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.]

 

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.

- 

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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Graduate Thesis on… Negative Numbers? November 22, 2010

My thesis is a study of the long-term effects the ZeroSum ruler has on eleventh grade student understanding of negative integers.  By eleventh grade, students should easily be able to answer “-22 + 5 =”, but on a diagnostic test given to 57 students, 40.35% of the students answered this problem incorrectly.  Why does this matter?  It matters because it shows that students did not learn the relationship between negative and positive numbers in elementary or middle school.  By the time they get to me in eleventh grade and need to be fluent in equation manipulation, answering “-22 + 5 = -27″ is a real problem.  

 

My thesis was set up the following way:

1: Diagnostic test: eight simple sums and differences of integers  (ie: ’22 + 5=”) without a ZeroSum ruler or calculator

2: Introduction to the ZeroSum ruler with examples

3: Three activities, spaced out over 2 weeks,  using the ZeroSum ruler

4: A post test within days of the last activity (no ZeroSum ruler or calculator)

5: A delayed retention test one month after the last activity (no ZeroSum ruler or calculator)

 

Because the attendance rates of students in Boston Public Schools is not the best, especially by the 11th and 12th grades,  a subgroup of 31 students was identified who took the diagnostic test, participated in at least 2 of the 3 activities with the ZeroSum ruler, took the post test, and took the delayed retention test.  The data shows a 62% decrease in student error from the diagnostic test to the delayed retention test.  These results indicate that the ZeroSum ruler works to improve student comprehension long-term even without the ruler.

 

Pretty exciting stuff.

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 “doublling” 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 gthe 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.

 

 

So, how much do I owe you? April 19, 2010

 

You friend borrows $22 from you.  He pays you back $15 the next day.  How much does he still owe you?  Asked this way, it’s obvious he owes you $7.  But give a kid the problem -22 + 15, and the answer mysteriously becomes, well, mysterious.  

 

WHY?

  

My students can certainly tell me how much I would still owe if I borrowed $22 and paid just $15 back. Like us, they’d probably count up from 15 to get to 22. But give a student the problem “-22 + 15″, and all bets are off.

  

For this number sentence, we are taught in school to find “-22″ on a number line and count to the right 15 spaces to find the number we land on. But this is not what we do in real life to find out how much someone still owes. There is a huge disconnect here.  In real life, we count up from 15 to 22, keeping a tally on our fingers of how many numbers we pass by.  We would never count up 15 from -22 to find how much someone owes us!  It’s no wonder students have difficulty with negative numbers with the way we are taught!

  

To plug my product, the ZeroSum ruler allows a student to count the spaces from 15 to -22 by folding the ruler in half at the pivot and counting from 15 to +22. When the positives are aligned with their negatives, they’re essentially finding the difference between the absolute values of -22 and 15.  This is the way we think and therefore a more natural way to learn.

 

 

 

Thank you Allan Cohen! April 5, 2010

 

Allan Cohen writes a blog called “Classrooms Without Walls” and gave me a shout out at:

 http://www.gather.com/viewArticle.action?articleId=281474978153616 .  How super!   Thank you Allan!