After exchanging money today on Florida street, we popped in to get an ice cream from the chocolate shop at the mall.  We couldn't pass up the chocolate Easter eggs on sale (post-holiday!)

"Easy" Problem
Eggs regularly cost $20 ARS each.  Mommy bought six.  But they're on sale now, three for the price of two.  So, how much did she pay?

A Little Harder
Mommy gave the lady $100 ARS, but was told she needed four more pesos.  Mommy's hearing is going -- turns out the original price wasn't $20 per egg... how much was it?

If You REALLY Want a Challenge
How much does one egg cost (both regular and discounted price) in USD if the blue rate today is 10.3?  9.6?  12.2?  Hehe.

Inspired by a TED talk I recently watched,  I threw together an impromptu math problem on the fly today while out with the boys.

We love Melona iced milk pops, available here in Barrio Chino.  Today on the way home from church, we stopped to pick up three to eat on our walk between train and subway (8 blocks).

I told the boys I had gotten 50 pesos change... so how much does each popsicle cost?

"How much did you give him?" piped up Simon, indicating his understanding of needing to obtain relevant information to the problem.  I responded, "$104".

From there, they were able to figure out that three pops cost $54, and so each one must be more than $15 but less than $20.  Eventually, the discerned that the cost per pop was $18.

Not to negate the beauty and importance of exposing children to pure math, but if we're going to teach at least part of our math program as "authentic" problem solving, then the questions ought to be, well, authentic, rather than contrived.  

Right?

Mmmmm.... Melona! Yum!

The internet is a beautiful thing!

I was sure I had packed cuisinaire rods for this year's home schooling adventures; alas, when we needed them for this morning's lesson on the length model of fractions, there were none to be found!

Undeterred, I had the boys search the NLVM site for some online rods.  Nothing!

Desperate to stick to the lesson, we searched online for another interactive cuisinaire rod, and stumbled across nrich.maths.org -- and they had virtual cuisinare rods!  FREE!

I'll be honest, I have no idea when this is.  And besides, should it be the 100th day of home schooling, or of school in general this year?  In any case, the boys were asking about doing something special, so here are a few things I plan to throw into the mix next week...

First, we'll spend 100 minutes on independent literacy in the morning (usually, the boys do reading, writing and word work for 60 mins... can they do it for 100?  We'll see!)  Some options for writing and word work will be:

• make a list of 100 nouns
  
• 100 adjectives
  
• 100 places you'd like to visit
  
• "100 years from now..."
  
• "If I live to be 100 years old..."
  
• What happened on this day in history 100 years ago?
  
• brainstorm 100 acts of kindness... make a list
  
• find out how to say "100" in 10 different languages
  

Then, we'll do 100 jumping jacks for DPA, following which, we'll read a little poem together... or sing a 100s day song if I can find one...

For recess snack, we'll eat 100 raisins and chocolate chips.  Or we'll drink 100 ml of milk or water!

After recess, we'll watch this video:

And play a few games: 

• Guess My Number
  
• Find the Bones

Finally, we'll count out 100 snapcubes and have them build a structure with them.

And for HOMEWORK, they'll play "100 Snowballs"!

As would be expected, the recently-released results of the PISA math test have resulted in a flurry of media responses.

Everyone's an expert, it seems, not so much on the math necessarily, but on the best remedy.  (Ontario students didn't do so well.)

This article from the Globe and Mail discusses the problematic "new math", often painted as less formal in structure...

"The OECD report noted that the top performers had more exposure to formal mathematics, algebra and geometry than the word problems used in “discovery learning,” which advocates that children can learn math more effectively when they are given opportunities to investigate ideas through problem-solving and open-ended investigations."

As someone keenly interested math education, especially at the elementary level, I am troubled by comments like the one above because it implies that a problem-based approach to teaching math is inherently inneffective in providing a rigourous mathematical education.

I would argue that the problem isn't so much the pedagogical approach itself, but rather the way in which it is manifested in a classroom.  Many teachers know the words "constructivist" and "problem based", but -- in addition to lacking confidence with the math itself -- they may lack a firm understanding of the theory behind these terms and practical experience in implementing these approaches effectively.

This doesn't necessarily mean that problem based learning is a bad thing in and of itself, but rather perhaps suggests  that teachers need to develop strengths and confidence in this area in order to teach math more effectively, incorporating both the mathematical skills and the broader learning skills into a rich, holistic program.

Interestingly, I attended an online professional learning session this evening with Cynthia Nicholson, on "Infusing Math Learning with Critical Thinking".

After exploring a quote from John Van de Walle, on how we teachers cannot "think for our students", one teacher commented that "if we can provide opportunities for our students to discover math truths, that will be more valuable and far reaching for them" than the skill and drill and kill model.

Of course, in order to provide said opportunities, teachers must be both masterful facilitators and knowledgeable math teachers.  

The truth is that teaching is complex! And the average teacher -- even a very committed "average" teacher who takes the odd PD workshop -- has a wee bit (or more) of math phobia.  This was confirmed in tonight's session, where several teachers "blanked out" and "shut down" (their words, not mine) when confronted with fairly typical math word problems incolving proportional reasoning and unit rate at about the Grade 4 level.

More rigorous mathematical instruction for pre-service teacher candidates is perhaps a critical componant.  (I shudder to think how little I understood about even basic mathematical concepts when I graduated from Teacher's College nearly two decades ago!)  
Still, being "good at math" is not enough.

If we don't equip our students with the tools for critical thinking and encourage them to develop a  growth mindset (Carol Dweck), then those who struggle with math may give up rather than persevere with mathematical challenges.  Taking the time to teach students "stick with-it-ness" as well as specific mathematical tools (how to draw an effective picture to model mathematical thinking, for example, or, how to ascertain when something "makes sense") will help those students who are not rote learners to stay "checked in" in the math classroom.

(Of course a reasonable depth of mathematical understanding on the part of the teacher is vital in order to be able to do this effectively.)

If we want students to fluently rattle off facts they have memorized but with minimal understanding or ability to apply, then we should "return to the basics" as some in the article mentioned above insist, and focus on traditional skill-based, rote-only teaching.  But if we want our students to "learn to think and think to learn", then we had better inform ourselves about the math, AND ALSO  nurture thoughtful communities, frame critical challenges, teach intellectual tools, and assess thinking and performance.

The recipe for solving the disappointing PISA results problem includes two main ingredients:  A more solid foundational understanding of mathematics for teachers, and a deep comprehension of how to effectively teach critical thinking in the classroom.   

Both are needed if we are to effectively educate students for success in the 21 century.

Sometimes in our quest to find the latest thing, we can overlook some old standards that do a great job.  Coming to Argentina, I brought a few concrete Math manipulatives with me, but I just didn't have the space to bring everything!  So I've relied heavily on apps.

But then this morning, as we embarked on our 3D geometry unit, I remembered the good old National Library of Virtual Manipulatives, a FREE online resource containing several dozen manipulatives and related learning activities, organized by grade level and strand/topic.

We spent a considerable amount of time on the site; the boys particularly enjoyed the platonic solids, and we definitely got in over our heads (i.e. well beyond grade level), but we were having so much fun, we kept at it!

Look closely at these two photos...  They were taken three days apart, at the same "Jumbo" store, at the same pumpkin display.  Notice anything odd?

Try not to get distracted by trivial things, like the giant cart of soy milk  or the hot babe with her twin boys in the first photo, and focus more on WTF? type stuff...

Yes, that's right kids, the price has changed dramatically!!!

On Saturday, carving pumpkins were $7.99 a piece, and now, a mere three days later, you will find that they are selling for the not-so-bargain price of $49.99 per kg!!!!  Wot?!

The gender binary is alive and well in the world of educational gaming, it would seem.

This morning, we started playing an online math game that the boys' class back home is playing.  It's a virtual world, cleverly cbuilt around the math curriculum expectations for a given grade level, where players work together or individually to complete various challenges by solving math related problems. We're actually just getting started, so although I was excited about the prospect of linking gaming (which the boys love) with the ability to keep connected with their peers from their classroom back home, we're really not far enough into the game yet for me to make any professional judgements, when it comes to academic value.

But we're into it far enough for me to be pissed of already...

No sooner did Alex and Simon register, using the login so generously provided by their teacher, then they were faced with having to categorize themselves by choosing to click one of two buttons in response to this question:  "Are you a boy or a girl?"

SERIOUSLY?!!!

I could almost feel the steam pouring out of my ears as I thought of every gender biased toy store, movie or picturebook we've worked so hard to avoid or at the very least, think through critically together, in the boys' last nine years.

WHO THE HELL CARES IF THE PLAYERS ARE "BOYS" OR "GIRLS"???!!!

Let me guess -- those who selected "girl" will be offered an array of pink, flashy, "girly" swag to buy with their acquired points in the game, while the "boys" will be offered more "manly" options?  Are you kidding me???!!! How can a so-called "educational" game force today's students into the gender binary within minutes of beginning to play?!

I am thinking of three particular students at my school last year who would be befuddled when they came to this screen.  All three are transgendered "boys", but while two of them are VERY comfortable with their feminine side (and would probably click on "girl" with little hesitation, given the opportunity), the third has spent his whole life fighting himself and his parents, getting relief only on Halloween, when -- once a year -- he freely dresses as a beautiful, bejewelled, flambouyant woman, as he has done every year.

I know Alex and Simon are fairly firmly rooted where they're "supposed" to be on the gender spectrum, so from a personal perspective, my concerns centred more around the perpetuation of a myth rather than their own personal gender struggle, which I'm pretty sure is non-existent.

But as a public educator, I am perplexed and deeply troubled that in this era of social justice and so-called equity, we are still confronted with widely supported resources for our classroom that so clearly defy everything we are striving to teach our students in the 21 Century.

A few deep breaths, and then I will calm down and find the "feedback" button in the game, and offer the developers a piece of my mind!!!

I blogged recently about downloading Ventura System's Coordinate Geometry app.  Since we have two copies, we decided to play a cooperative game suggested by the developer...




Procedure:

1. Two players sit facing each other but can see the other player's iPad.  
2. Player 1 draws a figure on his/her iPad.   For example a right triangle at (5,5), (0,0), and (5,0) using black.
3. Player 1 draws the reflection of the figure across the y-axis using red.
4. Player 1 gives verbal directions to player 2 to try to copy the figure and reflection onto his/her iPad.
5. Using math vocabulary, P1 describes the figure and transformation for P2.
6. When finished, the players compare drawings on each iPad.