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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My Old Schoolhouse review debut! January 27, 2011

The ZeroSum Ruler was recently reviewed by The Old Schoolhouse Magazine!  You can read their full review – and get a glimpse at my old address where I fought a slumlord to the death of my career and almost me – at: The ZeroSum Ruler’s Old Schoolhouse Review!

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“Math is a subject that students can sometimes fake their way through. They might not understand how a problem works, but given the formula, they can follow rules and get things to come out all right in the end. Faking can only get them so far, though. Eventually, they will either forget the formula or not be able to recognize it when arranged in an unfamiliar manner. What a math teacher wants to see is the light bulb moment–when a student doesn’t just use a formula but understands why it works.

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Working with negative numbers is an abstract concept that many students have a hard time visualizing. How does one visualize what isn’t there? According to the website, the ZeroSum Ruler naturally brings this abstract “knowing” into concrete “showing”! This simple little device helps students see not only the negative numbers but also their relationship with other numbers.

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For instance, a student might not see how subtracting 10 from 5 is actually the same as saying 5 + -10. Visualizing the process with the ZeroSum Ruler helps students see that when they are figuring out a real-life scenario, such as how much someone owes them, they are really counting forward in positive numbers.

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The ruler itself is of laminated cardstock and is hinged at zero so that it can be folded, making the positive numbers line up with the negative numbers. This allows students to count forward the number they are subtracting or adding.

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The true gem of the ZeroSum Ruler is the creator herself. Shana is passionate about math and making it reachable for students. Her website contains math videos and commentaries that help students see that math is fun, interesting, and relevant. She breaks things down in an easy-to-understand method, and she is also happy to help with math questions from students and teachers.

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The ZeroSum Ruler is a great asset for students struggling with the concept of negative numbers. And its creator is a great help to parents struggling to teach those students.

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Math is a subject that students can sometimes fake their way through. They might not understand how a problem works, but given the formula, they can follow rules and get things to come out all right in the end. Faking can only get them so far, though. Eventually, they will either forget the formula or not be able to recognize it when arranged in an unfamiliar manner. What a math teacher wants to see is the light bulb moment–when a student doesn’t just use a formula but understands why it works.

- 

Working with negative numbers is an abstract concept that many students have a hard time visualizing. How does one visualize what isn’t there? According to the website, the ZeroSum Ruler naturally brings this abstract “knowing” into concrete “showing”! This simple little device helps students see not only the negative numbers but also their relationship with other numbers.

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For instance, a student might not see how subtracting 10 from 5 is actually the same as saying 5 + -10. Visualizing the process with the ZeroSum Ruler helps students see that when they are figuring out a real-life scenario, such as how much someone owes them, they are really counting forward in positive numbers.

- 

The ruler itself is of laminated cardstock and is hinged at zero so that it can be folded, making the positive numbers line up with the negative numbers. This allows students to count forward the number they are subtracting or adding.

- 

The true gem of the ZeroSum Ruler is the creator herself. Shana is passionate about math and making it reachable for students. Her website contains math videos and commentaries that help students see that math is fun, interesting, and relevant. She breaks things down in an easy-to-understand method, and she is also happy to help with math questions from students and teachers.

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The ZeroSum Ruler is a great asset for students struggling with the concept of negative numbers. And its creator is a great help to parents struggling to teach those students.”

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Thank you, Old Schoolhouse Magazine! 

 

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You can purchase a ZeroSum Ruler eBook here: The ZeroSum Ruler on CurrClick or on my blog over there —>

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Patterns in i? November 29, 2010

You can imagine my surprise at the end of last school year when, on my tutoree’s final online examination, the imaginary number i was everywhere.  “WHAT?” I thought, “There was just one small section of one small chapter on i in the textbook and here it is, on my students’ final exam, EVERYWHERE.”  At best, it was frustrating.  Sure, math is math, but different publishers tend to focus on different topics, and i was not on of those topics Glencoe included much of.

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For five years, I had taught Algebra 1 and loved it.  The kids loved me and I loved all of their “ooooh, I get it!”s.  But this year had been different because I was moved up to Algebra 2.  So I set my mind to teach this slightly more advanced Algebra (at least with Glencoe it’s only slight), brushed up during the summer, got my curriculum down pat, taught a rough year right up until the final exam and….

  

BOOM!  i! 

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Patterns are everywhere, especially in math.  The imaginary number i is no exception.  The number’s value follows an interesting and very distinct pattern, repeating itself every fourth iteration.  The pattern it DOES NOT fit is into a regular one in Glencoe’s Algebra 2 textbook.  I was mad that my students and I had worked so hard only to be sidelined by a final exam not connected to Glencoe at all.  

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So this year I changed.  I taught i first!  We wouldn’t be stopped!  If the “patterns_of_i.xls” sheet over there in the margin for you to download and use in your classes is not enough, I’d be more than happy to email you more.  You can reach me at sdonohue@post.harvard.edu. 

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I quit this year, jumped a sinking ship, really.  It was horrible leaving the kids- like I was going on maternity leave and never coming back.  But it was the decision I had to make so that I could focus on my thesis, my health and on finding a job where I would be respected.  What they say about finding happiness first before you can pass it on is true.  What they also say about not doing school part-time unless your job is also part-time is also true.    

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Now I’m finishing my thesis and looking for a new job, oh, and emailing you files to use in your classes.  I have thousands that I’ve made over the years that I’d love to share with you. 

 

  

Go i!

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Negative Numbers. OH NO! October 6, 2010

In our BPS high school, there’s a big focus on the “broken window theory”, made famous recently in The Tipping Point.  One broken window we’ve identified in the school as far as discipline goes is hats and ipods.  So, there’s been a big push to get rid of them.

 

I’d like to mention to you a “broken window” that has somehow gotten lost in the mess of school closings, going charter, union fighting, pension plans, longer days, MCAS scores.  As a high school math teacher, the biggest broken window I face – in fact, it’s a gaping hole not even bothered to be temporarily covered with plastic- is… negative numbers.

 

What do I mean by negative numbers?  I’ve done my research as they’re the topic of my Harvard thesis.  Students using the TERC Investigations curriculum in Boston elementary schools do not do problems like “-22 + 5″.  One TERC representative told me they “leave that topic to middle school”.  So, I looked at the middle school Connected mathematics Project 2 (CMP2) curriculum, and negative integer problems, like “-22 + 7″ are taught for 20 days total in the 7th grade.  20 days.  From then on, students are assumed to know how positives and negatives interact and to be able to evaluate “-22 + 5″.

 

Then students get to me, their 11th grade Algebra 2 teacher, and they can’t solve for y in “y + 22x = 5x – 7″ because they don’t know what “5 – 22″ is.  The kids think -22 + 5 = -27.  Why?  Maybe the rules of multiplication get mixed in.  I don’t know.  Or maybe it’s because these problems were taught to them for a total of 20 days four years earlier and were never touched n again except in the context of other problems.  Understanding why and how kids think is beyond the scope of my thesis and my means for data collection.  What I can tell you is that because my students don’t know what “5 – 22″ is, they can’t solve y + 22x = 5x – 7 for y.  Because they can’t solve the equation for y, they can’t graph the equation.  I assume you know where I’m going with this.

 

Please, as someone on the front lines of math education in Boston, I’m telling you that the biggest difficulty our students have in math is adding and subtracting positive and negative integers.  It seems ridiculous and that there are bigger fish to fry, some of which I have listed, but if you want more competency in math, please, heighten the focus on negative numbers.  It will lead to better test scores, more understanding, but most of all, to students who feel good about themselves when they’re not still making silly 7th grade mistakes in high school.

calculators KILL negatives! (uh, raised to even exponents, that is:) May 17, 2010

 

What’s negative 2 to the fourth power?  16?  -16?  If you put “-2^4″ into the TI-83, you get -16.  But we know that (-2)(-2) = 4 and (-2)(-2) = 4, and (4)(4) = 16.  So why does the calculator give us -16?

 

This post is no doubt for the high schooler and not for someone addicted to the )( buttons on the calculator like I am.  I parenthesize.  It comes from a fear that something will go negative that should be positive.  I have reminded my students more times than I can count to parenthesize, so many times, in fact, that I am more than sure that most tune me out as soon as they hear the first syllable.  But still the negative raised to an even number sneaks past the best of ‘em.

 

The evil negative base reared its ugly head again today when I graded papers on the geometric sequence an = a1 • r^(n-1) where:

an = the value of the nth term

a1 = first term’s value

r = ratio of change (ie “doubling” would be 2)

n = the terms placement (ie: 5th term would be n = 5)

 

“Find a7 if a1 = 5 and r = -2.”  The answer I or course got more than the correct answer was ” -320″.  What should the answer be?  “320″.  The problem should be written out first as: 5(-2)^(7-1) to make the process clear.

 

At least no one gave -1,000,000 as an answer.  There’s still hope!

 

overkilling negatives? May 8, 2010

 

I know the ruler seems a bit overkill for a simple subject like adding positives and negatives, but I teach 11th grade in Boston and it’s the biggest stumbling block for even my students taking my advanced algebra class.

 

The problem is that kids are taught a “noun-verb” way of solving problems like “-12 + 7″. They are told to find -12 (noun, static number) and count up 7 spaces (verb, movement) to the right to see what number they land on. This is fine in a classroom with a number line taped to the desk, but it doesn’t teach the kids how to think about the numbers and a lot of kids will get this problem, and ones like it, wrong. It only gets worse with “x + 12 = 7 (solve for x)” or “y + 12x = 7x + 3 (solve for y)”. It’s the same problem over and over again, just disguised.

 

The problem with the number line and the “noun-verb” way of solving is that it’s not the way we think. It’s not even the way we are taught in school to solve these problems. In the Boston 7th grade curriculum is a book called “Accentuate the Negative” where the very first page of text has a caption over a kid’s head that reads something along the lines of “I owe my dad $4. I have -$4″. So this business of “owing” comes into play very early.

 

If I owed you $12 (-12) and I only paid you back 7 (+7), how much would I still owe you? Asked like this, it’s a simple problem. You’d count up from 7 until you got to 12, knowing that the answer would be in “owe”, or negative. In school however, the kids are told to start at -12 and count up 7 spaces. This is completely backwards from how we think.

 

So to get to my ruler…. The ZeroSum ruler allows a kid to find -12, find 7, fold the ruler in half and count the space between the two numbers’ absolute values. This is what we do when we are finding out how much someone owes us, and this is really the way we think. In time, and to answer your question about what a kid would do with numbers beyond -25 and +25, a kid would start to see the relationship between positives and negatives and that if you “owe” more than you “pay” (if the negative is further away from break even (zero) than the positive) then the answer will take a negative sign. But it’s really the space between the absolute values we are counting.