Equal Signs, Balance, and Timbits (#tmwyk)

Callum had to start taking the bus to Kindergarten.  When I picked him up the first day, I decided to get him some Timbits as a reward.

(official looking Timbit image from Tim Horton's website)

A few minutes of driving passed, and I looked in the rearview, and suggested he save some for his brother.  To which, Callum replied, "there's only one left."

Nobody intends their child to eat 9 Timbits in one sitting, but it seemed like a good time to talk some math.

The conversation went something like this:
-if 9 are in your stomach, how many are left for Alec?  (one)
-if you started with 10, and 1 is left for Alec, how many did you eat? (9)

I wanted to get him to add "9+1=10".  I was working on the concept of "one more than nine, one less than ten."

Here is where things get interesting.  I use "plus" or "add" as much as possible, because they are specific operational words he is going to need really soon.  But based on recent readings, and #tmwyk conversations, I restrained myself from summarizing the math we were discussing as "nine plus one equals ten".

We are doing some work on Marian Small's Uncomplicating Algebra book, and the Ontario "Paying Attention to Algebraic Reasoning" monograph.  One common stumbling block is seeing an equal sign as merely an indication to "get the answer", or, even worse, what you press when you're done pushing buttons on a calculator.  More properly (and foundational to algebraic reasoning), an equal sign should be a statement of equivalence, or balance.

So I tried this:  "9 Timbits in your belly, and 1 more for Alec, are the same as the 10 we started with".  I wondered if "the same as" could be a good substitute for "equals" in talking math with young children.  I think small things like one word choice can have large effects on how young children learn math.  The precision of our math language really matters when we are building the conceptual foundation.

Consider this example (this mistake happens all the time in upper elementary):

4 + 5 = __ + 3

Those with less of an understanding of balance will right exactly this:

4 + 5= 9 + 3= 12

In grades 6, and 7, we see this exact error all the time.

Here is a powerful Kindergarten example showing how you can build the concept of balance with young children:

RT“@PlayAndWonder: #engagemath talk about the conditions needed to balance the scale. @peelfdkmath pic.twitter.com/PfKYt64X7q” @MatthewOldridge
— Peel FDK Math (@peelfdkmath) October 9, 2014

Any thoughts on the uses and misuses of equal signs?

Educreations in the Math Classroom to Capture Mathematical Thinking

This summer we were chatting about our favourite digital tools on Twitter and sharing some tools we would like to explore this upcoming year.  For my #Peel21st blog hop post, I wanted to talk about finally breaking through with Educreations.

The screen capture apps are well-known, as is the Educreations vs. Explain Everything debate (both have their pros and cons). I also know you can produce beautiful finished products with screen cap apps, but that's not what I want to talk about. I do think screen cap apps are a new text form all to themselves, but that is a subject for another post.

Pattern Clip

In the sample above, not much more happens than a short discussion of a pattern they built, but you can see and hear their thinking about patterns start to develop. Educreations work can be messy, and "in the moment ", and that's what we want in the math classroom. I have seen students who own the app use it as a sort of notebook, capturing thinking on the spot, even taking a photo of a textbook page and drawing all over it.

Here's another video.

Pattern #2

At this point you know we haven't developed long lists of criteria or a learning goal yet. We are still playing.

The inquiry cycle I prefer is this:
-play with the math
-look at what we did (in this case we watched about 12 pattern Educreations)
-bring out common misconceptions
-do mini-lessons as needed (on finding a rule from t-table, or graphing a pattern, e.g.)
-look at more and different types of patterns (not just blocks)
-develop criteria for the assessments we will use

Third very unique pattern

What I am saying here is that I like screen cap apps more for "in the middle" assessments than for summatives. You want conversations and observations all in one app? You got it! Evidence of mathematical thinking? Sure.  I think screen cap apps could be one of the most powerful assessment tools we have.

Code.or by Erica Armstronghttp://msarmstrong.ca/2014/09/my-favourite-tech-tool-code-org/

Puppet Pals by Debbie Axiak http://debbieaxiak.blogspot.ca/ ​

Scratch Jr. by James Cash http://makelearn.org/2014/09/16/scratchjr-peel21st-blog-hop/

IFTTT by Jason Richea

beyondangrybirds.blogspot.ca

Photo Editor by Aviary by Tina Zita http://misszita.wordpress.com/2014/09/17/peel21st-blog-hop-my-tool-for-this-year-is-photo-editor-by-aviary/

Pocket by Greg Pearson http://leaderinadigitalworld.blogspot.ca/2014/09/cant-find-somethingyou-shouldve-put-it.html

Notability by Phil Young http://wp.me/p3RGo2-1Jb​

On a balanced math program, and knowing things, "all the way to the bottom"

You might know I got obssessed with Jordan Ellenberg's book, "How Not to Be Wrong."  It's not written for teachers, specifically, but it has lots of lessons and inspiration for us.

Here's one:
On Mathematical Knowing

I've spent lots of time thinking and wondering about why we spend so much time fighting about math. I jokingly talk about the #MathWars a lot, but in truth it's time to lay down our arms. One can stake out a position on the "back to basics" side, or the "discovery" side, but the truth is, and always will be, somewhere in the middle.

@PeelSchools, my employer, recognizes this-we have a balanced math instruction document now (as do several other boards).  There are many voices of moderation out there.  Practice needs to be balanced with problem-solving.  Number facts are the scaffolding upon which strong mathematical buildings are made, so they must be known.  Yes, students can "discover" a whole lot of math, but they usually need a lot of guidance to see what they found means.

One thing both "sides" agree on is- actually, forget that, there are no "sides". We ALL stand for student understanding, and being able to use math skills and concepts.  (One common debate is how and when we "know" a math  fact, versus how and when we "understand" a math fact- i'll leave that one to the researchers and cognitive scientists)

Perhaps the best strategy, in any given situation, to borrow from Mr. Ellenberg's quote, is the one that helps our students know the mathematical big idea or concept under study "all the way to the bottom."  If we are working with circles, that means the deep pleasure of finding pi using circles and string, AND developing, using, and applying the circumference formula.  Seeing how grade 3s can move beyond repeated adding as their schema of multiplication develops, AND help them start to know their facts with practice and games.  Watching junior students apply their sense of what proportionality means, and watch their toolbox full of strategies grow.

What we are not fighting about is the beauty and utility of mathematics-we all agree about that. Our methods and our means should help our students follow their thoughts about math deep down, all the way to the bottom.  Let them find the spark (with our guidance, of course)...      

The Next Best Thing

I was thinking today, when I attempted to tackle some yard work with a 2 year old and a 3 year old in tow, that we don't always get to do exactly what we want to do. I can have a bunch of energy, and set out to pull every weed in the lawn, cut the lawn, trim every bush, sweep the patio, etc., etc., and I will pretty much always fall short.

You know that sick overwhelming feeling that comes when you see the weeds choking out the garden, and the patio covered in pine needles?  It's a tough one to overcome.  "I can't do this." "It's too much." "May as well just not start."

Then, at the end of the day, we ask ourselves, "have I done enough?" @tina_zita covers this question in her blog post, "June Comes the Same Time Every Year."  Doubt tends to creep in, and negativity. 

Teachers know this: we sometimes come in to work with to-do lists as long as our arms, a set of "must be dones", that don't get done.  Parents know this-dinner to cook, laundry to do, kids to put to bed.  Baseball players should know this, but they often swing for the fences when a base hit would do. 

There can be kind of an analysis paralysis associated with trying to get things done.  Too many things to do, too little time, nothing gets done.  But those who say just getting started is the biggest thing are probably right.  

I'm no expert with lifehacks for all situations, but today, I asked myself, "if you can't do everything you want to do, what's the next best thing?"

Cutting the lawn and pulling all the weeds became cutting the lawn close so the weeds didn't show.

Whacking the weeds down in the garden as much as I could took the place of weeding the whole garden. 

The patio got swept, but there is still a bit of loose debris. 

So try this thought exercise the next time you feel overwhelmed.  Take your perfect world set of goals for each day.  We are humans, and we dream big, so some of them may be a bit out of reach. For each one, figure out what the next best thing is, that you can live with.  

Mark 2 sets of essays instead of three.  Finish 3 items on your to-do list instead of 5, but do them really well.  Adjust your expectations, and be careful of your perfectionism.  

The lawn looks fine.  There's still weeds, but it's done. (For now...)

CC image by katerha.

"Put them in a line and count them": Comments from the Real/Fake World of Math Class

I used this problem with grade 6s yesterday:

An interesting problem, or not? pic.twitter.com/7nB784XGR3

— Matthew Oldridge (@MatthewOldridge) June 20, 2014

I am not sure where this problem came from.  I do know that it stumped me for a while, although a decent number of students eventually got it, and the solution makes sense.

The first thing to note is that we are distinctly operating in the "fake world" here. No such club exists. There is no pressing need to do this particular math. There is no inquiry that can be done here. I defer, as always, to Dan Meyer on this topic.

 The problem is "word problemy", in the fact that it is asking for something simple- a single number of people, which it obscures through the design of its words.  That said, there are no dirty tricks here- it's a fraction problem, of the sort that uses a fraction of a missing whole, then a fraction of a slightly larger whole.

So why give this problem?  Here are some of our usual reasons:

1.  We are doing a fractions unit, and this problem can get us to think about fractions.
2.  To see what kind of thinking our students do on this problem.
3.  This problem is useful in its applications in the real world.
4. Because it's interesting, or beautiful.  These are our best reasons to do math, I think.

#1 didn't apply here.  #2 I am always interested in, although slightly less so when the pressure is off us in late June. As to #4, this problem did hold our interest for a while. It is a tricky one, and it engaged our desire to "puzzle it out".  There were some arguments interpreting the language of the problem, and trying and discarding some possible answers. Multiples of 4 and 7 were clearly involved, somehow.

As to #3, I will defer to my student's comments on the problem.

-you would just line the members of the club up in a line and count them. That's the solution. 

It's difficult to argue with that logic.

Reading, Writing, Coding, and "Absolutes" in Our Education System

I have taken a critical stance on the role of coding in schools.  Whenever I read an article with an underdeveloped thesis like "all kids must learn to code", I immediately ask "why"?  Often the articles are little more than a buzzy few paragraphs touting a great app. The "why" so often seems to be missing.

It doesn't help that these articles often come through aggregators like ASCD Smart Brief, Flipboard, or Zite with preposterous headlines like, "Is Coding as Important as Reading and Writing?"  Not only do the articles never make such grand claims, but also such absolutism tends to detract from the argument in the article.  (has anyone else noticed how headlines are little more than Buzzfeed style "clickbait" these days, even on reputable newspaper websites?)

I'm of the mindset that we never teach without the "why" in mind.  If we can't identify the critical skills, habits of mind, or plain old reasons for doing something, we probably shouldn't be doing it in the classroom. This post is an attempt to unravel the "why" of coding in the classroom.

My own experiences consist of a lone programming course in the Pascal language.  I am not sure that I finished my program for my project (a Minesweeper clone, that the teacher gave the perhaps politically incorrect name "Drunkard's Walk".) I do remember unraveling the mysteries of binary and other non base 10 number systems. That's a strong math connection right there. I would consider learning to write code for iOS, if I thought of a good app idea. Maybe that is a growth goal right there.

I've found lots of inspiring examples of code in the classroom, creative work in Scratch, for example, in primary.  My working theory, having seen but not worked in Scratch, is that it's an interesting and "new" form of visual storytelling.  I also think we should be listening to the words of people like athlete Chris Bosh, who talks about his formative experiences with code. I also usually tend to listen when the President of the United States stands up and asks all Americans to consider learning computer science.

This Mother Jones article is probably the best and most detailed read i've found on the topic.  The "why", if you accept the argument, and I do, is that we will all be the better for using the particular computational logic we can learn through becoming more computer literate.  Beginning from a "feat of imagination", and bringing creative, flexible and logical thinking leading to a task.  I think this is the true argument behind bringing coding to schools.

My colleague, @cashjim knows a lot more about this topic than I do, and here's what he has to say:

@MatthewOldridge Great article! There is a #coding bandwagon but you can encourage computational/procedural thinking without hard coding.
— Jim Cash (@cashjim) June 18, 2014

And further:

@MatthewOldridge ...not coding for coding's sake. However, some early coding experiences can lead to a lifelong interest. That is a fact.
— Jim Cash (@cashjim) June 18, 2014

I am left thinking, like I often am, about the role of "non-negotiables" or "absolutes", particularly in the elementary grades.  There is a long checklist of curriculum expectations that every student must work on, at the same time as everyone else in their grade.  The breakdown of traditional subjects still holds-math is still math, geography still geography, history still past.  Everything stays in its little container (except when true inquiry learning takes root, and new branches grow).

So should coding be added to a list of non-negotiables for the 21st century learner?  What would it replace? Or should we be giving children as young as Kindergarten choices about things to learn, including coding? I'm of the mind that more choice is always better, and yes, I advocate for giving young children choices in what they learn.  So we wouldn't always do "coding for coding's sake", as Jim said: rather, give kids a problem to solve, and let problem-solving with code be one of their options.  Keep it about questions, and the constant process of inquiry (across all subjects), and let computational thinking be one of our solutions.

"Be Like Mike": Michael Jordan, Albert Einstein, and the Growth Mindset

In our math session yesterday at #7to10PDSBnet, we talked a lot about how having a growth mindset impacts on education.

As we were watching Eduardo Briceno's TED talk, I got to thinking about the premier athlete of my childhood, Michael Jordan.  I think Michael's early "failure" is well-documented (getting cut from his grade 9 team), but we probably don't often think about the growth, reinvention, and improvement that happened over  the course of his career.

My thinking is that it's every so easy to look at the Michael Jordans around us as freaks of nature, the lucky few who are so prodigiously talented and gifted that they make things look easy.  The truth is, Michael was probably the hardest worker around, and he never settled into a fixed mindset about himself as a basketball player.

Two examples to prove this point:
-mid career he became an average 3 point shooter. Not great, like Reggie Miller, or Ray Allen, but average.  But this did not happen without taking many many thousands of 3 pointers.  It comes out of a dedication and belief in improvement, and persistent work.
-at some point he added the ridiculous iconic turnaround jumpshot that became his main weapon.  Again, this doesn't happen without a belief in your own ability to improve, to get better, to grow.

You could probably argue that it's easier to have a growth mindset as an athlete-you are always working on and developing skills that will help you play the game better.  You could also argue that Michael had the benefit of height, agility, spatial awareness, and leaping ability.  There may be some truth to that- but his success was all in how he leveraged his gifts and grew over time.

We all have our gifts.  We just need to nurture them and develop them.  But what difference is there between saying "I can't shoot a 3 pointer", "I can't do Math", or "I can't cook"?  If we believe that we, and our students have brains that are constantly growing and developing, then the sky is the limit, (nearly) anything is possible. Such is the wonder of the human brain.

Another "anything is possible" kind of guy was Albert Einstein.  One should be careful about adding to all the maybe true/maybe false Einsteinisms on the Internet, but I came across a statistic about how many times Albert was wrong in his equations.  Simply put, he couldn't do the math.  He couldn't make things work out, over and over again.  What he considered his most important work, proving the existence of a force or energy that is pushing spacetime to expand, was a total failure.  It was proven much later, and the Nobel Prize went to someone else for that.  I have heard it called his greatest disappointment.  There is another statistic floating around, where a physicist claims around 20% of Einstein's work had mistakes of various degrees.

So perhaps if we truly believe in having a growth mindset, we can stop saying, "he's so smart", or "she's just not good at math".  Perhaps then "Be like Mike" wouldn't mean becoming a super athlete, but rather being the best you can be, and growing, learning, and changing over time, with persistent work.  Perhaps then being an "Einstein" might mean being someone who perseveres in the face of great difficulty, rather than "being a genius."

"I'm not good at this ... yet!" Great discussions around growth mindset. #7to10PDSBNet
— Nicole Boujos (@MSSBoujos) May 22, 2014

#7to10PDSBNet "Even Einstein wasn't Einstein before he put in years of passionate, relentless effort" pic.twitter.com/zYXgdhN5Z7
— E. Boutros (@MrsBoutros) March 5, 2014