“…maths is really not about formulas, it’s about finding patterns in things, and how they influence people. And that’s a big part of art…” 

This statement from STEAM: How art can help math students really struck a chord for me:

“The big point of it is that maths is really not about formulas, it’s about finding patterns in things, and how they influence people. And that’s a big part of art,”Year 12 student Maxim Adams

I am currently teaching geometry (among other maths!) and we are in the sweet zone of shapes: triangles lead to parallelograms which lead from 2D to 3D to the idea of 4D (think twisted paper*). All of this incorporates the effects of angles on shape which affects (and effects) area. Some shapes are more ‘beautiful’ than others. Some shapes and forms lead us to wanting to know more…


I use paper folding to illustrate these shapes and connections. Patty paper is very light and a great beginning medium to illustrate the connections between shape and angle, how changing the angles changes the shape. Origami fascinates me, and I use it to fascinate my geometry students, and to lead them into exploring the relationships found there. The zig zag origami is complex and beautiful, and it takes a bit of time to teach, but as my students learn to fold, we are talking about math. Even their mistakes, when they have to go back and refold, teach them about the shapes and connections. And as STEAM director Melissa Silk says,

“I think it needs to be embedded from a very early age…When I’m teaching design, I’ll try to bring in a mathematical element. It’s worth students trying to make those connections without having to find it arbitrarily later in life.”

These ‘connections’ she speaks of are not innate. Students see without understanding. Teaching reveals and makes obvious; children begin to see math in everything in the world around them. It will allow them to make much deeper connections, more fluid connections, throughout their education and beyond.

 Art is not math, it’s not boring. Folding paper, for many math students, isn’t ‘doing math’. While they may struggle to fold straight lines, to make inverse folds, to make it look ‘right’, they are creating something beautiful, and if I can teach them nothing else, I can show them math can be beautiful!
 The idea, after introducing them to this complex looking, yet simple, folding technique, is to have them think about what it will take to replicate angles, curves and other shapes with their paper. Then they have to apply some design sense: how many folds? How long does the fold need to be? What direction do they need to fold? What angles will produce a curve? The folds are repetitions of a fold- what is the proportional relationship? In this image, there is a curve. Each fold is a point on that curve. It can be mapped on grid paper, it can be written as a formula. There is a connection!

Don’t be afraid to start with complex looking folds. It’s so powerful for students to struggle with something that seems beyond their abilities. Success is completely intrinsic and will lead to a student who is more confident. This feeling of success is habit forming. It is a primary building block in creating a student who perseveres!

(Check out 7 more art and math connections here.)

We can calculate the area of this beautiful 3D paper construction, we can create more of them by following the ‘formula’ or we can dramatically change or even destroy an image by changing the formula! This creative process stretches the imaginations of my students. The ‘what if’s are endless.

Many of my students will continue with this exercise beyond class, through drawings, foldings, and cut paper. There is a great simple foldable here that actually incorporates cuts to create a 3D effect. It’s a great way to jumpstart paper folding and cutting shapes to students that are not accustomed to paper-folding activities! I want my students to make graphic organizers that are beautiful to look at. Notes as art… What a concept!

Notes as Art… What a concept!

 

Pop Up Book graphic organizers

 

The possibilities for using this as a graphic organizer are endless!

I am an #mbtos -er…. 

I am an #mtbos -er because I finally found a place where I b

Continue reading “I am an #mbtos -er…. “

Are you ready to find out what misconceptions you have about your students misconceptions!?!

‘One colleague suggested turning to the calculator and using the answers as an investigation. Why does the calculator give this answer? What rules is it following? Can you write a set of rules? What would the calculator say for this sum?’by mrbartonmaths

Student misconceptions are critical to planning successful lessons! In a recent series of posts (I think he is up to 11 posts now) mrbartonmaths explains how he uses diagnostic questions to delve into the misconceptions his students have about basic arithmetic.

The quote above came from #10 in the Insight of the Week series: order of operations. As mrbartonmaths explains,

‘…the misconceptions I think students hold are different to the ones they actually make, and I want to put this to the test on a larger scale.’

While I found all of the responses interesting, I was surprised by the number of responses that indicated students were trying to place parentheses (brackets*) into the problem where none existed! These students were trying to make sense of the problem using familiar notation. The only problem I saw is that students didn’t know the ‘rules’ of brackets!

My comment to mrbartonmaths:

Teach them the rules for brackets!

To return to the thought at the beginning of this little essay, ‘have them use the calculator to evaluate the rules the calculator is using’ would require students to identify and investigate their own misconceptions- an idea I find ultimately rewarding!

For a great interactive lesson- which includes some much needed awareness of when parentheses are needed- try this Make This Number game.

As a further step in ‘teachers as lifelong learners’, I love that mrbartonmaths has embarked on something he calls Guess the Misconception, an email poll he sends out weekly to those who are signed up. What misconceptions are you holding about your student’s misconceptions!?!

I am reminded of the time I asked my Algebra I student, who was having a lot of trouble solving basic algebra problems in one variable (3x + 7 = 13 for example), why he kept wanting to start with 3x first. I had spent time working with him on the ‘unwrapping’ idea, without success. He pointed out that he was dividing by 3 because it was first. Headsmack! (Me, not him!)

Don’t assume! (You remember what that does, right? Makes an a– out of u and me!)

Thank you, mrbartonmaths, for giving us a little bit more insight (and some great ideas) into best teaching practices!

First post of 2016

This year: new ways to deliver PD; a different way of communicating with my students; improve my Spanish; a second semester in my ELL training (certification in my future!); but one thing at a time!

My #MTBoS mentees: hang on for the ride!

Great thoughts on PD: (credit goes to a brief tweet from @pamjwilson):

“stop [Tchrs] doing good things to give them time to do even better things”

From Sustaining Formative Assessment with Teacher Learning Communities

And…

Dan Meyer’s recent post on Swan’s idea that we ‘get worse’in the process of getting better.

Professional Development: Getting Worse Before We Get Better

Think about what happens when we make announcements to the whole class. Does ‘broadcasting’ really work? (Crediting another tweet: @justinaion):

#noTalkWC, Alice Keeler

I started with an expensive program, but ended up using a free app:

Duolingo

I love the way it has improved my accent! (Southerners speaking Spanish can be embarrassing!)

My ELL training is a grad class at UGA… Three semesters (I started this last fall, 2015) later, (and a certification test!) I will add a new certification to my license! #lovethedirectionmycareeristaking!

More than anything, this online PLC is about sharing and supporting great ideas, one teacher to another. I look forward to learning even more!

My newest inspirations are the three lovely people I get to mentor this year: Melanie, Doug(@freeejazzz) and Sandy, Welcome to the group! And thanks to everyone(!) for helping me become better. Happy New Year!

Let’s quit calling them 21st Century Skills; These babies are useful in any century!!!


Dan Meyer has struck again:

“I spent a year working on Dandy Candies with around 1,000 educators… In my year with Dandy Candies, there was one question that none of us solved, even in a crowd that included mathematics professors and Presidential teaching awardees. So now I’ll put that question to you.” Dan Meyer’s full post

What is it about a challenge like that?!? Of course I had to follow the thread! 

I read the over 100 comments as writers posed solutions, wrecked solutions posed by others, and even wrecked their own solutions! I watched as they systematically used the faults in their solutions as a springboard to better – but apparently still breakable – solutions.

I also heard a ghost of an admission that there may not be a single solution,

as timteachesmath writes, “Which broken algorithm is best so far? An algorithm that fails for ‘720’ but works for 95% of really composite numbers less than 720 might be better than one that works for ‘720’ but only works for 80% of really composite numbers less than 720.”

There is a Lesson here!!!!

I teach algenra 1 and geometry; that means 9th and 10th graders. I want to challenge them with Dan’s problem.

It’s simple, right? We are just talking about a box of CANDY!

I can see you now, shaking your head in disbelief: 9th and 10th graders able to frame an answer to a problem that even 1000 math teachers couldn’t solve.

Not only that, I can give this lesson to both algebra AND geometry!

Here is my explanation of the sequence of activities that would make the most sense to their budding understandings of math:

Essential Understanding: The best packaging involves the least surface area.

  1. The least surface area results from the tightest (closest) configuration of a cube’s side lengths.
  2. The surface area is a result of the combined areas of the six sides of the candy box.
  3. To find the minimum surface area for any number of candies, check for the following conditions: a) if the number is prime: 1, 1, the prime; b) a perfect cube: root squared times six; c) numbers with three primes: use the three primes; d) numbers with four or more primes: Multiply groups of the prime factors back together to find three products. These three products will be the three factors that will be the measurements of the box.
  4. Calculate the surface area from the measurements of the box.
  5. The box with the least surface area will have the factors that are closest to each other. It is possible for two of the factors to be the same number.

720 is a great example for (d):

Prime factors of 720 are 2, 2, 2, 2, 3, 3, 5

While they can be multiplied back together to create numerous factors, not all sets of three factors will give us minimal surface area.

Some of the sets of three that can be created are:

4, 4, and 45;

8, 6, and 15;

10, 6, 12;

And so on, until we get the multiples 8, 9, 10;

Checking for the optimal area involves a handshake (multiplying) among each of the three numbers – 8 times 9, 8 times 10, and 9 times 10, adding the products together and multiplying by 2.

Does anybody else see the individual lessons embedded in this process? This one problem is incredibly rich!

It’s not the solution, it’s the building of understanding!

The interactive process of doing this by hand is a wonderful opportunity to teach finding primes (6n-1), (6n+1). Students might also feel the need to learn how to find prime factors (and learning that all numbers are products of primes!). The question would arise about the geometry of area vs surface area. (Think of the manipulatives! I wonder of my kids would feel silly stacking cubes of jello!!!)

We also wouldn’t be able to ignore the eminently practical side of saving the planet through minimal packaging – not to mention the extension of how many candies we should pre-package for the best shipping (i.e, how many boxes can fit into a bigger box? Can we afford to package odd sizes and still keep our costs low enough to generate profit and sales?) (ooh! I can teach my kids to design boxes – quadratics, anyone?) Here we could also lead the class into the sales curve (parabolas – more quadratics! I’m in Heaven!)

By Jove! I think I figured out why Algebra and Geometry finally got together! They complete each other!!!

And I love the fact that once my students come to this understanding of the problem, they could begin to write a viable solution, either in algorithm or in code. Or maybe their understanding leads them to the conclusion that a single algorithm isn’t possible – did somebody just whisper the word “proof”? (You did just think that – you know you did!)

Just think of the STEM project ideas this activity could generate…

As many of Dan’s commenters pointed out, this is tedious by hand. But the truth of the matter is – they knew how to begin solving the problem by seeking to understand the problem to be solved! These are the skills our children need to learn. These are the lessons we need to teach. Let’s quit calling them 21st Century Skills; these skills really are useful for any age, anywhere. I’m living proof! (I’ve made it this far on those skills, haven’t I? LOL!)