Kiss those Math Headaches GOODBYE!

Posts tagged ‘Education’

CALCULATORS & “NUMBER SENSE”


Hi folks,

Now that summer has officially begun, I’m enjoying a certain distance from the heat of the school year, and that distance gives me a chance to reflect.

One set of ideas that my mind keeps poking around again and again is this:  a) the weakness in actual number sense among today’s elementary and secondary students,
b) the concomitant modern focus on teaching Number Sense during these school years, and c) the now-rampant overuse of calculators.

I find it interesting that Number Sense has become a “big important new topic” that math instructors are required to teach. I also find it interesting that the new focus on Number Sense has been growing steadily at the very same time that students in so many parts of our country have become more and more calculator dependent.

Could there be a connection?

Yes, undoubtedly! Back when I set up shop tutoring math, K-12, in 1990, Santa Fe (NM) Public School students were not permitted to use calculators willy-nilly. Because of that, our students were not calculator-dependent. Students were expected to know the truths of arithmetic forwards and backwards, and wouldn’t have dreamed of reaching for a calculator to find the value of something so simple as, say, 7 + 5, as happens routinely today. Yes, routinely! I should know; I’m a professional math tutor.

What’s more, I’d say that students in the 1990s generally understood concepts such as odd and even numbers, prime and composite numbers, how to prime factorize, how to find the GCF and the LCM, and the many other skills that are part of the “new area of math instruction we call Number Sense.

That’s because teachers used to require students to use their minds to work with numbers. Students used to grind out 7/18 + 5/12 by hand, not by pressing buttons. They used to figure out the LCM of 22 and 30 by using an algorithm rather than by tapping an app. They used to prime factorize numbers using the good old factor tree and simplify radicals by thinking rather than by pressing a sequence of buttons and scrolling through the numbers flashing across their LCDs.

You can probably see where I’m going with this. Today’s math students have become overly calculator dependent. That dependence on calculators, in turn, has made them deficient at the skills in the topic area we call Number Sense. And precisely because today’s students are so deficient at number sense, precisely because they have been allowed to become so dependent on their e-devices rather than on their mental devices, curriculum designers have devised this whole new area of math, Number Sense, that now gets taught as its own “thing” rather than being an integral thread of everyday math instruction. Number sense used to be something students developed naturally, by mentally working with numbers, day-in, day-out, using paper and pencil and mental math.

Lest I be called a Luddite, I’m not saying that calculators have no place in the math curriculum. But as a tutor who has helped students with math for some 27 years now, I can say with certainty that today’s students’ innate ability to work with numbers, play with numbers and calculate with numbers has been dulled and frankly allowed to atrophy because calculators have become an all-too-easy, all-too-available crutch.

In this way, math curricula and math educators who overly promote calculator usage have done a great disservice to students. The good news, though, is that  teachers could correct course without too much trouble.

Teachers could still allow students to use calculators, quite appropriately, for higher-order processes — such as graphing two functions to see where they intersect, and to see if the answer found that way comports with the answer attained by solving the systems simultaneously by hand — while at the same time disallowing calculator usage for arithmetic calculations.

I’d like to see teachers get their students back to basics in this way because, from my perspective, we’re raising a new generation of students, many of whom have little ability to calculate mentally and little understanding of how numbers work. As a result, these children (soon-to-be adults) are unnecessarily vulnerable.

They’re vulnerable because they cannot tell if they are receiving the correct change from a cashier. They’re vulnerable because they cannot tell if their car or home interest payment is correct. And they’re vulnerable in a larger sense because they lack the ability to easily think numerically, i.e., quantitatively. And when people lack the fundamental ability to think quantitatively, even having a calculator won’t save them in many situations. That’s because they might not even know what operation to do to find a solution in a real-world situation.

But in an even more direct and practical sense, the new calculator-dependent students are vulnerable because they have been set up to struggle mightily in their college math classes. That’s because nearly all U.S. colleges require students to take math tests without using calculators!

So I say let’s get back to basics, and let’s do it in a smart way. Let’s continue to let students use calculators for higher-order thinking skills, but let’s disallow calculators for ALL arithmetic so that students will be required to once again become strong in those critical fundamental skills and so that they will re-gain the natural form of Number Sense that is their right and their due.

 

 

 

 

 

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Everyone Wins When Teachers Listen


Recently I saw something on YouTube that points out how we sometimes miss interesting comments while teaching.

An elementary teacher was teaching about our base 10 number system, and she was showing her first grade students the number 1.

“What do we add to 1 to make 10?” she asked.

A girl answered:  “Zero.”

Instead of listening deeply to what this child said, the teacher plowed ahead,”Well, not quite. Who else has an idea?” And the teacher waited until a student volunteered the answer she was waiting for:  9, saying that 1 + 9 = 10.

(This part of the lesson focused on number pairs that sum to 10.)

What’s sad about this situation is that the girl who said “zero” had a good point that the teacher missed. The girl was saying that a zero, written to the right of a 1, creates the number 10.

This point, had it been explored, could have sparked excitement. The teacher could have pointed out how strange it is that a 1 plus a 0 can create the number 10, when most people would say that 1 + 0 = 1. The teacher could have posed this as a riddle and asked if any student could unravel it. That riddle, in turn, could have helped the whole class ponder how interesting it is that the mere position of a digit affects that digit’s value, in our base 10 system.

Not only did the teacher miss this opportunity, she also inadvertently missed an opportunity to validate the girl. We can’t know for sure, but when a teacher passes over a student, acting like her comment was “incorrect,” the child can feel rejected. Instead, the teacher could have pointed out that this girl’s comment was in fact “right” in a most interesting way.

Of course no teacher is perfect, and as teachers we all miss comments we later wish we had noticed. Still it is helpful to be aware that we might BE missing things. Only then will we be more open to the many surprisingly interesting, unscripted comments that children make every day.

Finding domain and range — with color!


Have you ever noticed that a lot of students struggle with the idea of domain and range? This concept, taught mostly in Algebra 2,  often confuses students to the point where they cannot even identify the domain and range of a simple, continuous function.

I don’t really understand why students struggle with this concept, but I recently found a way of showing the idea that makes it considerably easier — using color to mark up a function.

Here’s an example of a problem where students need to figure out the domain and range by looking at a graph, like this:

What I have students do is use two colors to sort of “box in” the function. With one color, green in this case, students mark the left bound and right bound of the function by drawing vertical lines. And with another color, red, students mark the lower bound and upper bound by  drawing horizontal lines. I have students write in the phrases:  left bound, right bound, lower bound, and upper bound, like this:

Finally I ask students to figure out the domain and range by writing three-part inequalities for x and y, respectively, like this:


I’ve used this approach with a number of students, and so far no one has been unable to find the domain and range when using it. So it appears to be a winner. Try it yourself, either as you teach a concept, or as you re-teach it to those who are struggling.

Common Algebra Mistake: Interpreting Negative Signs in Front of Parentheses


Certain areas of algebra are like pebbles in your shoe: looked at closely, they  are very small — tiny in the scheme of things — yet they are “oh-so-bothersome!”

As a tutor, I’ve long felt this way about negative signs before parentheses. It’s a small thing really, and it seems simple to grasp for those of us who get it. Yet students make so many mistakes when facing this situation; to them it’s quite irritating.

So there I was again, trying to help a girl see how to simplify the expression: – (– 5x + 3y – 7)

But this time I thought of something different, the word “opposite.”

I talked for a moment with my tutee about the idea of opposites, and then I started out like this:

Q:  So, what’s the opposite of black?

She replied:  White (with the teenage “that’s-totally-obvious-what are you-doing?-insulting-my-intelligence? accent)

I told her not to worry, this would lead back to the problem. Next I gave her two terms for which she were to find the opposite, as in:

Q:  opp (tall, happy)

She wrote:   (short, sad), still wondering …

And I continued:

Q:  opp (heavy, up)

She wrote:  (light, down), sighing.

Then I explained that in math we express the idea of “opposite” with nothing more than the negative sign.

Then I gave her some problems with the negative sign:

Q:  –  (cold, left)

A:  (hot, right)

and

Q:  –  (under, near)

A:  (over, far)

She was still giving me that “this-is-so-easy-I-could-die” kind of look. When I thought about that, I realized it was good!

Next I  explained that in math, just as in real life, there are opposites. And we find mathematical opposites by examining signs. For example, the opposite of 5 is – 5; opposite of – 3/4 is 3/4; opposite of – 3x is 3x; opposite of y is – y, and so on.

Then I gave her these problems:

Q:  – (+ 2x, – 5)

Still she was with me:  – 2x, + 5

and

Q:  – (– 4y, + 3x, – 6)

A:  + 4y, – 3x, + 6

The sighing was slowing down, finally. Then I simply told her that we’re going to “lose” the comma (how’s that for modern slang!), both in the original expression and in their answer. Then I gave her a new problem:

Q:  – (5a – 3a – 9)

This puzzled her a bit. So I explained that she needs to mentally group the term with the sign that lies to the left. And that if no sign is showing, as for leading positive terms, she needs to mentally insert the invisible positive sign:  5 becomes + 5;  2a becomes + 2a. Once she got that, she was able to proceed:

Q:  – (5a – 3a – 9)

A:  – 5a + 3a + 9

And so on … one success after another. The concept was sticking. And best of all, she had a conceptual framework — the concept of opposite — that she could “lean against” any time she got stuck.

The longer I tutor the more I realize that this kind of conceptual framework — a story or concept we know from everyday life, which relates to the algebra in a direct way — is a big key to helping students grasp algebra. I use these kinds of stories in my book, the Algebra Survival Guide, providing stories we know from everyday life, which serve as analogies that show how the math works. For example, in the Guide I use a “tug-of-war” analogy to show how you solve problems like:  – 3 + 8.

– 3 + 8

Tur-of-War Teaches – 3 + 8

I’ve had so much success with this “story”-approach to algebra that I am working on an eBook that provides a whole litany of stories that work for algebra. It is fun to work on, and kids like this approach because it gives them a new way — an everyday way — to relate to the math.

So in any case, my suggestion is that when you teach or review the concept of negative signs before parentheses, you might just try the “opposites”  approach and see how it works with your students.

Abbreviating the Order of Operations


My recent posts about “Dear Aunt Sally” have, I hope, shown how dangerous it is to teach the memory trick of Please Excuse My Dear Aunt Sally — at least without some additional explanation.

Today I propose a way to save Dear Aunt Sally, for those of you who still like her.

As you know, the memory sentence is often abbreviated PEMDAS, which stands for:  PARENTHESES, the EXPONENTS, then MULTIPLICATION, then DIVISION, then ADDITION, then SUBTRACTION.

The problem with PEMDAS is that it makes kids think they always multiply before dividing, and that they always add before subtracting.

For students attached to PEMDAS, I let them use it, but I have them write it a novel way, so they realize they must pay attention to the left-right orientation of the operation symbols.

In the new way of writing PEMDAS, I put M and D in the same place, separate by the word “or.” Then I do the same for the A and S. So the whole memory device looks like this:

Alternative for PEMDAS

My suggestion is that teachers who like PEMDAS try this and see if your students start making fewer mistakes with the order of operations.

For those of you who never liked PEMDAS on the first place, I recommend that you check out the order of operations as presently in a clearer way, as it is in my Algebra Survival Guide. To get a feel for that, you can download the chapter of the book on Positive and Negative numbers here. Just click where it says:  See Sample Chapters of the Algebra Survival Guide and Workbook.

Then you’ll want to get the Survival Guide for the chapter on Order of Operations, which you can do through Amazon.com, here. Sorry, I can’t make everything available for free … I do have a business to run, with new products I’d like to create and make available.

Best way to write PEMDAS

Hopes for Obama and Education


Here’s another interactive post for all of you blog readers.

It’s sort of the Pink Elephant that has landed in the living room, and I can’t just pretend it’s not here any more

The election of Barack Obama could portend significant changes in the system of public education in this country.

If you’re open to sharing, I’d really like to hear your thoughts.

What do you hope for in an Obama Administration, with regard to education? Research and new Grants? Changes to No Child Left Behind? Greater funding for urban and rural districts? Higher teacher pay? An end to vouchers? More accountability? Merit pay?

What are your concerns?

Do you have any ideas that you think Obama would do well to heed?

What might Obama ignore in the field of education that he would do well to pay close attention to?

I’m opening this up pretty wide. But with one restriction. All comments must be on the topic of education. Any that are not on education I will have to discard.

Your coments may be short, Just, what’s on the top of your your mind? I’d really like to know. And we can all benefit by hearing from one anoher.

Best

— Josh


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