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.

Is Quirky a scam? November 18, 2010

 

I paid my $10. 

I wrote all about the ZeroSum Ruler. 

I submitted a picture, wrote my bio, wrote all about the ruler, and…..

 

Nothing.

 

No really, what is “Quirky”??

 

Still, I did manage to see that five people, probably unemployed and bored, decided to click on my product number to see what it was.  Here are their comments and my replies:

 

Becky Blatchford:

“I like this. If it is inexpensive enough say under $10 I’d bet moms would buy this up!”

Me:

“Thanks Becky.  Hopefully they will when more people find out about it.”

 

socrtwo:

“Interesting and simple.”

Me:

“Thanks scortwo”

 

Michael Kloeckner:

“I love it, I cannot believe this is not out there somewhere”

Me:

“Thank you Michael. Now only to find a venture capitalist to back production and marketing!”

 

Dave McKey:

“I agree with Aaron. no need to fold the ruler.  good luck!”

Me:

“Thanks Dave.  I’ll think about the fold.”

 

Aaron Dale:

“I watched the video on your website and the only question I have is, why fold the ruler? Just make the ruler so it has negative numbers along one edge and positive along the other edge. But if a kid can use a tool to combine negative and positive numbers, they should use a calculator. It would be best for the kid to understand why they get the answer they get instead of counting the difference on the ruler. It seems like cheating.”

Me:

“I see what you’re saying about the folding.  it would be a lot easier to make without the fold.  The reason I made it fold was so that it starts out looking like the normal numberline the kids use in school.  The folding then shows the relationship between negatives and positives.  When a kid tells me -22 + 5 = -27, there’s a problem.  When this kid is in 11th grade, there’s a bigger problem.  When this kid is not just one kid but many, we’ve got a huge problem.  My thesis results are in and the ruler caused students to make 1/2 the mistakes as before the ruler.  True, data isnt’ everything, but I never thought it woule make such a difference. ”

 

So what is Quirky?  Can I build a website, undercut Quirky by charging anyone with an idea $8 to write all about it only to have it disappear into a “relist” button one week later?  I mean, everyone has an idea, most everyone has $8, this is the best scam going, right?

 

Am I missing something?

 

 

 

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.

 

 

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.