hows of using history in mathematics education

2) Swetz: Mathematical pedagogy from a historical context

3) Tzanakis & Arcavi: Integrating the history of mathematics in the classroom (from ICMI study)

Our readings for next week's class (Wed. March 25): Technology & embodiment

Hi everyone, and hope you are enjoying our excellent virtual class discussion on the topic of "attention". I am impressed by the high quality and wide range of the discussion!

Here are our three readings for next week's class, on Technology & Embodiment -- two contemporary themes in mathematics education that often come together.

1) Healy & Knigos, Charting microworlds

2) Smith, King & Hoyte, Learning angles through movement with Kinect

3) Ferrara, How multimodality works in mathematical activity




Hope you enjoy these, and looking forward to more good discussion via your blogs and in our class next week!

Our virtual class: On Attention and mathematical abstraction

Two articles by John Mason on attention:

1) Mason 1982

2) Mason 1989

Please read these two thought-provoking articles on mathematics, attention and abstraction.

Then, over the course of our virtual class (from today to our return on March 24), please post at least four substantive posts in the "comments" section of this posting. Make sure to read and consider the whole thread of the conversation to date and respond to what is being said, as well as adding your own ideas. Your four posts should be separated by at least a day each.

Enjoy this interesting conversation!

Lucie De Blois sent these two articles that you might be interested in!

Hi everyone. Lucie De Blois, our guest speaker from the Université Laval, sent these two articles as a follow-up to her talk about the 'false reality' of word problems, Guy Brousseau's important concept of the didactic contract, and ways that geometry can be a way to help students develop arithmetic problem solving skills.

The first article is in English; the second article is in French, but is just 4-5 pages long, so you might want to try out your reading skills in French with this.

Many thanks to Lucie for a fascinating and stimulating talk!

Walter Whiteley sent this link and powerpoint that you might be interested in

Hi all. Walter Whiteley has sent us an interesting link on using 3D printing with Grade 4 kids to design a boat that floats and bears weight, using geometry in the design. (I could certainly imagine doing this with physical materials apart from 3D printing as well...)

Walter also sent a brief 2-slide powerpoint about different kinds of geometric transformations (affine, projective, etc.) that preserve different kinds of invariants.

Many thanks for a wonderful guest talk,  Walter!

Readings/ textual analysis of issues 1, 50 and 100 of FLM for our class with David Pimm March 4

OH NO! I just realized that in (2) and (3) below, the scanner only scanned every second page!!


I will add a link to each of the three issues from the FLM website, so that you can see the missing pages of each article.
Here are the "readings" for next week's class and your blog posts:

1) FLM 1-1 (the first issue of the journal)
(pretty much complete here)

2) FLM 17-2 (counted as the 50th issue of the journal, as I didn't have 16-3!)
plus this link, to see the missing pages!

3) FLM 33-2 (the 100th issue of the journal)
(Sorry, the link on the FLM site does not include the actual articles! The site is not very consistent, unfortunately -- you will have to go with looking at every second page I'm afraid).

[After you have read (1), (2) or (3), everyone should also take a look at:

Bingjie Wang's Masters thesis, pp. 52-64 only
Bingjie presents data here on three Western math education journals (FLM, ESM and JRME) as part of her textual analysis.]

The idea here is to "read" the whole issue as a text, rather than reading every article in depth.

That means, for example:
•reading the table of contents, and thinking about the titles and topics of articles.

What levels of schooling or age groups do they have as their theme (if any at all!)

What kinds of issues are addressed? How are these distinctive?

•looking at the articles themselves:

How long are the articles?

Are they usually illustrated (and if so, how?)

Are there a lot of references cited?

Are there subheadings on the articles? If so, are they the subheadings that you expect, or not?

What language is the article in?

•looking at the issue as a whole:

What is on the front and back cover, and why?

What did you learn from the author identifications?

What about the material on the inside of the front and back covers?

Is there any material between the articles? If so, what is it? What kind of tone might it set?

There's lots more you might look at too. The idea is to get a holistic sense of the journal as an entity, with a history, a community of writers and readers, etc. You might try looking up the name of the journal to see if there's anything interesting written about it elsewhere too.


***Additionally, David Pimm has just sent a number of short (1-2 pg.) pieces from FLM 34(1), which turns out to be the actual 100th issue! 
Sorry that there has been some confusion about this week's readings...
If you are able to take a look at some or all of these very short pieces, which all talk about the nature and history of the journal, it would be some additional good background for our class next Wednesday.

a) Bartonlini-Bussi short piece FLM 34(1)
b) Barwell short editorial FLM 34(1)
c) D'Ambrosio short piece FLM 34(1)
d) Lee short piece FLM 34(1)
e) Pimm & Sinclair short piece FLM 34(1)
f) Sfard short piece FLM 34(1)
g) Sriraman short piece FLM 34(1)

Readings for our Wed. Feb. 25 class on Geometry with guest speaker (online), Walter Whiteley

Next Wednesday Feb. 25, geometer and mathematics educator Dr. Walter Whiteley of York University in Toronto will join us as a guest speaker via Skype from 6:00-7:00 PM!

Here are the three articles Walter would like us to read in preparation for our discussion and his visit:

1) Whiteley, The decline and rise of geometry

2) Whiteley, Why study geometry?

3) Tahta, Is there a geometric imperative?

Walter has so far sent me one question he would definitely like you to address as part of your response:

"Do you feel you know enough geometry to bring it in when it is relevant, to use it as an additional representation for the mathematics you are teaching (and using)?"

He would also be very interested to hear your questions about geometry in relation to other areas of the math curriculum in your teaching, and your ideas and opinions about recent trends to reduce or remove geometry from many school and university math programs.

*Please do invite Walter to be an 'author' on your blog, so that he will be able to take a look at what you've written before he talks with us. His email address is: whiteley@mathstat.yorku.ca

Is a dynamic geometry experiment a proof?

The following four topics can be approached in a number of interesting ways, including the use of dynamic geometry software (Geogebra, Geometers' Sketchpad, Cabri...)

Geogebra/ Geometers' Sketchpad circle geometry lesson 1: chords of a circle

Lesson 2: Angles in a circle

Lesson 3: Tangents

Lesson 4: Cyclic quadrilaterals

A link to Geogebra (a free shareware, open source dynamic geogebra app):

http://www.geogebra.org/cms/

The question: is a dynamic geometry experiment a proof? Is it convincing? valid? complete and watertight?

What would you say about a more traditional, deductive proof of the same theorems -- for example:

Proof of inscribed angle theorem

Proof that angles in the same segment of a circle are equal

Two readings on proof for discussion in today's class (and for our online participants!)

In today's class, we will take time to read and discuss excerpts from two classic and influential works about the nature of proof in mathematics & math education.

In the first, from Thinking Mathematically, John Mason discusses justifying conjectures -- that is, proving them. His 'big idea' suggestion (convince yourself, convince a friend, convince an enemy -- and then learn to be your own enemy) gives a very interesting and often quoted plain-language idea of mathematical proof. Mason refers to puzzles from elsewhere in the book, and I have included some of these at the end of the scanned
excerpt so that you can try one or two of them.

****************

The second, from Proofs and Refutations, is part of Imre Lakatos' influential doctoral thesis in the history of mathematics. It is heavier reading than the first, but is worth the effort for the insights it gives. Lakatos reconstructs a bit of the history of mathematics as if it were all happening in a classroom, with a Teacher and students Alpha, Beta, Gamma, Delta, etc.

The topic is Euler's Polyhedral Theorem, a lovely little bit of theory that says that, if you count the vertices, edges and faces of any polyhedron, the sum of the number of vertices is two more than the sum of the number of edges and faces: V-E+F = 2. To make sense of this theorem, it is very helpful to play around with virtual and/or actual models of polyhedra -- see previous posting. You can look at the simple, regular polyhedra (cube, tetrahedron, octahedron, dodecahedron and icosahedron -- the Dungeons & Dragons dice shapes!) and count the vertices, faces and edges to see if they work.

Lakatos shows that this seemingly-simple geometric theorem actually encountered a lot of opposition through counterexamples historically. Several kinds of responses to counterexamples are shown in this excerpt, including "the method of surrender" and "monster-barring". The main point for us reading this excerpt from Lakatos is to see that, historically, the task of creating mathematical definitions and theorems, proving theorems, and offering counterexamples was anything but clear, simple and straightforward! The idea of monster-barring (by changing our mathematical definitions to exclude "monstrous" cases) is something I have used in my secondary school math classes.

*****************

Instructions for Keri, Vanessa, Shan and Alex (and optionally, Jubilee): Take a bit of time to read and make sense of these excerpts, then post four substantive comments each over the course of the next week in an online discussion amongst yourselves, considering the idea of proof in mathematics learning and whether it is significant, important and/or understandable for learners at a variety of levels. Is there something about proof as a way of thinking logically and critically that is important for all learners -- or not?

Polyhedra to play with

In connection with our discussion of Lakatos' Proofs and Refutations, here are some links for polyhedra models to explore:

Animated polyhedra to play with on screen (also includes polyhedra 'nets' that you can print, cut out and construct).

Wikimedia page of animations of polyhedra

Modular origami polyhedra instructions

Pop-up dodecahedron model: How to

I've made one of these from a cereal box and rubber bands, and it's delightful! Here is the model:





Here's how to do it:

Cut out two copies of the "pentagon flower shape" from a cereal box or other similar cardboard. Make each model about 10 cm across. Crease the join lines so that the five pentagons bend freely from the centre one.

Place the two models on top of one another as in the centre diagram. Lace an elastic band alternately over and under the "petals" as shown in the right-hand diagram, holding the model down flat.

Then let go and -- with any luck -- up pops a 3D dodecahedron! Do try this at home before you bring it to class. It is quite wonderful when the 3D figure suddenly appears, and making the model helps students remember the composition of this polyhedron.

Adapted from Ian Stewart (2009). Professor Stewart's Cabinet of Mathematical Curiosities. NY: Basic Books, p. 7. (A highly recommended resource book for your classroom as well.)

Paulus Gerdes' books at Lulu.com --many available as free pdf downloads!

Very sad to say, Paulus Gerdes passed away this winter at a relatively young age, and the world has lost a great ethnomathematician.

Gerdes' books are all available at this Lulu.com site, many of them as inexpensive or free pdf downloads. Here is an example of one of them:

Explorations in ethnomathematics and ethnoscience in Mozambique

One of the ways to work against neocolonialism and ethnocentrism is to work reflexively -- that is, to view one's own culture as the subject of anthropological or other study, inspired by a sense of the importance of all cultures.

Reading chapters 1, 2, 3 or 5 from this book by Gerdes with your group, can you think of things (or activities) in your own cultural world (and/or that of your students) that might lend themselves to mathematical study, exploration or exemplification? Would you consider using these in your teaching? Why or why not? How, when, with what focus?

Readings for our Feb. 18 class

Hi everyone. Next week is UBC's Reading Week, and we will be meeting off campus at Terra Breads Café in Olympic Village: 1605 Manitoba Street, on Songbird Square (with the huge bird sculptures), about 1 block west of Science World. The nearest Skytrain station is Main Street Science World.

Here are our readings for next week's class, on the topic of Proof:

1) Gila Hanna & Ed Barbeau on proofs as bearers of mathematical knowledge

2) Vicki Zack on Grade 5 students' counterarguments and proofs

3) John Borwein on experimental computerized mathematics, the pleasure of discovery and the role of proof

Readings for our February 11 class: Ethnomathematics

Hi all! Here are our readings on ethnomathematics:



1) Paulus Gerdes on "frozen mathematics" and geometry

2) Carraher, Carraher & Schliemann on street mathematics and school mathematics 

3) Marcia Ascher on Marshall Island wave maps as an example of ethnomathematics

'Something to think about': Mathematical problems from Mason, Burton & Stacey, Thinking Mathematically (2nd Edition)

It helps to be doing mathematical problem solving to get a sense of how it works for both learners and teachers.

As you work on these puzzles, note with your 'teacher bird' awareness how your 'learner bird' is engaging with the problem solving!

1) Thirty-one:
Two players alternately name a number from 1, 2, 3, 4 or 5. The first player to bring the combined total of all the numbers announced to 31 wins. what is the best number to announce if you go first?

- Try playing it!
-What do you need to record?
-What totals enable you to win in one move? Generalize!
-Can you find a strategy for playing that will guarantee you to win?

extend:
•What if 31 is changed to some other number?
•What if the permitted numbers are 1, 2, 3, 4, 5 and 6?
•What if there are 3 players?
•What if the permitted numbers are 1, 3, 5 or 2, 3, 7?

2) Sticky Angles:
Given a supply of sticks, all the same length, and a supply of angles all the same size, can you join the sticks together end to end at the given angle to make a closed ring?

-Have you tried it physically?
-Introduce a way of making a supply of angles!
-Have you stayed in the plane?
-Have you tried folding a strip of paper appropriately to mimic many sticks joined at the correct angle?
-Will your method always work, or is your angle special in some way?

extend:
•What is the shortest such sequence when it is possible?
•What length sequences are possible?
•Does it help to have more than one angle available, particularly when you are confined to the plane?

3) Sequence:
Write down a sequence of 0s and 1s. Underneath each consecutive pair write a 0 if they are the same and a 1 if not. Repeat this process until you are left with a single digit. Can you predict what the final digit will be?

-Specialize systematically.
-Allow your system to alter as you begin to see what is going on.
-Try to be systematic about patterns and not about lengths of sequences.
-Try working backwards from the final digit.
-Find a convincing argument to support your conjecture.

extend:
•Write down a sequence of 0s and 1s in a circle and proceed as before.
•Set your result in a more general context by using 0, 1 and 2 with some appropriate rule.

4) Milk crate
A certain square milk crate can hold 36 bottles of milk. Can you arrange 14 bottles in the crate so that each row and column has an even number of bottles?

-Depict the crate. Find a way to be able to manipulate bottle substitutes.
-Specialize to crates of other sizes
-How many bottles might there be in each row and column?
-Specializing to larger crates might help.
-Eventually you will deal with the 36-bottle crate, but what about square crates in general?
-Can you build up new arrangements from old ones?

extend:
•Try to place other numbers of bottles.
•Try rectangular crates.
•What is the largest/ smallest number of bottles that can be suitably arranged in a given crate?
•How many ways are there to place the bottles?

Some real-life workplace mathematical problem solving from the 90s (from Colours of Infinity)

In this older but nonetheless fascinating film on fractals, Colours of Infinity (featuring Benoit Mandelbrot, Arthur C. Clarke and Ian Stewart), Michael Barnsley talks about how he made a breakthrough in fractal image compression for video and computer images.

35:30 - 42:00.

YouTube video: Watch kids solve a word problem -- "How old is the shepherd?"

This is a replication of the famous "what is the name of the captain?" word problem -- a problem with misleading or insufficient information, and which many kids attempt to solve nonetheless.

"How old is the shepherd?" video

What do you think of this? 

Readings for our February 4 class: Problem-solving, problem-posing and word problems!

Here are our three readings for next week:

1) Alan Schoenfeld (1996, FLM). In Fostering Communities of Inquiry, Must It Matter That the Teacher Knows "The Answer"?

2) Stephen Brown & Marion Walter on problem posing

3) Susan Gerofsky (1996, FLM): A linguistic and narrative view of word problems

SNACKS!!

1. Susan
2. Vanessa and David G.
3. Philipa and Keri
4. Jubilee and Alex
5. Conrad and Murugan
6. Dave H. and...
7. NO FOOD
8. Sophie
9. Alain
10. NO FOOD HERE EITHER
11. Kevin and Shan
12. Total Potluck
13. Total Potluck

Bingjie Wang's MA thesis (2012): Mathematics education journals

Here is a link to Bingjie's very interesting thesis!

It is also available here, at Circle UBC: <https://circle.ubc.ca/bitstream/handle/2429/38622/ubc_2012_spring_wang_bingjie.pdf?sequence=1>

Readings for our Jan. 21 class with guest speaker Beth Herbel-Eisenmann, Michigan State University

Focus for your blog post:
•What questions would/ will you ask the author about the article?
• Identify two or three "stops" in the article, and why they make you stop.

1) "Muddying the clear waters": Teachers' take-up of the linguistic idea of revoicing

2) Strong is the silence: Challenging interlocking systems of privilege and oppression in mathematics teacher education

3) From intended curriculum to written curriculum: 
Examining the "voice" of a mathematics textbook

Readings for our Jan. 28 class: The Psychology of Mathematics Education

Here are our three readings for next week's class, on a view of mathematics education in relationship to psychology:

1) Fischbein (1999). Psychology and mathematics education.

2) Sfard (1991). On the dual nature of mathematical conceptions.

3) Tahta (1993) Review and commentary on Victoire Sur Les Maths

*Note the change to (3) -- I decided to wait till a little later for us to look at Pimm's work.

Wed. Jan. 7 class readings for next class: Is mathematics education a 'field'?

1) Higginson: On the foundations of math education

2) Wheeler, Nesher, Bell, Gattegno: Research program for math education (1)



3) Kilpatrick: Reasonable ineffectiveness of research in math education

Writing responses to articles -- talking back to text

As you read academic books and articles, view videos, films and works of art, hear lectures, etc., it is helpful to engage with these 'texts' in a real or imagined conversation. Thinking and engaging critically doesn't necessarily mean being negative about what you're reading! It just means that you
ask questions, connect your reading to your own knowledge and experiences, think about what is not included or what could have been as well as what is included in the piece, etc.

Some good starting prompts for engaging in a critical dialogue with a text:

• I was surprised by...
• I was excited to read that ....
• I was annoyed to read ...
• I wholeheartedly agreed/ disagreed with the author on this point.... because....
• I wondered why ...
• I would illustrate this piece as follows ... (with a story, a diagram, a drawing, a comic strip, a graph, etc.)
• I could predict .... but not ... in this piece
• If I could talk to the author, I would ask them...
• This relates to my own life/ knowledge/ experiences in this way...
• Something that was not included in this piece was ...
• This piece reminded me of (something else I've seen/ heard/ experienced), in this way...
• If you extended this idea to the limit, it would result in ...
• This fits into the author's other work in this way...
• I see the following influences in this piece: ...
• The author interprets .... in the same (or a very different) way than I would: ...

"The Stop": Giving attention to things that stop you in your tracks as you read

David Appelbaum, The Stop (1995)


A "stop" is something that stops you (in your reading, in your observations, in your teaching/ learning). The metaphor is suddenly coming across a big rock in your path that stops you from smooth and continuous (and unconscious) walking. It must be attended to. It is a moment that allows for an opening to new paths, new ideas, new approaches if  you are able to give it the attention it demands.

A stop in your reading might be something you didn't expect -- something confusing, or difficult, or exceptionally beautiful, or something you strongly agree or disagree with, or an unknown word or phrase. A stop touches you deeply in some way (by irritating, or moving, or perplexing you, for example).

The stops allow for a change of heart, a change of mind, a political and/or intellectual engagement, a reconsideration of strategy, or even a reconsideration of world view.

From Lynn Fels, "Coming into presence: The unfolding of a moment" (Journal of Educational

Controversy):



"A stop is a calling to attention; a coming to the crossroads, in which a choice of action or direction must be taken, oft-times blindly, as experienced by Appelbaum’s (1995) blind man as he tap-taps the obstacles he encounters with his white cane—there are as yet unknown consequences of the subsequent action or decision as yet to be taken and embodied.


Between closing and beginning lives a gap, a caesura, a discontinuity.

The betweenness is a hinge that belongs to neither one nor the other.
It is neither poised nor unpoised, yet moves both ways . . .
It is the stop. (Applebaum, 1995, pp. 15-16)


A stop is a moment that tugs on our sleeve, a moment that arrests our habits of engagement, a moment within which horizons shift, and we experience our situation anew. A stop occurs when we come to see or experience things, events, or relationships from a different perspective or understanding; a stop is a moment that calls us to mindful awareness of Arendt’s appeal for renewal through action in the gap between past and future.


How we choose to respond and how that choice of action or non-action impacts on our lives and on the lives of those around us speaks to the risk, the opportunity, to the possibility of action. As media philosophers Taylor and Saarinen (1994) remind us, in spaces as familiar as the London tube, or as unmapped as cyberspace, we must “mind the gap” (p. 5). Applebaum’s moments of stop are moments that call our attention to the gap; moments that interrupt, that provoke new questioning, that evoke response, reflection, and hopefully, lead to meaningful and moral action."

Welcome to EDCP 550: Math education -- Origins and issues

I'm looking forward to meeting you, exploring the foundations and issues of our field together, and having lots of good discussions. I will post our draft course outline (and then the revised draft) here for future reference.

Here's to a great term and an interesting course together!