# 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

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!)

# A better question might be, “state your answer as a complete sentence.”

I originally published this on Better Q’s. More research is being published on placing things in context. There are lots of ways to make content “relevant.” The challenge outlined here is to reconnect the answer to the question….

I caught myself doing something terrible! Ok, not terrible, just really wrong. Well, maybe not wrong, but it is very common: I worked a problem out on the board and left the answer, 20, out there. All. By. Itself.

No context, no units, no connection to anything. It was just there. My co-teacher jumps in, “Is this the right answer?” The class chimes in with a chorus of yesses, but I knew it was wrong. I looked into each smiling face, one after the other, waiting for someone to notice what was missing. One of them came through for me! “It’s hours, it’s gotta say hours!”

My mistake is one we math teachers make all the time – we get the answer, the right number, and we stop there. And move on to the next problem. But, you say, there is no context, it’s just a practice problem.

Skill drill and…

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# A Lesson in Co-planning (or just planning!)

Let’s bake a cake. You start.
What kind of a cake are we making today? A chocolate cake? Good! I love chocolate!

We need a recipe.

How can we possibly choose! Which recipe will make the cake that we need? Which ones have the ingredients and flavorings we want to use? Which recipe is simple enough for our skill level?) I don’t know about you, but I am not ready for chocolate angel food!) Have any of us ever baked a cake before – any cake?

Is there a recipe that will push us and our students a little beyond what we already know; what we are comfortable with? And let them build on previous knowledge?

Do we set all of the ingredients out for them, or do we let them go to the cabinet and explore what is there to use? Do they have to follow the recipe exactly? Do we hover as they measure, or do we let them experience the trial and error that can be found in cake baking (or math sense making, or working as a group to design, build, or solve a problem together)? Will we need to premeasure for some students, but not for others? Do we know who will need more guidance than others? Or who will need the ingredients for a flourless cake because of food allergies? To pull this off, we will need to settle on a recipe, or two, or three (or maybe let our students find the recipes that they are most interested in).

Decide together how you will each handle supporting each step in the process.

The key word is together. I am a co-teacher with three wonderful, brilliant teachers. Who have already planned their lessons for the current unit. Without me.

I have to go find them to ask about the lessons, and ask about my role in the class, usually the day before, or morning of, the class. I don’t see the materials without asking for them. By not including me in the materials selection and planning process, they have created double work for themselves.  This is not how co-teaching becomes its effective best. It also shortchanges the very students it has been created to help.

A co-taught class is a regular education class with students who need extra support in class.

A co-taught class is a regular education class with special education students – who need extra support in class. In this environment, students with special needs have the opportunity to interact with other students in what is called a “least restrictive environment” or LRE. The key to making the LRE work is having two fully certified teachers, one of whom knows what needs to be done to help the students who need their lessons delivered with a little more support.

When one person handles ALL the planning, no provision will be made for the alternative forms of presentation/activities that may be needed to support ALL students in the class with opportunities to learn.

I don’t think this getting left out is being done on purpose.

Each of us has our own teaching style. We may plan our lessons informally, away from school, over the weekend, perhaps. Some teachers plan obsessively (in a good way-getting copies made ahead of time, or planning from the test backward – all excellent strategies). Letting someone else in on the process may feel like leaving oneself open to criticism.

Sticking with the baking analogy, some of us always make a chocolate cake. It’s the cake that everybody eats. It always has chocolate frosting. Everybody gets it, even if not everybody can eat chocolate. And there’s the rub. Regular Ed teachers are not aware of different needs/ limitations/ legal issues of special education. There is a misconception that the way they teach their lessons will work just fine (perhaps they have been successful in this way, and have test scores to prove it) for any student. And if the student isn’t “getting it” the fault is the student’s behavior, or that the student needs to listen more closely or take better notes.

Along comes the special education student.

This is a child who is perfectly capable, mentally, of keeping up with, or even surpassing the other students in the class, except for one thing – they need more processing time, or they don’t compute numbers the way others do, or they have problems focusing, or they learn by doing, instead of hearing, or they need larger print, or they don’t write fast (for notes – so they need pre-printed materials). Because of these reasons, or others like them, this child has been placed in a room with two teachers: the general Ed teacher and Me.

Here’s where I come in. I know the best ways to help these students learn, to get past whatever wall is causing them to need these extra services. I can build these supports into a lesson. I can make sure that there are activities to support these needs, but they have to be designed into the lessons during planning – with both of us buying in to what is going to be done in the classroom.

It does no good for me to walk into the room on the day the lesson is delivered and find out Marie (not her real name) can’t read the lesson worksheet because the teacher has printed multiple copies on a page to save paper and the print is so tiny we can’t tell whether that’s a division sign or a plus sign without checking what the answer is supposed to be! I will have no file on hand with which to print her a larger copy, because I was not involved or copied in on the materials for the class. The larger copy, by the way, is required for her by law, because it has been written into her individual education program document, or IEP. The reg Ed teacher should know this because she has a copy of the accommodations. I would have said something while the copies were being planned- and we would have been prepared.

The regular Ed teacher needs my knowledge for her classroom to be successful. She (or he) needs me to ensure each child is getting information about the lesson in the most optimal way. When a co-teacher is left out of the planning, the unfunny thing is that all children in the classroom suffer, because multiple learning styles are not being represented. All children benefit when lessons are learned in multiple formats. Not everybody likes chocolate cake every day, all the time.

Another benefit of the co-teacher model is that the reg Ed teacher isn’t carrying all the load, teaching the class, grading papers, and struggling with disruptive behaviors.

Is it hard to let another teacher in your space? A recent Education Week article on good co-teacher practices compared it to a marriage. I think it is more like two horses in harness. One cannot lead without tipping the wagon. We must pull together, in step, to get where we are going without upsetting the cargo. There is no room for personalities, and yet, we need to play to each other’s strengths.

Co-teaching takes communication.

Co-teaching requires a learning curve, and it requires both teachers putting all of the students in the room first. It requires letting go of pre-conceived notions about what special Ed students can and cannot do. These students are ours, not those are yours and these are mine. That diminishes our expectation that all students can learn the material. It causes us to view students who require support as being troublemakers. Did you know the statistics? The bottom line is that more special education students get written up for behavior problems in regular classrooms than other students! A well-run and executed team teaching plan can help, both in educating regular Ed teachers on what’s really going on when a student appears to be misbehaving AND reducing the amount of learning being interrupted for all the students in the class.

Co-taught classes often look and feel different than regular classes, for both teachers!

The reg Ed teacher has no experience of students that have a learning disability. For example, they have no knowledge of what teaching strategies are directly helpful for a student who is dyslexic (colored backgrounds for worksheets or PowerPoint displays reduce cognitive workload for these students, and do not negatively affect the regular students either – the different look is often welcomed by all). I have to make my co-teachers aware of the difficulties a child will face before we present the lesson. My responsibility is first the child, but also being a partner to my co-teacher: not withholding valuable information, or not impeding classroom discipline by making unnecessary exceptions, or undermining my partner’s authority.

Planning together allows the process of differentiation to happen without bringing undue attention to the disability.

This is a critical point: we cannot single out the special education student in these mixed classes. While privacy is the main issue, I believe we have a moral obligation to our children, as well. We are charged with creating a safe learning environment. Children do not learn when they are stressing over being or feeling “different.”

Planning together can go a long way towards eliminating stress – for students AND teachers. Like any good teaching strategy, the process takes work and practice! Cut your co-teacher some slack and treat him/her the way you would want them to treat you if you came into their classroom. Hey! That’s not a bad idea. Can we meet in my classroom tomorrow? I’ll need to get in some extra desks….

# Math Problem of the Week

I get to be the sponsor for my school’s chapter of Mu Alpha Theta. This is an awesome group! Unfortunately, all but three students graduated last year, and one of those transferred to the new school!😩.