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The Calculator Argument

Once upon a time, two mathematicians, Cal Q. Late and Tommy Go Figure, were having a argument, really.

"Calculators are terrific math tools," said one of the mathematicians.

"I agree, but they shouldn't be used in the classroom" said the other.

"But?" asked Tommy Go Figure, and this is when the argument started. "That is just crazy!  I agree that having a calculator to use is a convenience, but it does not replace knowing how to do something on your own with your own brain."

"Why should kids have to learn how to do something that they don't have to do, something that a calculator can always be used for?" Cal Q. Late argued.

Tommy retorted,  "Why should kids not have the advantage of knowing how to do math?  To me, a calculator is like having to carry an extra brain around in their pockets.  What if they had to do some figuring and did not have their calculators with them?  Or what if the batteries were dead? (Here's a good reason for solar calculators.) What about that?"

Cal reminded Tommy, "No one is ever in that much of a rush. Doing math computation is rarely an emergency situation. Having to wait to get a new battery would seem to take less time than all the time it would take to learn and practice how to do math. That takes years to do, years that kids could spend doing much more interesting things in math."

"Look," Tommy went on, exasperated, "kids need to depend on themselves to do jobs. Using a calculator is not bad, but it should not be the only way kids can do computation. It just doesn't make sense."

Cal would not budge in the argument. "The calculator is an important math tool. When you do a job, it makes sense to use the best tool there is to to that job. If you have a pencil sharpener, you don't use a knife to sharpen a pencil. If you are in a hurry, you don't walk; you go by car. You don't walk just because it is the way people used to travel long ago."

"Aha!" answered Tommy. "Walking is still useful. Just because we have cars, we don't discourage kids from learning how to walk. That is a ridiculous argument."

This argument went on and one and on...and to this day, it has not been resolved. So kids are still learning how to compute and do math with their brains, while some are also learning how to use calculators.  What about you?  Which mathematician, Cal Q. Late or Tommy Go Figure, do you agree with?


Of course, this argument was made up, but it is very much like the argument schools and teachers are having about what to do with kids and calculators. What do you think?  Leave your comment for others to read.

Give Reading A Helping Hand!

I believe the Conceptual Development Model should be constantly used when creating lessons for students, no matter what their age or grade level. (I even use it on the college level.) This particular model helps to bring structure and order to concepts found in almost any discipline. Here is an example of how I used the model in reading.

When I taught third grade, I noticed that my students often had difficulty identifying the different components of a story. I knew I needed a concrete/pictorial example that would help them to remember. Since we always had our hands with us, I decided to make something that would be worn on the hands. By associating the abstract story concepts with this concrete object, I hoped my third graders would make connections to help them visually organize a story's elements. I also suspected it would increase their ability to retell, summarize, and comprehend the story.

I purchased a pair of garden gloves and used fabric paint to write the five elements of a story on the fingers...**characters, setting, problem, events, and solution. In the middle of the glove I drew a heart and around it wrote, "The heart of the story." (theme) Towards the wrist was written "Author's Message." (What was the author saying?)

After we read a story, I would place the glove on my hand, and we would go through the parts of the story starting with the thumb or characters. (a person, animal, or imaginary creature in the story). We then proceeded to setting (where the story took place.) We did not progress through all the story elements every day, but would often focus on the specific part that was causing the most difficulty. The fun came when one of the children wore the glove (Yes, it was a little big, but they didn’t seem to mind) and became the "teacher” as the group discussed the story. As the student/teacher talked about each of the fingers, we would all use our bare hands without the glove.

I also made and copied smaller hands as story reminders. This hand would appear on worksheets, homework, bookmarks, desks, etc. Sometimes the hand contained all the elements; sometimes it was completely blank, and at other times only a few things would be missing. The hand became known as our famous and notorious Helping Hand.

Why would I allocate so much time to this part of the curriculum? Because…

1) If a student learned the elements of a story, then they understood and knew what was happening throughout the story.

2) If a child is aware of who the character(s) were, then they cab identify the character’s traits during the story.

3) If the child knows the setting of the story, then they recognize where an event was taking place.

4) If they know the problems that are taking place, then they can be a part of the story and feel like they are helping to solve it.

Such visual tools allow a teacher the flexibility to focus on one single story element or present a more complex or intricate view of all parts of a story. By knowing the components of a story, students are more engaged and connected to their reading. It’s as if they assimilate the story and become a part it. So, are you ready to Give Reading a Helping Hand in your classroom?

**The five parts of a story may be identified as introduction, rising action, climax, falling action, and resolution or other similar categories.

Setting Limits in the Classroom

One of the most practical books I have ever read is Setting Limits in the Classroom: A Complete Guide to Effective Classroom Management with a School-wide Discipline Plan (3rd Edition) by Robert J. Mackenzie. This year, many of our local schools are making it a require read and school wide book study. It will be used for daily group discussions as well as for application in the “real” classroom.

It is easy reading and contains many practical, no nonsense methods for classroom management that actually work. No theory here; just real life examples that can easily be applied in the classroom. Many of the chapters give effective ways to encourage the unmotivated child. (I'm sure that each year you have one or two sitting in your class.) It is a book worth purchasing, reading, and sharing. AND many of the suggestions carry over into managing your own children.

The paperback book can be purchased on for about $10.00. Mackenzie has written several books, one entitled: Setting Limits with Your Strong-Willed Child: Eliminating Conflict by Establishing Clear, Firm, and Respectful Boundaries. I haven't read this one, but I wish it had been available when I was raising my first son!

Finding the Greatest Common Factor and Least Common Multiple

The most common method to find the greatest common factor (GCF) is to list all of the factors of each number, then list the common factors and choose the largest one.  Example: Find the GCF of 36 and 54.

1) The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

2) The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, and 54.

Therefore, the common factor(s) of 36 and 54 are: 1, 2, 3, 6, 9, 18.  Although the numbers in bold are all common factors of 36 and 54, 18 is the greatest common factor.

To find the lowest common multiple (LCM), students are asked to list all of the factors of the given numbers. Let's say the numbers are 9 and 12.  

1) The multiples of 9 are: 9, 18, 27, 36, 45, 54.

2) The multiples of 12 are: 12, 24,  36, 48, 60.

As seen above, the least common multiple for these two numbers is 36.  I have seen this done on a large letter M as illustrated below.

We often instruct our students to first list the prime factors, then multiply the common prime factors to find the GCF. Often times, if just this rule is given, students become lost in the process. Utilizing a visual can achieve an understanding of any concept better than just a rule. A two circle Venn Diagram is such a visual and will allow students to follow the process as well as to understand the connection between each step. For example: Let’s suppose we have the numbers 18 and 12.

1) Using factor trees, the students list all the factors of each number.

2) Now they place all the common factors in the intersection of the two circles. In this case, it would be the numbers 2 and 3.

3) Now the students place the remaining factors in the correct big circle(s).

4) That leaves the 18 with a 3 all by itself in the big circle. The 12 has just a 2 in the big circle.

5) The intersection is the GCF; therefore, multiply 2 × 3 to find the GCF of  6.

6) To find the LCM, multiply the number(s) in the first big circle by the GCF (numbers in the intersection) times the number (s) in the second big circle.

3 × GCF × 2 = 3 × 6 × 2 = 36. The LCM is 36.

This is an effective method to use when teaching how to reduce fractions,

Finding GCF and LCM
I have turned this method into a resource for Teachers Pay Teachers. It is 16 pages and begins with defining the words factor, greatest common factor and least common multiple. What a factor tree is and how to construct and use a Venn Diagram as a graphic organizer is shown. Step-by-step examples are given as well as student practice pages. How to use a three circle Venn Diagram when given three different numbers is explained. Two pages of blank pages Venn Diagrams are included for classroom practice. To learn more, just click on the title under the resource cover on your right.

Next week's post - Finding the LCM and GCF for a set of algebraic terms.

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.

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

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.