Question about math programs and calculator usage

[Seasider]

I am posting this to the high school board, though my student about to enter pre-algebra is still middle school, I thought I'd get better info here than that part of the forum. The last time I taught/facilitated a kid all the way through, we stuck with Saxon. I am really a non-mathy person, so once was enough for me; the next couple of kids went out for their higher maths. As I look ahead for the youngest, this question arose.

 

I've been looking at what local area co-ops and friends are using for the higher maths. On the one hand, several use Saxon and some other maths and don't really talk about calculator use until about Algebra 2, when graphing calculators are introduced. Others specify the need for calculators (just basic scientific ones, not graphing quite yet) beginning in prealgebra. My question/concern is, am I seeing a bent towards teaching kids to use the technology more so than to understand the underlying mathematical concepts and arithmetic? One particular option I have looked at is the prealgebra/algebra/geometry/algebra 2 series of texts by Ron Larson.

 

I DO understand that calculator use is essential, and that calculator skills are essential; my concern is the too-early introduction. 

 

I also realize there are many pieces to this puzzle and that an answer may not be concrete, and there are likely varying opinions. It was just a thought that occurred to me and I am hoping for some input. I honestly do not want to teach math myself, it's really not my strength, so I want to start this kid on the right path with outside curriculum and instruction from prealgebra onwards. I did see quark's pinned post, that was great, thank you, quark!

[regentrude]

Calculator skills are not essential for math education.

Early calculator use is detrimental, because students never develop a number sense. We see the fallout in college every week, when students who have been trained to be calculator dependent are incapable of doing simplest arithmetic mentally. They are not familiar with perfect squares and cubes and factors and can't see simplifications.

 

The math program I use (AoPS) gets by basically without calculators through calculus, with the exception of a handful of problems each year where a calculator is explicitly permitted. A well designed curriculum does not need a calculator to teach students precisely what it wants to teach; the necessity for calculator use is an indicator of a poorly designed curriculum where the problem developers have been too lazy to ensure that the numbers work out without calculator.

 

My kids get to use a scientific calculator in their high school physics and chemistry classes. They are simple to use and do not require several years of practice.

A simple scientific calculator is sufficient on the ACT and SAT and even the SAT2 in Math; only AP Calculus stupidly requires a graphing calculator.

My DD is majoring in physics and has never had occasion to use a graphing calculator.

 

ETA: I see no need to introduce a graphing calculator in algebra 2. The kids should learn to graph functions by hand, because that is where the mathematical insight is. Punching a function into a calculator is not math.

Edited May 9, 2016 by regentrude

[Sunshine State Sue]

My question/concern is, am I seeing a bent towards teaching kids to use the technology more so than to understand the underlying mathematical concepts and arithmetic?

 

I wouldn't doubt it.

 

I tutor GED students in math.  I will tell you that the students who had calculators introduced early are at a severe disadvantage.  You are right to be cautious.  I think some use of a scientific calculator in Algebra 1 is acceptable.   Introducing graphing calculators in Algebra 2 is acceptable.  For myself, I would make sure that the student understands the underlying concepts and uses the calculator more for speed or checking their  work.

 

I took a Statistics class at CC a few summers ago.  I learned to start at the back of the chapter to learn how to use the graphing calculator first.  That was so I would know quickly how to use it on tests.  Then, I would learn the mathematical concepts and how to do the work without the calculator.  I would do the homework without the calculator, then use the calculator to check my work and practice for the tests.  On the tests, I would use the calculator, and if time permitted, do the work by hand to make sure I got the same answers.  I had a much better understanding of both the underlying math and the calculator usage than most students.

 

I can't say I know anything about the Larson texts, but I've read favorable reviews here on the forums.

 

Good luck!

[Arcadia]

For the Larson books, you could skip the calculator portion in the books and still cover everything.

 

I am neither pro nor against calculator. There is a place for using a scientific or graphing calculator when test questions are written that way like for AP Calc and AP Stats

 

However be aware that for SAT subject tests, most do not allow calculators.

 

"The only Subjects Tests for which calculators are allowed are Mathematics Level 1 and Mathematics Level 2. "

 

What I did was let my kids use calculator to check their work after they have done the calculations using pencil and paper. I let them use calculators for trigonometry and for logarithms unless its those common trigonometry ratios (30/45/60/90 degrees) which I expect my kids to know.

 

I also let them use calculators for physics and chemistry when need be which is for not many questions.

 

Whether the teacher focus on concepts or calculator usage of the Larson textbooks depends on how strong the teacher is in concepts I think. The calculator portions is what kids would play with at home anyway without the teacher having to go through in class. There are also plenty of YouTube videos on how to use a graphing calculator.

[JanetC]

I have had families go way overboard on the "no calculator" thing, too. There is no reason to be using log tables or trig tables in the scientific calculator age. Ending when you get to the expression 5 times the square root of 2 (for example) and not entering it in the calculator makes it much harder to sanity-check whether the answer makes sense in the context of the problem.

 

You don't have to allow the calculator on all assignments, but it does start to make sense when you are dealing with roots, exponents, logarithms, and/or trig functions.

[Julie of KY]

I have had families go way overboard on the "no calculator" thing, too. There is no reason to be using log tables or trig tables in the scientific calculator age. Ending when you get to the expression 5 times the square root of 2 (for example) and not entering it in the calculator makes it much harder to sanity-check whether the answer makes sense in the context of the problem.

 

You don't have to allow the calculator on all assignments, but it does start to make sense when you are dealing with roots, exponents, logarithms, and/or trig functions.

 

To teach MATH, you can not use a calculator at all depending on the program. As stated above, AoPS is an excellent program that teaches everything without a calculator. The answers are left in radical form which is an exact answer rather than a rounded off answer with a calculator. There are math programs that insist on calculator usage early on - answer to four decimal places on trig or logarithmic functions. There are other programs that design the problems to test mathematical understanding and not calculator usage and expect the answer to be ln2 rather than a decimal.

 

Of course it doesn't make sense to leave your answer in a mathematically accurate, but not a single number with dealing with real life problems such as how high or how far. No one measures 2 sqrt 3, they measure to a decimal or fraction of an inch or centimeter, etc. I like math done without calculators and science with calculators, however not everything is designed this way.

[rebcoola]

Not a lot to add but calculators are not essential.  Our foreign exchange student was quite frustrated that the Ap calc class insisted on a graphing calculator, she was quite a bit faster without it.  No she wasn't a genius that was just the way they were taught.

[regentrude]

Ending when you get to the expression 5 times the square root of 2 (for example) and not entering it in the calculator makes it much harder to sanity-check whether the answer makes sense in the context of the problem.

 

 

I disagree. 5 sqrt 2 is the exact answer and should in most problems remain like that and not be converted to a decimal approximation. (In college you'd likely lose points - if a calculator is allowed at all in math, which is typically not the case)

 

I would rather expect a student to be able to check whether it makes sense by knowing that it's approximately 7 because they are familiar with sqrt 2 being approximately 1.4. Kids who use calculators typically don't know that.

Edited May 9, 2016 by regentrude

[Deee]

I have science degree and worked in medical research and biostatistics. I've written a string of papers full of number crunching. I've never even seen a graphing calculator. They aren't part of the Australian maths curriculum.

D

[Mike in SA]

Yup, I would prefer no calculator at all until manual calculation detracts from the learning process.  A good math problem is designed to allow you to see the composition of the result, square roots and all.  That composition means something - if you aren't ever looking at it, you're missing part of the point of the exercise.

 

There is a point where the calculator becomes essential, and you will know it.  If trig is that point, then I suspect the problem is constructed carelessly.  Most trig problems in quality programs end up with a basic handful of results from the unit circle.  If you are using the calculator, you aren't using the unit circle as you should.  A trig book which deviates heavily from the 15 degree increments may not be a very well written textbook.  That said, the physics problems from a precalculus course are likely to require a calculator from time to time.

 

Funny thing is, a few places we require our kids to work without - long division and long multiplication - are becoming increasingly esoteric.

 

[PinkyandtheBrains.]

 

 

We are using AoPS here in high school, my tenn only uses a calculator for science.

 

I was(am) calculator dependent. I've worked hard to make sure my children aren't.

[HomeAgain]

Not a lot to add but calculators are not essential.  Our foreign exchange student was quite frustrated that the Ap calc class insisted on a graphing calculator, she was quite a bit faster without it.  No she wasn't a genius that was just the way they were taught.

 

My son felt the same. He just didn't need it after AOPS and still pulls it out sparingly.  He is faster without it for most calculations, and when he does have to use it he prefers Casio to TI because it's more intuitive for him.

[regentrude]

 

There is a point where the calculator becomes essential, and you will know it.  If trig is that point, then I suspect the problem is constructed carelessly.  Most trig problems in quality programs end up with a basic handful of results from the unit circle.  If you are using the calculator, you aren't using the unit circle as you should.  A trig book which deviates heavily from the 15 degree increments may not be a very well written textbook. 

 

And, for the trig, I want to add:

sometimes using a calculator completely misses the essence of the problem and what it is supposed to teach. The student may be supposed to find a sine by repeated application of addition theorems and half angle formulas.

(Or for a complicated logarithm, the objective of the problem might be to use laws of logarithms.)

Working with a calculator completely misses the goal there.

And leads to those college students who cannot identify a 3-4-5 triangle or figure out the sides in a 30-60-90 triangle.

[daijobu]

I have science degree and worked in medical research and biostatistics. I've written a string of papers full of number crunching. I've never even seen a graphing calculator. They aren't part of the Australian maths curriculum.

D

 

I suspect the only reason people buy and use graphing calculators is for standardized tests.  We just purchased a graphing calculator for AP statistics dd's taking next year.  

 

Heck, if you want to see what a hyperbolic paraboloid looks like, open up google and copy/paste in: z = x^2 -y^2 .   Seriously, do it.  It's beautiful.  Why would I use a graphing calculator IRL when I can use google?  

 

My kids didn't start using calculators until they began competing in MathCounts and then my dd went on to use one in AP chemistry.  (Not a graphing calculator, but a regular one.)  I can't remember now, but there was a time when they used a calculator in AoPS...I think it was for calculating compound interest.  The text makes it clear (most of the time) when a calculator should be used.  

 

If you do buy a regular calculator, I recommend one that uses RPN.  It's much faster, easier, and less frustrating.  

[raptor_dad]

 

If you do buy a regular calculator, I recommend one that uses RPN.  It's much faster, easier, and less frustrating.  

 

And you won't have problems with unauthorized people "borrowing" your calculator :)

[Seasider]

....

 

sometimes using a calculator completely misses the essence of the problem and what it is supposed to teach.

 

....

Working with a calculator completely misses the goal there.

 

...

Yes, this is my concern. I thank you all for your replies. Before deciding which tutor to use I will be making inquiries about calculator use. Also, I've heard so many good things about AoPS that I am really going to take a serious look at that and see if I can trickle some in at home.

[Seasider]

I will look into this RPN (a link would help!).

 

Also, I wanted to add that for my oldest we also did what someone mentioned above, calculator use in science (*setting up* those equations/conversions is what we focused on their), but not in math until the highest levels.

[EKS]

If you do buy a regular calculator, I recommend one that uses RPN.  It's much faster, easier, and less frustrating.  

 

:iagree:

[daijobu]

I will look into this RPN (a link would help!).

 

 

 

This is one I bought a couple of years ago.  This manual has a detailed tutorial.

[wapiti]

If you do buy a regular calculator, I recommend one that uses RPN.  It's much faster, easier, and less frustrating.  

 

I have a kiddo with handwriting issues who is new to the graphing calculator (thanks to school) and he's doing a lot on it.  I'm not familiar with RPN - can anyone comment on whether RPN might be easier for an especially visual-spatial thinker?  What makes RPN less frustrating for you?  Just wondering!

[justasque]

I am posting this to the high school board, though my student about to enter pre-algebra is still middle school, I thought I'd get better info here than that part of the forum. The last time I taught/facilitated a kid all the way through, we stuck with Saxon. I am really a non-mathy person, so once was enough for me; the next couple of kids went out for their higher maths. As I look ahead for the youngest, this question arose.

 

I've been looking at what local area co-ops and friends are using for the higher maths. On the one hand, several use Saxon and some other maths and don't really talk about calculator use until about Algebra 2, when graphing calculators are introduced. Others specify the need for calculators (just basic scientific ones, not graphing quite yet) beginning in prealgebra. My question/concern is, am I seeing a bent towards teaching kids to use the technology more so than to understand the underlying mathematical concepts and arithmetic? One particular option I have looked at is the prealgebra/algebra/geometry/algebra 2 series of texts by Ron Larson.

 

I DO understand that calculator use is essential, and that calculator skills are essential; my concern is the too-early introduction. 

 

I also realize there are many pieces to this puzzle and that an answer may not be concrete, and there are likely varying opinions. It was just a thought that occurred to me and I am hoping for some input. I honestly do not want to teach math myself, it's really not my strength, so I want to start this kid on the right path with outside curriculum and instruction from prealgebra onwards. I did see quark's pinned post, that was great, thank you, quark!

 

 

I love the Larson texts, especially with the tests and worksheets that come with the EasyPlanner software which allow you to customize the program for all kinds of learners.  They are used by the local private schools for Alg I&II, and I have used Math I through Alg I several times over with my own kids and numerous tutoring students.  They can be done at least up through Alg I without using a calculator.  In the Alg I book there is typically one word problem per section (if that) which uses a calculator.  

 

 

Calculator skills are not essential for math education.

Early calculator use is detrimental, because students never develop a number sense. We see the fallout in college every week, when students who have been trained to be calculator dependent are incapable of doing simplest arithmetic mentally. They are not familiar with perfect squares and cubes and factors and can't see simplifications.

...

ETA: I see no need to introduce a graphing calculator in algebra 2. The kids should learn to graph functions by hand, because that is where the mathematical insight is. Punching a function into a calculator is not math.

 

I agree with every word of this.  Most especially the idea that using a calculator - even for simple multiplication and division - robs the student of the opportunity to train their brain to "see" relationships between numbers.  Yes, it can be slow going at first, but with practice comes much insight, so that when it comes time for, say, factoring polynomials, the student has the mental math skills to "see" the most likely solution without a lot of trial and error.  Not to mention seeing these relationships is key to maximizing scores on standardized tests.

 

For the Larson books, you could skip the calculator portion in the books and still cover everything.

...

 

Absolutely.  In fact, I'd argue that you'd cover it BETTER without the calculator.

 

I have had families go way overboard on the "no calculator" thing, too. There is no reason to be using log tables or trig tables in the scientific calculator age. Ending when you get to the expression 5 times the square root of 2 (for example) and not entering it in the calculator makes it much harder to sanity-check whether the answer makes sense in the context of the problem.

 

You don't have to allow the calculator on all assignments, but it does start to make sense when you are dealing with roots, exponents, logarithms, and/or trig functions.

 

I agree that there comes a time that having a calculator becomes useful, but I disagree with the example of 5 sqrt2.  I expect my students to be able to estimate things like sqrt2 - it's easy - think about C-rods - sqrt4 is 2, because if you make a square with 4 1-rods, the side is two.  If you make a square with 1 1-rod, the side is 1.  So sqrt2 has to be somewhere between 1 and 2 - one and a bit more, if you will, which is decent for many needs.  

 

You can get more specific by thinking about making a square with 2 1-rods and what it would look like if you sawed up one of the rods to fit around the other.  If you cut one in half, and put it on two sides of the other, it wouldn't completely fill the square - you'd need to shave off a bit more to fill in the little space left.  SO - 1.5 is too much, so sqrt is one-and-a-bit, more specifically it's more than one and less than one-half.  That's usually good enough for sanity-check purposes.  And of course, once you've used sqrt2 a bunch of times, you begin to memorize it, which is much more useful than having to use the calculator every time.

 

I disagree. 5 sqrt 2 is the exact answer and should in most problems remain like that and not be converted to a decimal approximation. (In college you'd likely lose points - if a calculator is allowed at all in math, which is typically not the case)

 

I would rather expect a student to be able to check whether it makes sense by knowing that it's approximately 7 because they are familiar with sqrt 2 being approximately 1.4. Kids who use calculators typically don't know that.

 

And of course there's this.  Quite often sqrt2 needs to be carried as such, because it cancels with something else along the way.  Kids who don't have access to calculators are quite willing to carry such things in hopes of cancelling, and usually keep a sharp eye out for such possibilities!

 

 

Yup, I would prefer no calculator at all until manual calculation detracts from the learning process.  A good math problem is designed to allow you to see the composition of the result, square roots and all.  That composition means something - if you aren't ever looking at it, you're missing part of the point of the exercise.

 

There is a point where the calculator becomes essential, and you will know it.  If trig is that point, then I suspect the problem is constructed carelessly.  Most trig problems in quality programs end up with a basic handful of results from the unit circle.  If you are using the calculator, you aren't using the unit circle as you should.  A trig book which deviates heavily from the 15 degree increments may not be a very well written textbook.  That said, the physics problems from a precalculus course are likely to require a calculator from time to time.

 

 

Both of these things.

 

And, for the trig, I want to add:

sometimes using a calculator completely misses the essence of the problem and what it is supposed to teach. The student may be supposed to find a sine by repeated application of addition theorems and half angle formulas.

(Or for a complicated logarithm, the objective of the problem might be to use laws of logarithms.)

Working with a calculator completely misses the goal there.

And leads to those college students who cannot identify a 3-4-5 triangle or figure out the sides in a 30-60-90 triangle.

 

Yes.

 

 

If you do buy a regular calculator, I recommend one that uses RPN.  It's much faster, easier, and less frustrating.  

 

Yes, yes, yes!!!

[regentrude]

 I expect my students to be able to estimate things like sqrt2 - it's easy - think about C-rods - sqrt4 is 2, because if you make a square with 4 1-rods, the side is two.  If you make a square with 1 1-rod, the side is 1.  So sqrt2 has to be somewhere between 1 and 2 - one and a bit more, if you will, which is decent for many needs.  

 

If the student has been made to memorize the perfect squares up to 20x20 (around 4th grade), she should have no difficulty recognizing that, since 14x14=196 which is very close to 200,  sqrt 200 has to be a bit more than 14, and hence sqrt 2 a  bit more than 1.4

Edited May 9, 2016 by regentrude

[daijobu]

I have a kiddo with handwriting issues who is new to the graphing calculator (thanks to school) and he's doing a lot on it.  I'm not familiar with RPN - can anyone comment on whether RPN might be easier for an especially visual-spatial thinker?  What makes RPN less frustrating for you?  Just wondering!

 

 

If you are doing multiple operations on multiple operands, the calculator will keep track of your intermediate quantities for you in its stack of memory.  With a regular computer you would need to write down all the intermediate quantities in order to reuse them.    

 

Here's a good explanation.  There's a bit of a learning curve, but not as steep as you would think.  Once you get the hang of it, you'll never go back.  

Edited May 9, 2016 by daijobu

[wapiti]

Excellent!  Thanks, daijobu.  I will tell ds.  This might be helpful for him instead of inputting lengthy calculations into the graphing calculator algebraically.  His refusal to write down the intermediate quantities has been driving me crazy!  When I was poking around earlier, I saw that there might be computer apps that function as an RPN calculator, so he could try it out before we bother to buy another calculator.

[daijobu]

 When I was poking around earlier, I saw that there might be computer apps that function as an RPN calculator, so he could try it out before we bother to buy another calculator.

 

I have one installed on my android phone that is called a41CV.  

[SparklyUnicorn]

I have a kiddo with handwriting issues who is new to the graphing calculator (thanks to school) and he's doing a lot on it.  I'm not familiar with RPN - can anyone comment on whether RPN might be easier for an especially visual-spatial thinker?  What makes RPN less frustrating for you?  Just wondering!

 

I have no idea, but my son loves RPN.  I don't get the allure.  And he too has handwriting issues.

 

He keeps trying to convince me I'm supposed to like RPN more.  LOL

[SparklyUnicorn]

Excellent!  Thanks, daijobu.  I will tell ds.  This might be helpful for him instead of inputting lengthy calculations into the graphing calculator algebraically.  His refusal to write down the intermediate quantities has been driving me crazy!  When I was poking around earlier, I saw that there might be computer apps that function as an RPN calculator, so he could try it out before we bother to buy another calculator.

 

There are some that can do both.  We have one that has three choices for method of entry.  It's the HP 50G.  You can't use it on stuff like the ACT (in case that matters to you), but his instructor lets him use it.

[daijobu]

I have no idea, but my son loves RPN.  I don't get the allure.  And he too has handwriting issues.

 

He keeps trying to convince me I'm supposed to like RPN more.  LOL

 

He's right.  You do not like RPN enough.   :rolleyes:

Edited May 10, 2016 by daijobu

[SparklyUnicorn]

He's right.  You do not like RPN enough.   :rolleyes:

 

LOL

 

I am wildly proud of myself for being able to figure out a graphing calculator at all! 

[justasque]

I'm just wondering: Does something like this affect anyone's decision to use a curriculum? I mean, if you were set on using it next fall, would you still use it, or would this be a deal breaker?

 

 

There is an app for iPhones/iPods that reproduces a classic HP 15C calculator which uses RPN.  It's called Free 15C.  It's my go-to calculator.

 

When learning RPN, it does help to visualize the numbers as being in a "stack" of boxes.  Putting in a number and pressing enter "pushes" the number into the bottom box of the stack, moving all previous numbers each into the next higher box.  Doing a calculation combines the bottom two numbers into the lowest box (using the desired operation; the answer is what into the box) and all of the other numbers in the stack move down a box.

If your stack looks like this:

 

2

3

4

5

 

and you use the "+" button, you get this:

 

2

3

9

 

Takes a bit to learn but it's very powerful once you get it.  And people who don't know RPN won't borrow your calculator once they see it doesn't have an equals button.  :-) 

Edited May 10, 2016 by justasque