It has been far too long since I have posted here, and for that I am sorry.  To tell you the truth I have been busy, but more than I busy I can admit that I have been a bit down on myself.  I still have so much to do to become the nguyeningest teacher that I want to be, and I have been a bit under a rock by recognizing that I am not there.

We can talk a big game on these blogs and share all these awesome lessons that work (when they work), but even those great ideas can fall or be used improperly, or rushed to the point that no meaning is taken from them.  I am a the point right now where I find myself trying to catch up to all the curriculum that needs to be done, and I don't like that, but I have a professional responsibility to make sure that kids are prepared, so I have to work better and harder than I do at this point.  Have I led students through inquiry? Are they engaging with their world?  Why can't I be more like (insert awesome blogger/teacher/awesome person name here)?  I feel as though I am not measuring up, and that's a hard place to be.

So I am introducing volume and I asked them to notice and wonder, and this is what I got...

This board of notice and wonder seemed a little too good to be true, so I asked the class (a class who I trust to be honest) this:*  

The majority of the students put their hand up for option A.  They looked at pop-cans and wanted to know legitimately awesome things.  They were curious, and they were taking it seriously.  They had cool discussions on why the companies would make these decisions.  Now granted, we have talked about pop companies before when we did surface area, and I walked them through that, BUT I can see that they have caught a bit of this math bug.  They have caught a bit of the curiosity that I had when driving home and watching road lines whiz by.

I am not a perfect teacher by any means.  I lose control of my class. I take forever to assess.  I give them notes, and I even *dun dun dun* lecture.  Not every class is inquiry based.  I am not always fully prepared.  I am very disorganized. I have more faults than I care to name, but somewhere, hidden under the heap of my self-deprecation is something that has attracted kids toward curiosity. We often say to ourselves "If I but do this ONE thing, it will have all been worth it."  Today this board represented my one thing.  Today I saw that I am not in the wrong profession; I am not the worst person in the world; I am not a terrible teacher. Today I cast off my self-deprecation and embrace encouragement.


*Keep in mind this is ONE of my classes.  My morning class was not as indepth.

In my last post I proposed a possible introduction to order of operations.  It was by no means a complete planned lesson, and neither shall this post be. What happened after, I posted on Twitter "how do we teach order of operations as a concept, rather than a rule to be memorised?"  My goal was to teach students a reason why multiplication precedes addition, rather than simply saying...

Or is that all we can do? It is a convention right?

The first thing I thought was that the order of operations are not simply a random guess in the dark convention.  I do not imagine the world's greatest mathematicians developing this necessary convention by the toss of a coin.  Rather, I expected that there is some reason the great minds of the past did in fact decide that there was an optimal way to follow the procedure of calculation now infamously known as BEDMAS or PEMDAS depending on the mnemonic device you have learned.

For me the order of operations has always made sense in terms of algebra, and that is exactly what Dr. Math (referenced in paper) claims. Well, great, I want to teach students order of operations conceptually, but they do not know algebra yet, what can be made of this?

My algbraic understanding moves me to discuss terms.  The standard convention makes sense in my mind when we discuss terms: 6x+6y.  I know that whatever x is, I have 6 of them, same with y, but since I do not know the values of x and y, therefore I cannot complete and simplify this term any further, because I am not given enough information. 6x+6y is necessarily simplified at this moment.

This is a great way to think of the operations, if you know the basics of algebra, but when we teach order of operations, students do not have a strong (if any) concept of a variable and especially like terms.  My struggle over the last few days has been this: "How do I reconcile my algebraic understanding of the order of operations within arithmetic?"

I think I just heard @fnoschese cheer, but it really comes down to understanding our models of addition and multiplication. When I think of my conceptual understanding of addition, this comes to mind...

That is, addition is largely a combining operation.  I have this many things here, I have this many things there, I bring them together and TADA!  However when I think of multiplication, I think of this...

When I think of multiplication I often think in groupings.  I do not take a quantity and combine it (like addition), but I create a quantity (seemingly from nowhere) via grouping.

The point that expresses this for me the most is when we look at the number line.  With addition we start at a particular number, and then move onward.  With multiplication we start at zero! Multiplication is an ex nihilo process, we get something from nothing. With addition we combine two quantities that are there, whereas with multiplication we simply start making groups. I think this is the "multiplication is a stronger" operation argument, but I just had to put it in my terms.

This is not an airtight argument for multiplication to necessarily precede addition, but since order of operations is a convention it need not be.  However I think this is my first fruits of grasping why this is a well defined convention, not a coin flip.  If I am uncertain of which order to use when given the problem 3x4+2x5, it makes sense to me that I cannot do any addition until I am guaranteed to have separated values that I can combine, multiplication doesn't have that requirement, so logically I can start with that multiplication.

I'd like to hear all of your thoughts on this though. Is there a point to all of this or should we just say, "Bedmas everyone! Look it's bedmas!"?  Do you want to get mad at me for my lame models or start the war on multiplication is not repeated addition?  Let's talk this through so we can make the order of operations a less daunting, more natural task for students...

I love my nephews and nieces.  They are great fun, and there is nothing more fun then getting them to do math.  I don’t know why I do this, but I love having them do little things here and there.  Count this, count that, what is this plus that?  At one point on my spring break my nieces were doing addition practice all on their own and showing me their work!  Seriously, what’s wrong with me?  I was really intrigued by my niece’s ability to add large numbers together and it has caused me to reflect on the nature of language and our ability to reason.

Ones, Tens, Hundreds, Problems?

My niece is in grade 1, and she is adept at adding single digits.  With little hesitation she can do her basic addition.  She even showed me that she could do things like add 100 + 100.  I thought this was really neat so I asked her some questions.

Me: What’s 1+1?
Niece: That’s easy it’s two.

Me: What’s 100+100?
Niece: It’s 200 duh!

Me: What’s 1000 + 1000?
Niece: 2,000 these are easy!

Me: What’s 10 + 10?
Niece: … I don’t know.

She has not learned place value, and therefore her ability to reason this was not developed.  I tried to walk her through it.  I got her to draw dots, but miscounting the dots led to incorrect answers.  I tried having her write out the answers to see a pattern.

1 + 1 = 2
10 + 10 = ?
100 + 100 = 200
1000 + 1000 = 2000

She found the process confusing.  I did not know what to do.  I’ve never taught math to someone this young.  This was just some fun we were having, and then I felt that her confusion might result in a math phobia that would predominate the rest of her life (I may be a bit overdramatic, but I was worried). We decided to take a break from math for the moment, and worry about that problem later, but I find myself still thinking (months past) about that moment.

Language and Thought

I remember in my intro to psychology class where we talked about what came first the ability to think or the ability to talk.  Can we share thought or elaborate our thought without language?  How does language affect our ability to think? Can we even think without language? Are we limited in thought by the language that we are given?

My niece gave a curious example for me to ponder upon.  She was able to reason, with ease, how to add hundreds and thousands together, even though she has little understanding of place value.  My hypothesis is that she can do this because the language of hundreds and thousands still uses our single digit counting; she is simply counting a unit that happens to be named hundred.  One plus one equals two, one apple plus one apple equals two apples, one hundred plus one hundred equals two hundred.  For my niece hundred was just another quantity that we can add together.  Ten, however, did not have the single digit number in front of it. We do not call ten one-ten, or twenty two-ten, and she was thus unable to reason it in any way.  Why is that? Why do the tens have this magical new way of naming numbers? Every other number that does not require a ten type number, uses single digits (i.e. 1,000,000 - one million; 100,000 - one-hundred thousand; 1,000,000,000 - one billion). Is this a fault in our language of mathematics, or just another hurdle that we must overcome?  Is this language deficiency making our sense of place value more difficult to grasp?

I don’t know, but it’s interesting to ponder... 

As I have been writing more and more entries I have noticed at the top of my site the ridiculous title “Welcome to Mr. Piccini’s Mathtabulous (and more) site.” First, *gag*, second, I realised that I had created this as a joke when I was first playing around with the weebly editor, and I never changed it.  I realised that this title, for me at least, was the Comic Sans of blog titles.  I started asking myself, “What is in a name?” This question was first famously asked by the Shakespearean character (like you didn’t already know) Juliet:

And this answer to the question, that a rose is just as fair under another name, could not be any more wrong!  Sure objectively if a rose was called by another name, this fact would not change the chemical structure of the flower that produces such pleasing odours, but if we were to name them something different, something that leaves an off putting taste in your mouth when you say it, perhaps “butt demons,” the flowers would lose the sense of class and allure that the term “rose” provokes.

When we change the name of something, whether we like it or not, we affect our perception of it positively or negatively.  It’s the same reason when I was a child I refused to eat aspic; regardless of how wonderful it tasted, I could not bring myself to try the dish.

Naming, can instill or revoke power from something.  Ancient cultures named their children with purpose.  If you were destined to be a warrior, you were named appropriately.  In the Bible people constantly had their names changed to better represent their job description. The First Nations used names to describe spiritual or physical attributes of their people.  In our culture, we choose a name because it sounds “pretty” or “nice.”  It is a less profound reason for naming, but  the affect of the prettiness of a name plays a role in our selection.  Some names have such an emotional connection that when we hear them we have a distinct, involuntary reaction: George Bush, President Obama, Justin Bieber.  These names depending on your politics,  nationality, and taste in music may present joy, wonder, hate, or depression. It doesn’t matter if they are rulers of a music genre, a country, or the entire world (Bieber) we have gut reactions that these names effect in us, whether we like it or not.  This is no different when it comes to the naming of a blog.

Where does one start?

I love the names that I see around me.  They are short, to the point, and they usually show an amount of introspection and humour that I appreciate.  Function of Time, Action Reaction, A Recursive Process, just to name a few, reflect the connection to these teachers’ passions as well as their philosophy of the vocation of teacher.  I get the sense that these are teachers who are forming themselves as people, as well as professionals.  They recognise the ongoing process of becoming great teachers without using buzzwords like “life-long learner” (no offense to anyone that has chosen that as their blog name), and that is what I wanted out of my new name.

I wanted my name to say “I am a math teacher and I am clever,” because well you know, that’s important right?  Being clever? (Past Timon didn’t read the first part of this blog)  I tossed out some ideas like “Guess and Check: An anthology of my problems and solutions” and gave them to Twitter to see the reaction.  As soon as I posted them I realised that I had put no heart in them, they were just goofy puns that I thought were clever.  I decided to add the nearly self-deprecating tweet:

I waited for responses from Twitter, and it seemed like one person liked each of my suggestions, and then I got this tweet.

David is usually fond of being a smart donkey, so I assumed this was cheekiness and I discounted this tweet (sorry).  Then another person chimed in by agreeing with David. I replied to them essentially with, “Really?! That’s a crazy blog name.”  David brilliantly said how it was more indicative of what I try to do on this blog and I had to agree.  It stuck to me, and I realised that this is where I needed to go.

I struggled with the image.  What did I want it to say for my blog?  Returning to the drawing board felt like I am constantly failing (it can feel like that at times), and I did not want to destine myself to constantly re-writing everything that I have (Mr. Sisyphus? No thanks).  I realised what I want to do in my career is to stay at the drawing board.  Re-writing has this feel of once you get it right you are done, but staying at the drawing board seems much more organic, and creative.  Coming back to the drawing board implies a lack of creativity when away, staying at the drawing board means creativity is continual and growing.  Returning to the drawing board implies that planning and teaching are two distinct processes, but staying at the drawing board means that my entire being as a teacher is looking towards growth at all times.  I realised that in my name I wanted to encourage the drawing board; I wanted to symbolise the realisation of what the drawing board should mean for a person: not a place wear we mourn our failures and start over, but where we create our masterpieces. I thus hereby declare that henceforth my blog will be named...

Embrace the Drawing Board

It’s not a math pun.  It’s not “clever” in the vain sense that I had originally wanted to be.  What it represents, however, is a symbol of who I want to be as a teacher, and who I want my students to be as persons.  I am not now nor will I ever be perfect.  I learn from my mistakes, which means I learn a lot.  I do not want to be afraid of those falls and I want my students to see this as a model.  The name also acts as a reminder to myself.  When I wish to avoid the drawing board, I need to see that only when actively building myself as a person and as a teacher will I find satisfaction and fulfillment in this crazy world of education.  Finally, I hope it can be an encouragement for you.  May you be able to go to the drawing board not as a punishment or as a reminder of your failures, but as a springboard to your potential.  

Completely Nonsensical Epilogue

There was some awesome ridiculousness as awesome people were helping me think through these steps and here they are.

• dy/drawingboard
• Between the Drawing Board and the Waterboard
• f(drawing board)
• Continuously at the Drawing Board, But Drawing Nothing (honestly how I feel most times)
• Drawing everywhere, and Board Nowhere
• Action-Drawing Board
• Lost in the Drawing Board
• A Recursive Drawing Board

And the winner...

• Draw-Drew-Drunk!

Dan Meyer opened his site 101qs.com some time ago now, and I have to say it has been rather engaging to see what the results have been.  You enter the site thinking you have some pretty top notch photo or video prompts that are bound to produce wonder and amazement, and yet somehow you find your perplexity score slowly going the way of the buffalo. At first it was disheartening.  You think to yourself how could my amazing collection of gumballs stacked on dominoes in the shape of a sierpinski triangle at the burning man festival not provoke the question, “How can we derive the gravitational constant?” The question now becomes, "Is that the point of 101qs?"  What do we want out of these videos and pictures?  Do we want everyone to be on the same page or do we want multiples questions to stem from each video?  Do we want both?  Is 101qs giving us what we want?

Clash of the Titans

 Over at Dan’s site people have been discussing these last set of questions and we find, naturally, Dan promoting his brand of “Make the prompt scream the question you are looking for” and Karim Ani saying, “There are more interesting questions that go beyond a one minute clip or picture.”

I think many of us, including Dan and Karim, find ourselves right in the middle of these two conflicting axioms. On one hand we desire that students seek for themselves.  We desire that they personally invest in interesting questions that provoke grand thoughts about life the universe and everything.  Yet we also recognize that they are young whippersnappers who have little experience beyond their hometown, school, and even neighbourhood; they may not have the capacity to think beyond the world of their home.  We then as teachers must make the hard decision of how to lead them to these wonderings.

How many “how many” questions can we put up with?
One of the biggest critiques of 101qs right now is that the questions are too simple.

• How many dominoes?
• How many combinations are possible?
• How many tickets?
• How many gumballs?

These questions have all been taken from the top of our list of most perplexing images.  Sure they represent different math (in this selection alone we have rates, combinations, volume, and area covered!), but the depth of questioning is shallow to say the least, and this is coming from teachers, who have experienced this world and are confronted with questions that affect our humanity and interactions with the world at large.  “How many dominoes?” just doesn’t cut it when we live in a world that is torn apart by famine, poverty, super-consumption, and an economy that seems to be hanging over us like the sword of Damocles.  We the teachers need to see how we can bridge this gap.  Students are certainly engaged by these problems, but how do we turn them, and mathematics as a whole, into more than a daily puzzler? If our subject, and this website, is turned into only a collection of neat little brain teasers, then we have missed the point of our roles as teachers.

Finding the Middle Ground

Just looking at the top of my site you will see clearly that I have drunk Dan’s Kool-Aid.  I love anyqs, and the whole three act process.  I know Dan's framework has directly made me a better teacher, and helped me to focus on more engaging class discussion, but I am beginning to find myself wanting more out of it.  In my comment on Dan’s blog I express that my main answer for this question is that we must understand that  101qs.com, in its infancy, is only presenting the first act of a fuller narrative.  We must use these videos as starters to the greater questions.  Students want to experience mastery of concepts, they want to feel positive about their abilities, and we can enter these students much easier into this subject that we love so much with a question like “How many gumballs are in that dang machine?”  It is fun, it’s nice, and students say to themselves, “I can do this!”  Then they take to the math, and they realize, “I need to know a lot of information to solve this.  What shape is that thing?  Is it a prism?  No, it’s a ball shape!  What is the volume of a ball?”  We enter with our sequels to these stories to propel their thinking, “How many gumballs could fit into this room?  Into this whole school?”  Students start adapting their thought to new situations, and they begin to see that this one little problem can extend far beyond just this picture.

Going for the Home Run

Once we have given students these chances to use their knowledge in differing situations we unload the brain busters.  We ask the hard questions, we develop projects from here. After seeing so many of these “How many little things can fit inside one big thing (gummy bears, tickets, dominoes, gumballs, teacups...) the question that has been arriving in my mind is this, “Why do we measure things in g, mL, etc. and not gummy bears, or gumballs?”  Get students to debate, and then create their own measurement systems.  Have them create conversion charts, and think about how to create a system that measures mass, volume, as well as linear measurement.  Have them reflect on the advantages and disadvantages of having their own measurement system.  Have them present their measurement systems to the class, have them create lessons about the Billy-Standard-System.  Then show them why the metric system is so awesome! They started their journey in the “How many?” river, but they are now swimming in the “What if?” ocean.

Where to Now?
Have I pulled this stuff off? Not yet, I’m still young, but I know that this whole 101qs thing is Dan Meyer’s attempt to pull the stunt on us as I want to do with my students. “Take some pictures,”  he says, but underneath it all is the possibility to go wherever we want.  So yes, if we discount these “How many?” questions as paltry, then that is all they will ever be, but if we keep pushing, we can turn these low floor questions into high ceiling discussions and projects.  So grab your camera, grab your phone, and get some first acts loaded.  We’ll work on the rest together later.

Being a second year teacher seems much like being a middle school aged adolescent.  All at once you have great vision for the future, unquenchable desire to see wrongs made right, and only half the necessary skills to see those grand desires fulfilled.  I am able to see the mistakes and the immaturities that I had as a first year teacher, and much of my life is spent redesigning what never should have happened.  This means I get to see not only my students grow, but I get to see myself grow. There are two sides to this coin however, and for each strength that I hone, a gap forms that causes the weaknesses of last year to be that much more apparent.  All of this means that I get the chance to see the good, the bad, and the ugly of my career, and so far here it is.

The Good

Having had one year to work through the curriculum has given me insight into student understanding, and how they approach the subjects and develop their mind.  Especially with Math, I have been able to develop curriculum that promotes student intuition and development of concepts.  Many lessons last year where I told students, “This is how you do this!” through boring direct instruction have been transformed to clear tasks that ask a more intuitive question where students get to reason, debate, argue, and explore. Blogging and being on Twitter has been the catalyst for all this.  I have a much clearer view of how I want to teach, and what effective teaching looks like.  If not for finding blogs and educators like Dan Meyer, Frank Noschese, and Shawn Cornally (and everyone else that I follow on Twitter), I would not have had the starting point for authentic learning and problem solving that I feel I have now.

This year I have taken Standards Based Grading by the horns. I am no longer marking students on assignments, but on the achievement of standards.  Now more than ever I can look at my grade book and know exactly where students struggle and where they need to focus their attention.  This was hugely helpful with Parent Teachcer conferences, and for once I felt as if I had something worthwhile to contribute, and to share with parents.

My parent communication has been much more open this year as well.  Every week I send an update e-mail for my classes, just to let them know what the next week will look like, and any tests or assignments that are due in the week.  I am also using Edmodo to post grades (obviously using the same standards as my gradebook). They love it and appreciate it, and I feel that keeping that connection with parents has established a stronger healthier classroom environment.

The Bad

Knowing exactly what I want out of classes are great, but since my expertise (what little expertise I have) lies mainly in Math (it is also my greatest passion) I become strapped for ideas in French and Science.  I am constantly finding cool new #anyqs and #wcydwt moments, but I do not know how to turn those into Science and French lessons, especially ones that work together as a whole.  In fact my greatest moment in Science class ever has only occurred when I stole, pretty much verbatim, Jason Buell’s introduction to matter.  It was excellent, and I need to find more of those units. I need to find more reflections of great educators that post these, so I can creatively steal and implement their ideas into my curriculum, but right now I am strapped for creating my own.  The story is even sadder in French.  I put so much love into Math, and so much time into Science, my French classes get the shaft.  This is what hurts the most because I do not want any of my classes to suck, but as many people tell me, sometimes you just have to make your one course be amazing, and work on the others later.  That feels like a cop out, and I don’t want to do that.  I want to find worthwhile resources that help me.

Since being a second year teacher means that I have some experience, but very little, I find myself reworking much of my curriculum. It is like I am a still first year teacher, just with a little more focus.  This means I am still making up a lot as I go, and in my classes I have a lot of need for educational assistance.  Working with my EA’s is fantastic, but it seems like every time I go to collaborate with them, I have nothing worthwhile to offer.  Our school has a beautiful inclusive model, and learning to operate within those parameters is very difficult, but extremely rewarding.  I enjoy the challenge, but I feel that week after week, I am struggling to develop curriculum that is accessible to all abilities and clear enough that my EA’s can understand as well and help where they need to.

The Redemptive

I have caught myself looking at other educators in my school and online, and thinking, “I will never be as good as them!” and it keeps me from moving forward.  That is not the attitude that I want my students to have and therefore it is an attitude that I sure want eradicate from my own thought.  The more that I develop these inquiry based problems, and the more that I think about SBG (specifically) the more I realize that I am simply on a journey towards being one of those teachers whom I admire.  I heard the other day somebody say, “We cannot compare ourselves to other people, because either we rationalize ourselves to inactivity or we send ourselves to despair.  What we must do is compare ourselves to our potential.” I love that idea, because if I judge myself based on MY potential I can only improve.  I can push myself beyond where I am, but I don’t have to make an unreasonable goal to be someone else. The gains that I have made this year tell me that I do have potential, and what I need is time to develop all of my abilities as a teacher.  I can’t wait to do that, and do it well!

I had a thought last night.  This happens from time to time, and I am only learning to post my thoughts as they occur.  I have been busying myself so much with Standards based grading, and I have been loving every second of it.  Let me tell you the truth SBG, to me, is not a reform because it is how I was taught to assess in my professional development.  I had a great professor that has dealt among her faculty many of the same things I hear strong advocates of SBG encounter.  "What about numbers?"  "Retakes just give out grades like candy!"  Blah-blah-blah.  This all being said I am a definite fan of SBG. I feel like it is powerful not simply as an assessor, but also as a curriculum guide.  When I make topic scales and outline specifically what I expect students to learn, I create a clear vision in my head not only what I want to teach, but how I want to teach it and how I want students to learn it!  What a fantastic method indeed.  This is where my thought comes in.  How do I turn the great assessment force of SBG back on myself? How do I get the super power inducing radioactive SBG spider to bite me, not just my student Peter Parkers?

I have become a firm believer in teacher modeling (no, I don't mean modeling instruction, though I love that too!), I simply mean that we must show kids that when teach them any important skill, we the teacher still hold that skill valuable in our own lives.  If not we are continuing the trend of getting grades for the sake of grades.  I had a great evaluation last year (one that I still need to write my reflections for... eek!), but it was a one time assessment!  I did not take into my own hands my learning, and outline very clear standards for myself that I could measure and improve.  I must out on a great chance to learn, to adapt, and to grow, and I also missed out on a chance to show my students that this skill is useful for the rest of their lives not just their K-12 existence.

So that is what I plan to do!  I need to become a self assessor just as much as my students do!  I need to track my progress, to see that each unit, I am learning from the mistakes that I have made, so that I can improve!  What is more, I do not want fluffy light-hearted goals.  No I want to make a scale of clear documented learning and skills outcomes that will guide me as a teacher, that I can assess myself (with the aid of my students), and that I can make a sweet set of skills.

These I believe will be my topics...

Instruction
Relationship
Assessment
Parent Communication

I almost included a half assed list, that would completely destroy my true goal, but I need to think about this more and stew on it before I 'git 'er done.  So instead I want to ask you, my superiors, what are your standards?  I know this is probably not a new idea for you, so please inform me.