The Tool Must Fit The Task- or, Physical vs. Digital Manipulation #SAMR

You might know that I love snapcubes. I think they are the most versatile type of thinking tool out there. I love how you aren't closed in to a single function, like many commercial manipulatives, and how tactile they are, how satisfying it is to snap them together and make something.They can be used in algebra, geometry, 2D and 3D measurement, playing with numbers, basically anything,

Cube tasks are their own genre of task, I think. There are lots just in the Ontario Spatial Reasoning monograph. 

Here is a version of the 5 cube task you can use with most grades, with interesting results:

Here's the thing with that picture: cubes are something Minecraft does really well, seeing as how they are the basic unit in Minecraft (1 m^3, supposedly, in the actual game!)

So you might think, as per the SAMR model, that Minecraft is automatically better than using snapcubes. Isn't it fresher, newer, shinier, more "21st century"?  

Not necessarily. Sometimes the original tools are better. 

I came to @MrSoClassroom with this task in mind:  

You have 24 cubes what is the least surface area? Greatest? @MatthewOldridge #engagemath

— Jonathan So (@MrSoclassroom) January 13, 2016

I thought, sure, we should offer Minecraft as an option, as well as snapcubes. I was persuaded otherwise?  Why? One tool was enough, and it was the one built exactly for the job. Sometimes (actually, many times), physical manipulation is better than digital manipulation. Figuring out how to do the task in Minecraft (for non-players) would have slowed them down too much.

Further, one face of each cube placed on the ground would be hidden from view. That is a big problem with trying to do this task in Minecraft.

But you know what, about the Minecraft picture above? It just might be easier in Minecraft. Why? You will never run out of cubes, while trying to build your structures, in an infinite world...

Behold, The School of the Future!

One of the best stories you can read about schooling was written in 1951, by Isaac Asimov. Called "The Fun They Had", it's about a generation that knows nothing other than teaching machines, and reminiscences about a physical gathering place called "school". Asimov knew futurism (and partly invented science fiction), but he, so far, has proven to be way off with this fiction.

You can read it, and see what you think. Certainly we all have our opinions about what the "school of the future" will or should look like, based on our own experiences, biases, and interests. A large part of what we think should happen in schools in the future is, I think, dictated by how disposed we are to see the past as better, or worse, than the present time. (This is mirrored in the larger debates over traditional/progressive schooling, and probably conservatism/progressivism in politics).

Here is a current Australian vision for the "school of the future". It frankly doesn't seem much different than the type of schooling predicted by Asimov: "electronic with a capital E!" But then you actually read the article, and it''s talking about things like" VCRs, laserdiscs, and telephones. Yep, the "school of the future" is already out of date.

I used to teach science fiction to kids, and I like to use images of what people thought the future would look like. This Reddit has some nice posts that you can look at. The human mind can always leap farther with its dreams and ideas than technology can. Filmmakers like Stanley Kubrick never thought to go past 2001 with his space odyssey. The Back to the Future movies had their own ideas about what 2015 would look like (and, strangely, hoverboards are here now).

I think technological progress is slow, relentless, and sometimes, paradigm shifting. Iteration upon iteration seems to be the way a lot of technologies progress. A lot of things in my house would be unrecognizable to my great-grandparents, but many would not.

The question to ask yourself is this: would Dewey in 1915, recognize your classroom as a place of learning, in 2015? My guess is yes. Yes, he would also recognize "21st century skills" such as creativity and critical thinking. You could teach Mr. Time Traveller Dewey how to use an iPad, and chances are, he would enjoy your class. I suspect he would quite like the opportunity to learn with modern tools.

Here are my predictions for the school of the future:

• kids will still learn in a room with teachers
• schools will continue to use the technology of their day (hopefully very effectively)

I am comfortable saying very little else about the school of the future.

As for "22nd century teaching and learning", i'm sure it will look a lot like "21st century teaching and learning". 21st century teaching and learning is really teaching for "the now", as my friend Mary says. What I am trying to say here is that human interactions and relationships will still be paramount. The "'school of the future" will "Human with a capital H".

The actual "School of the Future" in Manhattan

On Student Mathematical Thinking

Keith Devlin's perspective on mathematical thinking, as presented in this blog post, has been influential in my own thinking about how students interact with mathematics in our classroom.

I am happy to accept this as a working definition for myself:

Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns."'

This is not to say that I think novices (students) are always thinking like experts (mathematicians). I don't. School math and professional math are not the same things. But I do think that students can adopt the identities of (young) mathematicians in the same way that they can adopt the identities of (young) scientists: by using some of the tools of the disciplines as they work at their own levels. Students of all ages are encouraged to construct scientific hypotheses about the world, without controversy, whereas I think it is far too easy for us to present all school math as axiomatic, or even universal, without giving students a chance to see into the mathematical world themselves.

Keeping the expert/novice divide in mind, then, here are some things that students will generally NOT be doing:

-constructing proofs by contradiction
-developing theorems
-proceeding from rigorous logical structures like "there exists a..."
-using set theory

Some things that students will, and I believe should, be doing:

-developing their own reasoning, and "proving" things as true as they can, for any given task
-exploring truths about arithmetic, like commutativity
-systematically exploring why things like the formula for area of a triangle work for all triangles
-making generalizations using algebra, for example about a given linear pattern
-choosing the best "tool" for the job, whether it be adding, subtracting, multiplying, dividing, or higher powered tools, as they get older
-constructing models for how things "might" work, in the context of a given problem

Here is a nice piece that argues that mathematical thinking doesn't look anything like mathematics. If you don't believe that, I give your Andrew Wiles' proof for Fermat's Last Theorem: "Modular elliptic curves and Fermat's Last Theorem". Probably only a few hundred people in the world can read that.

For those keeping track, I am absolutely not arguing that any of these things are done in the absence of content.
I do believe that the content itself is the vehicle for thinking mathematically. I am also unabashedly on the side that says students aren't given enough chances to develop their own reasoning. Mathematical tools that just sit in the mental toolbox are no better than dusty old tools in my basement. They should be used.

I also think mathematical content should be presented for what it is, in discipline-based language. Even the viral Common Core problem on "5x3" was an ideal opportunity to teach kids of any age about commutativity. You might say kids can't handle that- I myself have seen grade 2s in @MrSoClassroom use the phrase "distributive property". A lot of our coaches are also exploring arithmetic in interesting ways with number talks. And yes, they are teaching standard algorithms as well, because that's part of the journey. But number talks might be an example of what Mr. Devlin is talking about-stripping down simple arithmetic into structural elements, then putting the elements together.

Students must be given the chance to work with lots of problems, in order be able to strip them down to their mathematical elements. As they learn more mathematical content, as the grades progress, they will be more able to do so. Sometimes we do that in a more structured way, such as giving a few problems that are obviously "about" Pythagorean Theorem. Other times, we might let students play with the structure of problems (I used to like having students write their own Pythagoras problems, to see what they could do with the problem structure).

Why not? Students come up with things much more interesting than the old "ladder against a wall". 

Even better is using that structure to explain something in the world, in this case, how televisions are sold:

A few pictures I still have and like that I hope illustrate some of these points about student mathematical thinking:

 A student created their own model for the final career point tally of 3 NHL players. He had to decide what method for projecting their career points, including a calculation method and what assumptions to use.

This one represents an attempt to systematize the thousand locker problem.  The deep structure of this problem is that all square numbered lockers are the only ones touched an odd number of times. For any student to understand that, they have to first come up with a tool that will help them find the structure. 

This problem is something about numbers on horse stable stalls. I forget. They made it all the way through a systematic organized list before realizing they could start to generalize about their results. 

This one represents a students' understanding of a dot pattern. She saw the structure as adding a group of three dots to the side each time. It was a big "a-ha" moment, and led her toward the algebraic expression that models this pattern.

This picture represents a students' choice of tool for a problem involving using a certain amount of fencing to enclose as much area as possible. Students were generalizing about the meaning of "squareness" with regard to area and perimeter.

This picture is one of my favourites, It represents a kid exploring the class of polygon called a "trapezoid", in order to attempt to figure out how you might get its area. It's exciting because it led to...

...the beginning of a generalization about trapezoids. In my experience, making statements about all cases, beyond ones that they have found themselves, is tricky. A nice article about how generalization develops in early algebraic reasoning is here. 

This one represents a generalization with algebra about a version of the tables and chair problem. Gradually students realize that the "+2" stands in for the two chairs on the end, that are always there, even as more tables are added. 

In summary, I believe that students are very much capable of mathematical thinking, even if they don't always "think like mathematicians". They need lots and lots of chances to break down problems into their basic structural elements, to explore their own reasoning, and to start to make interesting generalizations on their own. 

Hope #oneword

When I think of my #oneword for 2016, I don't think of anything specifically related to teaching and learning. I don't think of "rigour", "grit", "moxie" (apparently a thing  now?), or anything to do with technology or mathematics.

I won't take a stand for something new and trendy, and I won't take a stand against something I see as stale and old. In 2016 I would rather be for things, rather than against things, anyway. I am for optimism, and I am for hope. I don't believe the world is getting worse every day. It's getting better. I am conscious now, on the planet, and my consciousness, as of all humans, goes back and forth between hope and despair, between the darkness that we stare into, and the light that comes from within (or without, depending on your spiritual beliefs).

I am for being hopeful. I am for supporting each other, when we're struggling, or feeling down, hurt, or broken-hearted. I am for the mental health of our teachers, administrators and support staff. I am for supporting our students to be hopeful, even in the face of trauma and despair. Classrooms are places of hope. (You can ignore trendy arguments against "safe spaces", classrooms must, always, be safe spaces).

This will probably be seen as another post about putting human relationships above all else in our education systems, and it is. It's about sustaining hope in each other. It will probably be seen as another mental health advocacy post, and it is. Most struggles are struggles of the mind. Look around you- are you aware of the struggles inside each and everyone around you, even if they don't show on their faces, or come through in their words?

There are people around you who:

-stare down the vicious monster called depression on a daily basis
-struggle with dependency (particularly on alcohol- if you are an adult in this world, you know someone who struggles with alcohol- you just don't KNOW it)
-are critically ill, or have loved ones who are, or who struggle with chronic pain
-have unresolved trauma in their past
-have suffered through broken relationships, heartbreak, heartache
-everyday swallow the bitter pill of disappointment over unrealized hopes, dreams, aspirations

I could go on, but I won't. Be the "light in the darkness of misery" (E. Costello) for others, but above all, take care of yourself. As much as is possible, be hopeful. Be not so fearful, and know that everything has its equal, and its opposite. Light has darkness, darkness light. If you feel despair, you can feel hope. Be ever hopeful.

Light vs. Darkness by Ritesh Man Tamrakar 

(CC, Flickr: https://www.flickr.com/photos/_rmt_/5569820904)