Kiss those Math Headaches GOODBYE!

Archive for the ‘Math Instruction Techniques’ Category

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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“Algebra Survival” Program, v. 2.0, has just arrived!


The Second Edition of both the Algebra Survival Guide and its companion Workbook are officially here!

Check out this video for a full run-down on the new books, and see how — for a limited time — you can get them for a great discount at the Singing Turtle website.

 

Here’s the PDF with sample pages from the books: SAMPLER ASG2, ASW2.

And here’s the website where you can check out the books more fully and purchase the books.

 

 

 

 

 

 

 

Factor Quadratic Trinomials, Part 2


This video shows the fastest and easiest way I know of for factoring quadratic trinomials. Give it a watch and see if you agree.

“Unpacking” Terms from Parentheses


How do you get math terms out of parentheses? And what happens to those terms when you remove the parentheses?

It seems like the process should be simple. But this issue often plagues students; they keep getting points off on tests, quizzes, homework assignments.  What’s the deal?

The deal is that there’s a specific process you need to follow when taking terms out of parentheses, and what you do hinges on whether there’s a positive sign (+) or a negative sign (–) in front of the parentheses.

But not to worry. This video on this page settles the question once and for all. Not only that, but the video provides a story-based approach that you can teach (if you’re an instructor) or learn (if you’re a student) and remember (no matter who you are). Why? Because stories are FUN and MEMORABLE.

So kick back and relax (yes, it’s math, but you have a right to relax) and let the video show you how this process is done.

And in customary style, I present practice problems (along with the answers, too) at the end of the video so you can be sure you understand what you believe you understand.

 

 

 

 

 

Algebra Mistake #3: (x + y)^2 = what?


Ever thought this after you got back a math test … ?

“Why did I do that? I used a rule where it doesn’t apply!”

Yep, that’s exactly what we’re looking at in Algebra Mistake #3, a case of “overgeneralizing.”

The situation we’re dealing with involves over-generalizing everyone’s “favorite” property, the distributive property!

How’s that? Well, you’re supposed to use the distributive property when a number multiplies terms inside parentheses.

But sometimes students get a little bit — shall we say — “carried away” — and use the distributive property principle in other situations, too. The results are a tad bit comic, if you’re the teacher, but not so funny if you’re the student and you’ve made the mistake 19 times on a test with 20 problems.

Anyhow, after you watch the following video you shouldn’t have to worry about this again because we’ll get the two wires in your mind untangled so you never make this mistake again. So just relax, watch and learn.

And oh yes, don’t forget that we’ve provided some practice problems at the end of the video to help you make sure you’ve got the concept nailed down.

 

 

Algebra Mistake #2: Does a x a = 2 x a?


Now that you’ve gotten a taste for the benefits of analyzing algebraic mistakes, it’s time to explore a second common mistake. This one is so common that nearly every student commits it at least once on the road to algebra success.

As you watch the video, notice how by thinking hard about two expressions, we can think this mistake through to its very root, thus discovering the core difference between two similar-looking algebraic expressions.

And along the road, we’ll learn a general strategy for decoding the meaning of algebraic expressions. What I like about this strategy is that you can use it to understand the meaning of pretty much any algebraic expression, and you’ll see that it’s not a hard thing to do. In fact, it just involves using numbers in a nifty way.

Best of all, students usually find this approach interesting, convincing and even a bit fun. So here goes, Common Algebra Mistake #2 …

 

Algebra Mistake #1: – 1^2 vs. (–1)^2


Welcome, welcome, welcome to my series on COMMON ALGEBRA MISTAKES!

We’re going to have some fun spotting, analyzing, dissecting, exploring, explaining and fixing those COMMON ALGEBRA MISTAKES, the ones that drive students and teachers UP THE WALL!

I’ve had so much experience tutoring that I find these mistakes fascinating, and I intend to share my (ok, bizarre) fascination in this series of videos.

Also, be aware that I’m very much OPEN to suggestions from you folks on mistakes that you’d like me to explore. I highly value the experience and wisdom of you students and educators, and I want to do all I can to work with you to un-earth the mistakes of algebra, and bring them to the light of day so we can find ways to stay out of their way!

Here’s the first video on these mesmerizing mistakes. Could any mistake be more classic than this very one? I doubt it. But watch the video and form your own opinion …