5 Lessons from Living in Paris – as applied to teaching August 19, 2012

I have never lived in Paris.  A good friend of mine went to school in Paris for a semester, but that’s about as close as I have ever gotten to the City of Lights.  I know about the Eiffel Tower and that Parisians eat all day and never get fat, but I had never been drawn to the City until I read this article:  “Five Lessons from living in Paris” by A. Pawlowski chronicling the learned lessons Jennifer L. Scott. 

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As I read about Scott’s experience, I think about how her lessons apply to being a teacher: 

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1: “Live a passionate Life” - When we’re passionate about Shakespeare, Roosevelt, Telophase or Acute Angles, our kids will be, too.  You know that person who can tell a story about going to the grocery store to get a Redbox movie so passionately that you can’t help but listen?  There is nothing in his mind screaming, “this story is not worth telling!” so the story, as mundane as it may be, comes off worthwhile and super interesting.

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2: “Cultivate an air of mystery” - This comes back to Maxwell’s Demon’s principle 4 of his Five Principles of the Evil Teaching Guru.  As much as it sometimes seems as if kids just want the answer, it’s so important for them to find, with some help, answers to their questions on their own.  They’ll be more interested and invested in what they are doing.

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3: “Look presentable always” – I immediately think of three people when I read this: my Brother, a Math Teacher in Worcester, Massachusetts; 2012 Massachusetts Teacher of the Year Adam Gray who teaches Math in Boston; and my husband, a wonderful Third Grade Teacher in Waltham, Massachusetts.  All three wear ties everyday.  It may seem simple, but dressing up says to the kids, “Hey, you’re worth looking my best.”

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4: “Don’t forget the simple pleasures” - This one is all for us.  I will never forget a lesson I learned during my first year teaching in Boston.  This lesson came from Mike Pfaff, then a Biology Teacher and now a Hollywood Actor, who told me to “Hold on to the small victories”.  Some days when its easy to fixate on the kid who refuses to open a notebook, we have to remember the kid who thanked us yesterday for a good class or got his first homework 100% of the week.

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5: “Make Life a formal affair” -  A former principal of mine told me a story about a kid who had been sent to his office.  When the principal asked him why he didn’t want to be in class, the kid explained that he was uncomfortable because there was free-seating and he had no friends to sit next to.  For this kid, free-seating was an issue.  As mundane as it seems, kids find comfort in knowing that when they walk into a classroom there will be a warm-up, followed by homework review, followed by a mini-lesson, followed by a class assignment, followed by homework assignment, and they are doing all of these things from the comfort of their assigned seat.  Kids will never admit it, but they want and need this formality.  When the daily routine is set, kids are more willing to take risks in the class material (lesson #2).

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Each summer, like all teachers, I set goals for myself for the next school year.  This year I want to remember these lessons from Paris.

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This is why kids hate math July 3, 2011

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A former student is studying for his college placement test and I am helping him.  Below is one of the questions on BHCC’s online practice test:

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• In triangle ABC, a = 12, b = 16, and sinB = B. What is the measure of angle A in degrees?

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The college gives the “correct answer” as 30 degrees.  Is this mathematically possible?

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UPDATE:  The college emailed me back and will be taking down this question from their online practice placement test.  It is a faulty question.  I wonder how many kids have seen this and thought they were too dumb to answer it.  These are the types of things that make kids hate math and my blood boil!

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Dividing Fractions With Pictures! June 8, 2011

Of all my posts, this one gets the most hits.  I think I know why.  Fraction division seems like it should be simple.  Afterall, ”flip the second fraction and multiply across” is a complete cake walk.  But when we have to explain the process to a kid (or an overly-inflated interviewer), things can go very wrong.  Why is it so hard?  Recently, I met a new friend, Chris Fink, through my blog.  Chris teaches Math in the California penal system.  Through a series of emails back and forth, we both came to a better understanding of this tricky process.  She was able to explain fraction division to her inmates (they all clapped and thanked her - yes, her - afterwards!) and I came to understand how to show the process through pictures a lot better thanks to her.  I left my old post underneath the new stuff because, though wordy, it does give a bit more explanation.  The following three screenshots (you can download the pdf here or by clicking on one of the three screenshots) are a decent start to How we show fraction division thorough pictures.  Thanks Chris!

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Fraction Division: Not Just a How

Dividing fractions has got to be the algorithm we most often take at face value.  The How – flip the second fraction and multiply across – is easy, while the Why can fill an entire chapter.

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Whenever we divide, we’re asking “How many groups of this will fit into that?”  With, for example,

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10 ÷ 2, we’re asking “How many groups of 2 will fit into 10?”  This is easy:

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Here’s 10:

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Here’s a group of 2:

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We can easily see that 5 groups of 2 will fit into 10:

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Unlike multiplication, division is not commutative.  We cannot divide backward and forwards and expect to get the same result.  For example:

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10 ÷ 2 ≠ 2 ÷ 10

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We always “flip the second fraction” in the fraction division algorithm as, contrary to logic, flipping the first fraction instead will not yield the same result.  For example:

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The first number in a division problem is simply more important than the second number.  The first number sets the stage while the second number asks, “How many groups of me will fit into your first number?”  In 10 ÷ 2, we weren’t putting groups of 2 into any old number; we were putting groups of 2 into 10.  We needed to keep the 10 in mind as we bundled our groups of 2.  Division with fractions operates in the exact same way.  Whenever a fraction is divided by another fraction, one of two possible outcomes occurs: a fraction less than 1 or a fraction greater than 1.  Of the fractions greater than one, answers can be either whole numbers or mixed numbers.  Whenever we deal with parts of wholes, things get interesting.

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We’ll start with a simple example where the result is a nice, easy whole number:

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If we ask “How many of the green pie piece will fit into the blue half of the circle?” we can see pretty easily that 3 will fit in perfectly.  If we were to superimpose the green pie over the blue one, the centerlines on both pies would line up nicely, creating a common denominator of 6.  Unfortunately, not all fractions superimpose over each other so nicely.  To develop a pattern that we can use with more difficult fraction division problems, let’s look at 1/2  ÷ 1/6  in a slightly different way.  First, we’ll set the stage with  1/2:

Here we have a circle and we colored half of it.  This next part is where things can get weird.  Remember how, in 10 ÷ 2, the 10 set the stage before we began bundling groups of 2?  If we instead thought about 10 ÷ 2  as  10/1  ÷  2/1 , we can begin to see why this problem was so easy: the 10 and the 2 already shared a common denominator.   Just as we did there, we’ll create a common denominator in this problem.  The easiest way to do this is to superimpose the 1/6′s denominator atop the 1/2 and see what shakes out:

When we divide the entire region into 6 equal pieces, essentially turning  1/2  into 3/6  , it will become very easy to then take  1/6 :

Just like in 10 ÷ 2, we now ask “How many  1/2 fit in  1/6 ?”  In other words, how many green pie pieces fit into the original blue  1/2  ?

3 do.  And in fact,  1/2  ÷  1/6    =   1/2  •  6/1    =    3.

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Fraction division hasn’t earned its own post based on easy problems like  1/2  ÷ 1/6 .  This problem was easy for a couple reasons: the answer was a whole number and it was very easy to create a common denominator.  Next, let’s look at a slightly harder fraction division problem in a still slightly different way:

This problem is more difficult for a few reasons.  First, the result will not be a whole number.  Second, the result will be a fraction less than one.  Third, the denominators 4 and 6 don’t overlap very easily, so we’ll need to create a common denominator that is larger than both 4 and 6.  We’ll deal with these first two reasons as we work through the problem.  To mitigate the third reason this problem is more difficult, we’ll create a larger common denominator.  Fortunately, this larger common denominator will appear naturally as we begin to draw the problem.

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First, 3/4 :

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To show 3/4 , we divided an area into 4 columns and pink-boxed 3 of them.  Keeping the entire area in mind as we have done before, we will now get ready to take 5/6  by first creating 6 rows:

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and then coloring in 5 of the rows:

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By using columns to show the first fraction, and rows to show the second fraction, we naturally created a common denominator of 24.  This will happen every time we use the column and row method.

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Now let’s ask our question: “How many   5/6′s (orange boxes) fit into  3/4 (pink outline) ?”  In other words, how many of the orange boxes from the group of 20 will fit into the pink-boxed 18 area?  So it’s a bit easier to visualize, let’s move as many orange boxes inside as will fit:

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18 out of the group of 20 orange boxes will fit.  And in fact,  3/4  ÷ 5/6    =    3/4   •  6/5    =    18/20

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So far we have seen a whole number answer and a fractional answer less than 1.  In this third example, we’ll look at the last type of fraction division problem – one that yields a mixed number.  This next problem was asked of me twice during two different interviews for middle school Math teaching positions in the Boston Public Schools:

Because we’ll need to create a larger common denominator here, as 2 and 3 don’t easily overlap, we will use the column and row method.  Starting with  1/2  :

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we’ll get ready to take  1/3  by dividing the entire area into 3 equal rows:

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and coloring 1 of the 3 rows:

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We see a group of 2 blue boxes.  As we’ve done before, let’s move the one on the outside into the inside:

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2 of the 2 blue boxes (2/2) will fit into the pink-boxed area.  Additionally, another 1 out of the 2 blue boxes (1/2) will also fit:

And in fact,  1/2  ÷  1/3     =    1/2  •  3/1     =    3/2

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With the new Common Core Standards, kids are being asked to divide fractions beginning in 5^th grade.  As with anything, once we develop a pattern for fraction division, showing the process with pictures becomes easy.  Once a kid can see and feel what is happening in these problems, the process of dividing fractions will begin to make more sense.

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Please check the comments below for some good additional information on why the division algorithm works.

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Also see Multiplying Fractions with Pictures!

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Also see Multiplying Fractions with Pictures! and Differences of Squares with Pictures!

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You can download a PDF ebook that uses pictures to explain fraction division, multiplication and addition on CurrClick at Fractions: A Picture Book!

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contact blog author Shana Donohue: shanadonohue@gmail.com

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My Harvard Math for Teaching Thesis: Complete! And ready to share… March 20, 2011

After many many years of jumping through many many hoops, I am finally graduating with my MA in Mathematics for Teaching in May.  My thesis, Negative Number Misconceptions in High School: An Intervention Using the ZeroSum Ruler is right now at the printers being printed and bound.  I don’t know about you, but that instantaneous feeling of relief after taking a final exam or passing in a final paper stopped hitting me sometime in college.  So now, I’m just feeling a bit burnt out.  OK, completely burnt out.  But I’m sure it will hit me soon since it kind of needs to; I need to now get in a post-Bach program to get my Initial teaching license.  I like to do things backwards.

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So here it is for download!  For all to read!  Or maybe to just glance.  In my study, the ZeroSum ruler proved effective in reducing eleventh grade error on integer addition and subtraction problems (especially with negative integers).  If I wasn’t so burnt out, I’d want to test it with younger kids.  Imagine how our world would be if my eleventh graders actually mastered integers when they learned them in, and only in, 7th grade.  But that’s in my thesis.]

 

ZeroSum ruler’s 62% success rate! March 9, 2011

 

The ZeroSum ruler improved my student’s understanding

of integers

by 62%

in a very short 2 weeks. 

Surpassed even my high expectations!

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HaPpY Calculating!

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Public school? Supersize my class! January 25, 2011

There are two upsides of a larger class: more diversity among students and the drive to succeed (and/or not act out in front of peers).  I found that when I was teaching if too many students were out on a day, the kids who were in school felt it should be a “free day”.  Kids weren’t there to bounce ideas off each other or to push each other. 

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So does class size matter?  It definitely does.  Even when I had less students in class and the ones who were there were less motivated to do work, I was better able to connect to the students who were there.  There were less kids to reach.  And isn’t that what good teaching is half about?

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Read the article The Class Size Debate on Huffington Post.

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Let’s grow some grass! November 7, 2010

 

Who knew?  Students love growing grass.  Let me elaborate…. Students love taking care of their grass as they watch it grow- enough to get them to do some pretty complicated algebra.

 

What started out as a simple week-long project that incorporated a bit of environmental sciences (my undergrad background) into algebra became a 10-week long project spanning the curriculum from ecology to linear extrapolation.  It’s been a long time in the making, but beyond a shadow of a doubt this lesson is one of the most engaging that I have created.   My students love the life aspect of the project and hardly complain about doing some pretty complex algebra.   

 

Now, with the magic of a WordPress widget called ”My Shared Files    BOX” (on the sidebar), I was able to upload the Growing Grass Project files onto my blog for all to use! 

 

All three files are important, but the excel workbook includes everything the student needs to create a final portfolio piece, including a formatted final excel sheet that the student can type into and cell directions. 

  

I’m very excited about this and hope that if you do use the project, that you will add a comment to this post on how it went.  I also have other files that go along with the starter ones I posted. 

 

I guarantee that your students will be engaged in their learning and that you will find ways to link most of algebra 1  to what comes up along the way.  

 

WARNING!  This lesson takes on a life of its own!  Proceed with caution! 

 

Now Let’s grow some grass!

 

 

p.s. For supplementary files, you can email me at ZeroSumRuler@gmail.com.  The files include ones on scatter plots and lines of fit as well as a PowerPoint and activity on linear inter- and extrapolation.