Mean Math

I don't understand why the average is called the mean in math.  I'm probably missing something big and obvious, but really, "mean"?  No wonder math gets a bad rep.

When I teach this, it's like teaching a foundational skill.  There's a specific process for finding the mean, median and mode of a set of numbers, so I would need to be thick in the skull to say "EXPLORE!"  They would have no clue what they were looking for.

But even when instructing directly, I love this bit of learning because I can make the direct instruction interactive and fun.

First, I gave every kid a number on a paper.  That peaked their interest immediately.  The whispers started ... debates over what would be done. "It's centres!"  "No, these are our groups!"

I had them tape the number onto their shirts and go to the back wall.  So much confusion.  It was awesome.

I love confusion, because when I'm in control of it, the confusion breaks like clouds to the sun, and then a quick and clear understanding can emerge, giving the illusion of PURE GENIUS.  Read that as: the kids feel so dang smart and successful, that they're set up for success throughout the rest of the lesson.

So once I had them good and rustled up, standing all out of order, I told them, "Okay, so we don't really NEED to learn about mode and median this year (it's not in the curriculum), but to get to the MEAN, I think we need to see what these are.  So, I'd like you to get in order, so that you can figure out the mode a little more easily."

They looked confused, but did as I said.

How adorable are these mathematicians?

I said, "The mode is the number that repeats most often.  Look at this data set.  You guys are the values.  You're each a different number.  Which number repeats the most often?"

They answered: "FOUR!"

"Excellent.  You're BRILLIANT."  Uncomfortable laughter.

"Now, find the median.  Anyone know what that is?"

STUDENTS: "The number that repeats the LEAST often?"  ...  "No, no one cares about that number." .... "Maybe it's the biggest number?"

ME:  "All great guesses!  The median is the middle.  If we line you up in order, it's the number in the middle.  When I point to you, you're out.  You need to squat down."  I systematically pointed to the end people on either end until we reached the middle person/number.  Fortunately, we have 21 students, so there IS a middle number.

Hmmm ... what will the median be?  I'm picking them off, one at a time!

They were able to see the last person standing was the middle number.  BOOM.  Done.  Check.

But what if the middle number is an even number?  I took one student out and repeated the process.  They were (surprisingly) shocked to find that there was "no" middle number, because there were two people left.  And so began the real lesson: how to find the mean, because the mean of those two middle numbers would give them the median number.

Rather than go right into the math demo, I had them go back to their spots and watch this video:

<iframe width="420" height="315" src="http://www.youtube.com/embed/QH2obAPwfqk" frameborder="0" allowfullscreen></iframe>

That was really to grab them even more and brief them on the process.  They thought it was funny and cheesy, which it is.

I gave them the demonstration of finding the mean flat-out, no secrets, and did two examples.  Then, I asked them to watch another video:

<iframe width="420" height="315" src="http://www.youtube.com/embed/oNdVynH6hcY" frameborder="0" allowfullscreen></iframe>

This was even more clear on the steps: Add up the numbers in the data set, divide the answer by the number of values.

We did another example, this time where I threw in a number to throw off the average.  I wanted them to think about the fairness of the average/mean.  The numbers were representative of their summer jobs:

Lawn Mowers: $30/day
Babysitters: $25/day
Dog Walkers: $20/day
Loungers: $1/day

The loungers really messed it up.  They found (with me guiding the SMARTPen) the mean to be $19/day.  I asked them if that really represented what most of them made.  The answer was NO! Because, it was less than three of the jobs could earn a day.

So, we removed the loungers and found a mean of $25/day.  They agreed that this one made sense.

Then, we talked about averages in school, especially in high school, helping you decide whether or not you will get into certain universities or colleges.  We brainstormed eight classes they could take (including "Potato Shop" ... I have some weird kids), and I assigned grades to them all.  I asked them to find the mean and tell me whether or not they'd get into their dream school, if they needed to have an average of 85.

For some, it was a one-minute find.  For others, it took ten.  Some wanted to show everything, including the mode and the median.  In general, we're off to a good start.  Yahoo!

Financial Literacy (Making it Real)

I believe that part of teaching math is making it real.  Numbers and concepts can get lofty and lose meaning very quickly.  We are always saying that we need to make the learning authentic, but I have to wonder how much we truly follow that idea beyond our first year in the classroom.

As we have been working with equations, I found my students enjoying and finding success in the ones that were related to money: how much could you make if, or how much was earned, and so on.  I decided to take it a step further.



I put my personal monthly expenses on the board.  As in, I put my ACTUAL monthly payments on the board.  Rather than just some random numbers, I wanted the kids to see how much an adult spends, without the fun.  If I've learned anything from financial guru Gail Vaz Oxlade, then it's that to be financially successful you need to be honest, share your finances with people, and learn early.  I don't know many professionals who would put it all out there, but I like to live wild, so I did.





The job of the students was to find out how much I spent in a month, and how much I spent in a year.  The operations were fairly simple, but the take-home pay would be anything but.  The kids jaws were literally (not literally) on the floor when they got through adding it all up.  Then, the discussions started on how to find the total for the year.  After some debating over how to get there (aka, multiply by 12) in some of the groups, and the kids feeling the need to recheck because the outgoing expenses for a year were SO HIGH that they couldn't fathom the numbers, they came up with their totals, wrote the equation to represent the work, and explained how they knew they had come out with a logical answer.

This was incredibly eye-opening for all of the kids.  Most had no idea that adults spent so much.  But I wasn't finished.  They couldn't believe it when I projected my last paycheck up onto the board (note: there's some media literacy ... reading a complicated check like that!) and showed them how much I make in a year.
Their jaws dropped!  They were amazed to see such a large number.  "So, I'm rich, right?!"  But that's when my critical thinkers piped in, "No, you're not.  Your bills for the year are XXXX and you only make (insert "large" number here)."


To which I replied, "Why yes!  You're correct.  And let's look at my take-home pay for this 2-week period."  So we went through it all, including ALL of the deductions (nearly half the check) and eventually got to the amount that was deposited into my account.

We had the richest discussions after this.  One boy, who had for weeks earlier this year complained about how much teachers make, said passionately, "I always thought you made so much BUT I NEVER THOUGHT ABOUT HOW MUCH YOU PAY EVERY MONTH!"  He was in shock!  Another girl asked, "If you were to buy a bigger, newer house, and your mortgage went up, would your house insurance go up, too?" Why yes, yes it would.  They all gasped.  Then I said, "Oh, I didn't include the amount I pay on my credit card."  We discussed credit cards and interest, student loans and debt, and at the end of the period, it would be hard to describe the general air.  The kids were intrigued, but troubled.  They were feeling ... I don't ... at the thought that their teacher had showed them his pay check and shared his monthly expenses with them.  They were talking amongst themselves about needing to save money.



I think that this was a great introduction into financial literacy.  The idea that at ten years old, they can think independently about money and how to use it wisely is inspiring to me, but I think it takes authentic, real connections.  It's not enough to say "What if you made X and bought X?" or "In your babysitting job you make X, so how much can you make in X # of months?"  That may be something interesting to get them find patterns and equations, but it's not going to make them critical thinkers.  I'm not saying don't do that - I do it all the time.  But in this case, it was about teaching them to think critically about money ... even if they are ten.  It's never too early to start making good choices when it comes to money, especially in today's gotta-have-it-and-pay-with-credit world.

Parameters in Problems (That's the Problem)

The problem with math is the wording of problems.

The longer I teach, the more I realize that I don't know what I'm doing.  No - wait.  That's not flattering, is it?

I do know what I'm doing when it comes to: planning and plotting, marking and giving feedback, building healthy, positive relationships with every student.

But, I don't know what I'm doing when it comes to: letting them find the way.

I think that the biggest problem we, as teachers, have created for ourselves is that we don't let them find their way.  In fact, we have created so many parameters that we have all but cornered ourselves into teaching.

Yes, there is a curriculum.

No, it doesn't need to be followed from expectation #1 to expectation #1million or whatever the last one is.

Parameters is something that clicked for me during a Hub.  I realized the problem with our questions was that we:

- assume that they know the parameters that we know, because we looked at the curriculum and "designed" the question in a brilliant way

- get upset when they build their own parameters and don't go the direction we were expecting

- get upset when we want them to build parameters and they don't because they don't know how or think that they're allowed, because we got upset when they did it before


Phew.  Take a moment to let that all soak in.


So what do we do?

I've started trying to design questions that have missing information.  I'm hoping that by allowing for - and discussing with the students - the parameters that are missing, that are present, and that are IMPORTANT, some of this silliness can be avoided.  Afterall, if I pose a question and the students solve it according to their parameters - they do the leg work, can prove their thinking, and are finished shortly thereafter ... what's the problem?  It didn't hit on the right curriculum expectations?  Not to be crass, but hey, shut up and be happy.  Make a note in your anecdotal records, where ever you keep them (mine are in my phone under the Memo app), identify what expectations they DID meet, and move on.

Here's an example of a problem I gave with little parameters.  I wanted to see where they went:

Create an equation to show how many days of school the students attended, if there are 194 days of school.

Note: We've been working on equations, symbols and missing values, constants and variables.  This was all highlighted before we got to the question.

I wanted to see where they would go.  I knew some would do this:
- how many days did ONE student go to school?
- how many days did OUR CLASS go to school by the end of the year?
- Where is the attendance?

None found the answer for the students in the entire school.  Most focused on the FINAL ANSWER and NOT the equation part, which was really the only concrete parameter I gave.  But that's okay, I took the info and incorporated it into the next day's problem, and identified it during our wrap-up.

Most students looked at attendance, made tables and charts, and found out how many days of school had been attended to date.  THAT is good, since we obviously don't know about future snow days or sick days.

One group identified this equation: 21 (students) x 194 - DM (days missed)  Perfection. Sorta. 

This question is WIDE OPEN.  On purpose.

Here is a LESS wide open question:

Build an equation to show how much Mr. Patrick makes, if he earns $10 an hour.  (Note: You can change it to challenge yourself.)  How much will he make in:
- a day's work?
- a week's work?
- a school year's work?

This was surprisingly difficult for the kids.  Not the finding the answer part, but the finding the equation part (which in this case, is the answer, actually).  Of course, I know I can say "If I make $10 an hour and work 7 hours a day, I will make $70/day.  So, take the days worked and multiply that by the amount I make: D (days) x $70 = answer.  But, different people set up different parameters.

They wanted to know:
- how long do you work?
- how long is the school day?
- does it include breaks?
- do you get paid if you do more?
- do you get paid if you stay late? come in early?
- do we count weekends?
- are holidays part of the 194 days at school? should I take them out? what about snow days?

And so on.  It was impressive.  They built rules for themselves, and found answers that suited their own set of rules.  I still got what I needed from them, even though many didn't quite make it to that FINAL ANSWER.  It's okay.  It was a positive experience, and I got to see (from the two days featuring these questions) that they have solid skills to find answers, even if it is not as quick and efficient as using an equation with constants and variables.

End of the world?  No way.

Let them eat cake! I say.  Oh, and let them build their own parameters, which are suitable and informed and make sense and are important to the problem, and then we'll all be happy ... and we can eat cake.  Who wants to split it into fractions for me?

Phrases That Worry Me

When teaching in general, there are certain words, phrases or combinations of them that worry me.  I can apply these directly to teaching math, as I myself have used them in the past, and I hear them from others, either when passing the doorway in the hall, or in conversation about what the students are doing.

These are some of my favourites; the ones that worry and bother me the most (especially when it slips out of my own mouth):

- "This isn't hard."  Maybe it is.  It isn't hard for us, because we're adults.  But maybe the concept, the numbers, or something else is hard.  Maybe there are outside forces that we don't know about making it difficult for the child.  Maybe it shouldn't be hard, in our perspective, but if a child struggles over something, simple or not, then isn't it my job to make it not hard?  Isn't the root word of "teacher": TEACH? 

- "I've already taught them this, so ..."  So what?  Maybe you taught it, but did they learn it?  Did they really, deeply learn it?  Did they experience it beyond a lecture, a trite activity, or something that made sense in my/your mind?  You may have already presented it, but it doesn't mean they know it.  And that is the rest of the job.  A teacher might teach something, but until a teacher has observed, made adjustments, and ensured a solid understanding of the materials, the job isn't done.  Even if the lesson was already taught.

- "I haven't taught this yet, so ..." So what?  Maybe they know it.  Maybe it's simple for some of them.  Maybe our words convolute things and confuse them, and then we start saying, "But I already taught them this," even though they already knew it.  Meaningful, open diagnostic assessment not only saves time, but can act as a teacher in itself, if not for the students, then for us as teachers; in fact, something that a child creates may end up teaching the other students in some way, at some point.

- "That child is a 3." Or a 2, or a 1, or a 4.  In other words, a B, or a C, D or A.  You know what?  You're probably wrong.  Maybe your assignments only allow the child to achieve that.  Maybe the environment, the classroom climate, only allows it.  Maybe that child normally scores a C, but actually gets it.  Our jobs as teachers are to teach kids, not to keep their level steady with what every teacher before us has pegged the child as.

There are millions more, but I'll leave it at that.  I know I've said these things in the past, but the more I learn about teaching, the more I learn that I don't know as much as I once thought I did, and the more I learn that teaching is a careful art - a dance that requires two-way communication, flexibility and empathy. 

What ARE We Teaching?

Today I attended a hub.  If you don't know what that is, here's the quick & dirty explanation:
Small groups of teachers, from various schools, meet up monthly for a day to inquire about learning.

Being inquiry based, we start with some sort of a plan, and led by fearless leaders, we go into a classroom and observe the students.  Then, we come back to the table, discuss what we saw, and usually make some minor changes to the lesson, and try it all again.  Today we really got into looking at "what comes next?"  I'm happy about that.  I needed to know what came next.  I feel comfortable with the classroom climate, student voice and problem based learning in my classroom.  What I need as an adult learner now is just that - now that I have this data, what do I do next?  And am I really seeing what I think I'm seeing?

Today revealed some amazing things.  We were in a Gr. 7 classroom, and the whole learning continuum really presented itself today.  The students were looking at a problem with fractions, but the question didn't explicitly state that it was about fractions.  I wonder if some of them even knew it was?  As we let them tackle the problem, we realized that there were some missing pieces.

This group found out from another that they could divide the two numbers to find the decimal.  But, they didn't have enough fundamental understandings to be able to identify what each number meant, or what to do once they'd found every decimal.  For this problem, and for their fractional understandings, that's a big gap that we can now fix, because we found out about it.

We reconvened and discussed the triumphs and struggles.  We got into smaller groups and presented our own observations, and then categorized them.  My group came up with headings like "Classroom Climate," and "Assumptions," and "Misconceptions" and "THE Math - Number Sense."  It was fairly obvious that as a whole, the class was missing major fundamental pieces, which were required in order to do THE math.

Working together to look at our observations, wonders & student quotes to find some common themes.
Quotes like "I'm so confused ... I'm getting mad.  I'm gonna spazz out!" were very telling, and informed us that some students are putting up walls ... so how do we target that now for them, and how do we target that for students who are beginning to feel that way now, in primary & junior grades? 

After our small group sharing and categorizing, we got together to see if we could group what we'd found:
were there common themes happening?  And then, how could we possibly target them?
WHAT COMES NEXT?

After much tribulation, we went back in with the prompt:

Design something to help the Gr. 4 students at (my school, not the highschool) learn about fractions.


Well.

It was a dud, I would say.  While some groups had some ideas, I think it is fair to say that they couldn't access the information without going to the internet.  That's a great strategy, except ... I don't think they were doing much other than regurgitating the information they'd found.

So now I'm left with these wonderings ...

- What ARE we teaching in primary & junior?
- Why AREN'T the fundamental concepts sticking with them?
- Is our teaching and their learning TOO compartmentalized?
- Is this because of text books?
- Is this because many of us have a discomfort with things like fractions, so we spit out the words, they "parrot" it back (to borrow the term from one of the resources we looked at today!), and then we say "Check! Moving on!"?
- Is this because the climate surrounding math has been, as a whole, stuffy?  Overwhelming?  Anxiety-ridden?  Why?
- Is this because they just can't remember?
- Do they have the language to get to the language or strategies that they need to understand it all?
- Are they worried about using the right words and pictures to please the teachers?
- Do we even know what kind of understanding we want to see from them?  I'm not sure I did.

I wonder if manipulatives (for those non-teachers reading, I mean the thingies they use to solve problems, like counters and blocks and so on) are the issue, too?  Do the kids stop using them in junior, and then not want to use them in intermediate, because it means that they're unable to solve the problem with their heads?  If that's it, we've got some major work to do around our classroom cultures.  I wonder if manipulatives looked cooler and more "adult" (maybe metal or polished fine wood blocks and counters instead of primary red, yellow and blue), would older kids be more comfortable with them?  I wonder if these kids were uncomfortable because the problem wasn't the kind that they are used to, and so they have trouble accessing the strategies that they need to solve it?  I don't know - I'm not the teacher and I don't work in the school.  I wonder if the kids would be more able to solve the problem if there was something more motivating about it?  I don't know - I don't know them and I don't know what motivates a 13 or 14 year old - I've only taught up to 11 years old.

So many wonders.  So little time.