Randomly Grouping Students

My college students are like charter members of a church. They claim their seat on the first day of class, and from then on, no one else better take it! Since we journal every day, often in groups, the same people were sharing with the same people day after day. This meant the students were not getting to know each other; they were unaware of how others were problem solving, and they were way too comfortable in their group. Things had to change!

So-o-o I asked my students to make name tents by folding a large file card in half. (I know it sounds elementary, but it does help this "old" teacher to quickly learn who is who). On the outside, they printed their name and on the inside, I placed a variety of stickers.  (My students didn't seem to mind.) Based on the sticker I called, the students would group by pairs, threes, fours or groups of eight.  Now, the students would divide up into groups based on something other than their preference.

On Monday, we grouped by 2's according to the color of the dog and cat stickers. Right away I noticed that the dominance of the group had changed, and more dialogue was going on between the partners. We then regrouped to present problems using the order of operations, only this time I used the stars to make groups of 3's. (I used this same idea when I taught math in middle school and high school only the stickers were put directly onto the students' journals which always stayed in the room with me.)  

I also used this strategy when I taught elementary (back in Noah's Day, after the flood), but there was always one or two "sticker pickers" in my class which seemed, in some magical way, to remove the sticker from their desk. To alleviate this problem, I placed the stickers on the desks and then covered them with clear packing tape or contact paper which was not easily removed. If a student moved away, I simply gave the new student the vacant desk or grouped the remaining students according to a different number.

Want to give this a try this in your classroom? Just purchase a variety of stickers. Then decide on the size of the groups you want such as 2's, 3's, 4's, 5's etc. and get to work!

How Gritty Are You?

What are the causes of success? My college students in my Math Study Skills class have been researching this topic since each one of them desires to be successful at math. We watched a six minute video by Angela Lee Duckworth: The Key to Success? Grit on You Tube.  She relates how she left a top paying job in consulting, to teach math to seventh graders in a New York public school. She soon realized that IQ wasn't the only thing separating her successful students from those who struggled. In the video, she describes her theory of "grit" as a predictor of success.  Below is a summary of what she says.

At first glance, the answer is easy: success is about talent. It’s about being able to do something – hit a baseball, play chess, write a blog – better than most anyone else. But what is talent? How did that person get so good at hitting a baseball or playing chess? For a long time, talent seemed to be about inheritance, about the blessed set of genes that gave rise to some particular skill. Einstein had the physics gene, Beethoven had the symphony gene, and Tiger Woods (at least until his car crash) had the golf swing gene. The outcome, of course, is that you and I can’t become chess grandmasters or composers or golf pros because we don’t have the necessary anatomy. Endless hours of hard work won’t compensate for our biological limitations.

But think about this - Beethoven wasn’t born Beethoven.  He had to work extremely hard to become Beethoven. Talent is about practice. Talent takes effort. Talent requires a good coach. But these answers only raise more questions. What, for instance, allows someone to practice for so long? Why are some people so much better at deliberate practice? If talent is about hard work, then what factors influence how hard we can work?

It is deliberate (conscious, intentional, planned) practice that spells success. In other words, deliberate practice works. People who spend more time in deliberate practice mode perform much better. The bad news is that deliberate practice isn't fun and is consistently rated as the least enjoyable form of self-improvement. Nevertheless, as golfers, musicians, etc. gain experience, they devote increasing amounts of time to deliberate practice, and consistent, deliberate practice is done by grit. Not surprisingly, those with grit are more single-minded about their goals – they tend to get obsessed with certain activities – and also more likely to persist in the face of struggle and failure. Woody Allen famously declared that "Eighty percent of success is showing up." Grit is what allows you to show up again and again

While grit has little or nothing to do with intelligence (as measured by IQ scores), it often explains why an individual is successful. Thomas Edison was right: "Even genius is mostly just perspiration."

Our most important talent is having a talent for working hard, for practicing even when practice isn't fun. It’s about putting in the hours when we’d much rather be watching TV, or drilling ourselves with note cards filled with obscure words instead of getting quizzed by a friend. Success is never easy. That’s why talent requires grit.

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Duckworth, A.L., & Gross, J.J. (2014). Self-control and grit: Related but separable

determinants of success. Current Directions in

Psychological Science, 23(5), 319-325

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How does your grit compare with others? I had my students take the 12 point survey developed by Duckworth to see how they rated. Some were surprised while others were well aware of their grit level. I even took it!  Want to give it a try or have your students see how gritty they are?  Just click on the word "survey."  When you have completed the survey, fill in the score grid below to find out just how gritty you truly are.

Faux Diamonds

In some preschool and kindergarten classes across the country, the geometric shape formerly known as a diamond is now being called a rhombus.  Why?  Does it matter? 

To be honest, a diamond is not technically a mathematical shape whereas a rhombus is.  When someone says the word rhombus, you know they are referring to a quadrilateral that has all four sides the same length; the opposite sides are parallel, and the opposite angles are equal.  (Mathematical Warning: A rhombus is not thinner than a diamond, AND the plural form, rhombi, is not a dance performed on the program Dancing With the Stars.)  

But what comes to mind when you hear the word diamond?  If you are a woman, you might envision a large sparkling gem setting on the ring finger of your left hand.  If you are a guy, you might think of a baseball infield. (The distance between each base is the same, making the shape a diamond.)  If you play cards, the word might bring to mind a suit of playing cards, OR you might recall a line in the song, Twinkle, Twinkle, Little Star.  Calling a rhombus a diamond is similar to calling a child a "kid" (could be a baby goat), or a home your "pad" (might be a notebook).  The first is an accurate term, the second one is not. 

So how does this affect you as a teacher?  It doesn't, unless rhombus is on a local benchmark or state test.  But if you are an elementary grade teacher, please use the correct mathematical language because a middle school math teacher will thank you; a high school geometry teacher will sing your praises, (see song below) and a college math teacher, like me, will absolutely love you for it!

Rhombus, Rhombus, Rhombus

  (sung to the "Conga" tune)
(The song where everyone is in a line with their hands on each other's shoulders)

 Rhombus, rhombus, rhombus;

Rhombus, rhombus, rhombus

Once it was diamond;

Now it's called a rhombus.

Much Ado About Nothing

I have decided to post (this is an updated previous post from 2011) some questions about zero that my college students have asked me in class.  I will say this, "Zero can surely give you a severe headache unless one knows its properties."  

Question #1 - Do you know why zero is an even number?    All mathematics is based on patterns.  Because of this, I know that an even plus an even number will always give me an even answer; an odd number added to an odd also gives me an even answer, and an odd number plus an even gives me an odd answer. In other words:    E + E = E     O + O = E     O + E = O



The numbers 4 and -4 are both even numbers. If we add them together, their sum is zero.  Based on the math pattern of  E + E = E,  then zero has to be even as well.  If we substitute zero in other problems such as 1 + 0 = 1, it fits the O + E = O  rule just as 2 + 0 = 2 fits the E + E = E  rule.

In Algebra, even numbers can be written as 2 x n where n is an integer.  Odd numbers can be written in the form of  2 x n + 1.  If we have n represent 0, then  2 x n = 0 (even) and  2 x n + 1 = 1. (odd)

I say all of this to relate an actual incident that occurred in my classroom.  I wrote the number 934 on the white board, and commented that since it was even it was divisible by 2.  One of my students was perplexed because he did not understand how 934 could be even when it contained two odd numbers and only one even number.  He actually thought that all the digits of a number had to be even for the number to be even.  Funny?  Not really!  Amazingly, he had made it through 12 years of school without understanding Place value as it relates to even numbers. Unfortunately, I had assumed that everyone (especially my college students) knew what an even number was.  I no longer make assumptions about students and their math knowledge!

Question #2 - Is zero positive or negative? The definition for positive numbers is all numbers greater than zero, and the definition for negative numbers is all numbers less than zero. Therefore zero can be neither positive or negative.

Question #3 - Is zero a prime or composite number? To be a prime number, a number must have only two positive divisors, itself and one. Zero has an infinite number of divisors so it is not prime. A composite number can be written as a product of two factors, neither of which is itself. Since zero cannot be written as a product of two factors without including itself, zero, it is not composite.

Question #4 - Why can't you divide by zero? I love this question. Back in the dark ages when I asked it, I was always told, "Because I said so." Being an inquisitive student was not a blessing when I was growing up. Math teachers who knew all did not want to be questioned!!!! Anyway, I don't mind the question, and here is my practical answer.

First, we must understand division. Division means putting or separating a number of items into a number of specific groups or sets. When you divide, such as in the problem 12 divided by 2, you are really putting 12 things into two groups or two sets. Therefore, if you have the problem 8 divided by 0, it is impossible to put eight things into no groups. You cannot put something into nothing!



Hopefully, this clears up a few things about zero.  I leave you with this math cheer.  (I always wanted to be a cheerleader!)

                            Zero, two, four, six, eight,

Who do we appreciate?

Even numbers! Even Numbers! Even Numbers!