Good calculator for moving into Algebra?

[Dmmetler]

CW's post on the Explore prep reminded me that I probably should get DD7 a calculator. She uses an app on her iPod occasionally for some of the harder LoF problems, but she won't be allowed to use that in the EXPLORE. I figure Christmas is a good excuse :).

 

Anyone have a recommendation that would be a good one for moving into middle/high school math and (more importantly) testing? Pretty girl colors/bling would be appreciated by my almost 8 yr old magpie, too!

[Crimson Wife]

I don't allow calculator use in general school work just yet (algebra 2 is when I will permit it), so I just had DD use a basic 2 line calculator for the EXPLORE. But when I get her a graphing calculator, I'm planning on the TI-89 Titanium because that's the one I've heard is best for calculus-based courses.

[ccolopy]

DS used his TI-30X IIS for Explore and it's served him well through algebra 2. It comes in pink. :)

[mathwonk]

I don't support having young children use calculators at all. I realize I could be wrong, but I have held this view through many years of college math teaching. the only time a calculator is needed is for problems where the numbers are so huge as to make hand calculation impractical, and this almost never happens.

 

The reason I oppose calculators is that hand calculation teaches children the properties of numbers in way calculator use does not. Short cuts in mental calculation also reinforce properties like distributivity and commutativity. I.e. to multiply 39 times 15, one can do 40 times 15 which is easily seen to be 600 (= 4x15x10) and subtract 15, getting 585.

 

One reason IU think kids struggle in advanced math classes like algebra, or later abstract algebra, which focus on properties of number systems, is a lack of hands on experience with actual concrete number systems, which is obtained by doing computations by hand.

 

It is really head shakingly frustrating to deal with college students in calculus classes who cannot take the cube root of 8 without a calculator, nor multiply 13 by 65 even with a pencil and paper. I have met some bright high school students who knew how to use calculators in useful ways,. but I have also seen a lot of college students who did not understand anything about arithmetic operations, having always depended on the crutch of a calculator.

 

Now I realize you may not have a choice. Some classes and some schools systems require calculators, as do some standardized exams. But in my experience, and I suspect many professional mathematicians agree with me, calculators are actually a hindrance to learning math. At best they should be introduced only after the child has mastered all the basic subjects, say in college.

 

If this battle is already lost, and the child must have one at age 8, I would work extra hard to require many numerical manipulations without it, so as to not sacrifice the many benefits of doing the math oneself.

 

 

The problem is, although there are exceptions, you often don't learn anything by using a calculator, you don't build any mathematical muscles.

[Dmmetler]

The EXPLORE officially calls for use of a calculator-so I feel it's better for her to have some experience using one rather than not.

[hsbeth]

DS used his TI-30X IIS for Explore and it's served him well through algebra 2. It comes in pink. :)

 

 

I second this recommendation. Starting with a Texas Instruments from the get go is the best idea as it helps make the transition to a graphic calculator easier later on. Many of the key strokes have some similarity.

[hsbeth]

I don't support having young children use calculators at all. I realize I could be wrong, but I have held this view through many years of college math teaching. the only time a calculator is needed is for problems where the numbers are so huge as to make hand calculation impractical, and this almost never happens.

 

The reason I oppose calculators is that hand calculation teaches children the properties of numbers in way calculator use does not. Short cuts in mental calculation also reinforce properties like distributivity and commutativity. I.e. to multiply 39 times 15, one can do 40 times 15 which is easily seen to be 600 (= 4x15x10) and subtract 15, getting 585.

 

One reason IU think kids struggle in advanced math classes like algebra, or later abstract algebra, which focus on properties of number systems, is a lack of hands on experience with actual concrete number systems, which is obtained by doing computations by hand.

 

It is really head shakingly frustrating to deal with college students in calculus classes who cannot take the cube root of 8 without a calculator, nor multiply 13 by 65 even with a pencil and paper. I have met some bright high school students who knew how to use calculators in useful ways,. but I have also seen a lot of college students who did not understand anything about arithmetic operations, having always depended on the crutch of a calculator.

 

Now I realize you may not have a choice. Some classes and some schools systems require calculators, as do some standardized exams. But in my experience, and I suspect many professional mathematicians agree with me, calculators are actually a hindrance to learning math. At best they should be introduced only after the child has mastered all the basic subjects, say in college.

 

If this battle is already lost, and the child must have one at age 8, I would work extra hard to require many numerical manipulations without it, so as to not sacrifice the many benefits of doing the math oneself.

 

 

The problem is, although there are exceptions, you often don't learn anything by using a calculator, you don't build any mathematical muscles.

 

I heartily agree with this and wish that more educators, particularly those at the university level, would speak out against calculator use. My dh, who is a physics teacher, shares many of your frustrations.

 

We discouraged calculator use with our oldest at first, but later felt blindsided by the requirements of many upper level standardized exams that were designed specifically to require and test the use of calculators. SAT 2 subject tests and the AP Calc exam require the use of a graphing calculator. My accelerated student was at a significant disadvantage b/c we hadn't started her with one early enough.

 

She is now in 11th grade and is a dual enrollment student (at a U of GA system school incidentally). She has been required to use one in her college level classes thus far. She's been quite successful on the SAT and ACT without the use of a graphing calculator, but has had some difficulty adapting to its use in her college coursework. She's still been successful, but it has felt like an uphill battle at times. They also do not use them on her competitive math team.

 

She's finally getting more comfortable in using one, but I do wish we had started earlier and have begun this process of introduction with our younger kids after completion of Alg. 1.

 

Honestly, I think the College Board is largely to blame here. Thankfully, my dd decided to continue her math studies through dual enrollment instead of attempting AP exams.

[quark]

I gave DS a very simple desktop calculator for EXPLORE. I taught him how to use it. He practiced for about 2 weeks and then did not use the calculator during the test! :lol: He didn't finish on time, got the lowest score in math among all the subtests but did well enough to qualify as a high scorer overall. I'm not certain that having the calc would have helped him because he always likes taking his time with math.

 

We finally allowed calculator use (a TI Nspire) after feeling blindsided by requirements too. We didn't use a calc for Algebra 1 but I also saw the need for him to start learning to use it so what I do is allow it for his algebra/trig-based physics course but not for math for now. The same with graphs. I want him to hand-draw graphs and get used to sketching them smoothly. I learned a lot from doing it by hand and I think he will gain a lot from the experience, if not the math then at least the patience and care required.

 

A big caveat though is that for some kids, calculator use can help alleviate frustration with computation. I have not experienced this personally with my son but I've heard about this from friends. I think with a little careful monitoring by the parents, you can avoid kids using the calc as a crutch. Every kid has different needs after all.

Edited November 8, 2012 by quark
extra information

[lewelma]

No calculator here and ds has gotten through Algebra 1 and 2. However, having said that, if there is a really ugly problem, he can ask me to use my 32 year old calculator, and if I am in a good mood, I'll let him use it. :001_smile:

 

I have taught math in high school, and it is *shocking* how calculator use many most kids into idiots. They just get lazy and put even the easiest calculations in, and they don't bother to estimate first, so they have no idea when the answer from the calculator is wrong. You don't want to know how many times I have heard "but that is what the calculator said."

 

Ruth in NZ

Edited November 8, 2012 by lewelma

[8filltheheart]

I heartily agree with this and wish that more educators, particularly those at the university level, would speak out against calculator use. My dh, who is a physics teacher, shares many of your frustrations.

 

We discouraged calculator use with our oldest at first, but later felt blindsided by the requirements of many upper level standardized exams that were designed specifically to require and test the use of calculators. SAT 2 subject tests and the AP Calc exam require the use of a graphing calculator. My accelerated student was at a significant disadvantage b/c we hadn't started her with one early enough.

 

She is now in 11th grade and is a dual enrollment student (at a U of GA system school incidentally). She has been required to use one in her college level classes thus far. She's been quite successful on the SAT and ACT without the use of a graphing calculator, but has had some difficulty adapting to its use in her college coursework. She's still been successful, but it has felt like an uphill battle at times. They also do not use them on her competitive math team.

 

She's finally getting more comfortable in using one, but I do wish we had started earlier and have begun this process of introduction with our younger kids after completion of Alg. 1.

 

Honestly, I think the College Board is largely to blame here. Thankfully, my dd decided to continue her math studies through dual enrollment instead of attempting AP exams.

 

For a different perspective, both of my STEM ds's have not been allowed to use a calculator in their cal classes. I don't remember when our oldest was actually allowed to use a calculator, but I know definitely not for cal 1. Our 11th grader is not allowed to use one in multivariable cal or in cal physics.

 

We don't let our kids use calculators w/the exception of the rare problem until alg 2 (and even then it is rare.) AoPS is meant to be completed w/o a calculator, so ds really didn't use a calculator at all through cal last yr. Mid-March last yr he started working w/Kathy in Richmond in prepping specifically for the AP BC exam and learning how to use his calculator was a big part of the prep, but it certainly wasn't an issue and not one that would make me want to allow a calculator earlier on.

 

They do use a calculator when taking the practice tests for the SAT/ACT so that they can be familiar w/which problems are faster to complete w/o the cal and which ones are faster w/.

[Dmmetler]

They do use a calculator when taking the practice tests for the SAT/ACT so that they can be familiar w/which problems are faster to complete w/o the cal and which ones are faster w/.

 

 

And all the debate on whether or not calculator use is a good thing is fine-but ultimately, DD7 is going to be going into a room and sitting down to take a test that allows and expects calculators. If she doesn't have one, even if she really doesn't need one, she's going to freak out because she doesn't. If she DOES have one, but has never used one, she's likely to spend the entire test period playing with the pretty buttons. Which is why I've let her pull out the iPod after she's done a few CWP or LoF problems to check her work, and why I was asking for suggestions on what might be a good calculator just so I can give her some familiarity with what she needs for the specific situation.

 

Truthfully, I suspect she's most likely to not bother to use it-this is the kid who regularly does multi-step algebra problems in her head because she finds it faster than writing them down, and who has been able to maintain a running total of my shopping list down to two decimal places since she was about 4.

[regentrude]

And all the debate on whether or not calculator use is a good thing is fine-but ultimately, DD7 is going to be going into a room and sitting down to take a test that allows and expects calculators. If she doesn't have one, even if she really doesn't need one, she's going to freak out because she doesn't. If she DOES have one, but has never used one, she's likely to spend the entire test period playing with the pretty buttons. Which is why I've let her pull out the iPod after she's done a few CWP or LoF problems to check her work, and why I was asking for suggestions on what might be a good calculator just so I can give her some familiarity with what she needs for the specific situation.

 

 

I recommend getting a simple scientific calculator for $10: basic, easy to use, one that has also trig functions, square roots, exponents. Not that she needs them now - but it will be useful for high school sciences and trig, and much easier to use than a graphing calculator. We have the casio fx 260-solar, and it has been sufficient for my students' needs through high school and for some college courses - anything that does not require a graphing calculator. In fact, I have not had need for anything more sophisticated through graduate school in physics.

 

http://www.officedepot.com/a/products/121121/Casio-fx-260-Solar-Scientific-Calculator/?Channel=Google&mr:trackingCode=6931DB1A-EC81-DE11-B7F3-0019B9C043EB&mr:referralID=NA&mr:adType=pla&mr:ad=22395426956&mr:keyword=&mr:match=&mr:filter=20224360076&cm_mmc=Mercent-_-Googlepla-_-Technology+Office_Machines-_-121121

[lewelma]

And all the debate on whether or not calculator use is a good thing is fine-but ultimately, DD7 is going to be going into a room and sitting down to take a test that allows and expects calculators.

 

Is she taking a standardized, calculator-based test in the near future? She is 7, right?

 

The one big thing I recommend for calculator use is to ALWAYS estimate your answer first. It must be a rigid habit -- painstakingly trained over months until it is unconsciously done EVERY time. Obviously, this will tell you if you have hit the wrong buttons by accident.

 

Ruth in NZ

[Dana]

I agree with getting the TI30XIIS.

It's under $20 and is a two line model so you can catch if you mistype something. Keys are in roughly the same place as on the graphing models of TI, making a later transition to a graphing calculator easier.

[KAR120C]

I agree with getting the TI30XIIS.

It's under $20 and is a two line model so you can catch if you mistype something. Keys are in roughly the same place as on the graphing models of TI, making a later transition to a graphing calculator easier.

This! The TI30XIIS is by far my favorite cheap scientific calculator. If it's not pretty enough (LOL) get stickers.

 

DS only got his TI30XIIS to take the Explore, and practiced with it for a bit before the exam... seven questions a day (add, subtract, multiply, divide, something with decimals, fractions, and percents). We didn't use a calculator for math in general, just for standardized testing when it was recommended.

[Dmmetler]

Is she taking a standardized, calculator-based test in the near future? She is 7, right?

 

The one big thing I recommend for calculator use is to ALWAYS estimate your answer first. It must be a rigid habit -- painstakingly trained over months until it is unconsciously done EVERY time. Obviously, this will tell you if you have hit the wrong buttons by accident.

 

Ruth in NZ

 

 

She is taking the EXPLORE after the holidays for Talent Search. The EXPLORE is designed for 8th/9th graders but is also used for younger students who are likely to benefit from higher level programs. Most standardized tests in the USA now expect/require/allow (officially, I think it's the last-in practice, it's more the first two) a calculator. I suspect this has more to do with TI lobbying than anything else.

[go_go_gadget]

I don't know that anyone's arguing against use of a calculator on a standardized test designed for that type of test-taking (and for this test, the simplest calculator would likely be fine), just against the use of one in Algebra I or II, which has come up sort of tangentially. In my experience as a math major, a calculator is only helpful when working with absurdly large numbers (so rare I can't even remember the last time), and under certain other, relatively unusual circumstances. I recently used one when needing to quickly find the trig values of non-standard angles such as φ=4/3 and the like in the context of spherical coordinates, but otherwise haven't touched one in months. As others have said, calculators weren't even allowed in most of my lower-division courses, and in the uppers they're an afterthought at best.

[abbeyej]

I also vote for a very basic scientific calculator for this purpose. Eventually she'll need a graphing calculator (at least most high school math programs use them), but she won't need it for a few years, even at an accelerated pace.

[mathwonk]

the problem is that a calculator is not needed except when required, and math understanding is needed all the time. Overuse and too early use of a calculator harms learning math ideas. advanced math classes never use calculators. so if you give in to the relatively minor demands to have a calculator for one or two standardized tests in high school, you have to figure out a way not to handicap your child from actually learning the subject later on. I did not allow calculators in any of my college classes, and certainly they were never used in graduate classes. Be careful not to sacrifice learning math for satisfying a trivial requirement that will soon go away. This is not the basic skills forum, this is the accelerated forum, including presumably people who may become mathematicians, if not handicapped early. It is a good idea to know how to use a calculator, but even more important to know how to get along entirely without one.

[Woodland Mist Academy]

so if you give in to the relatively minor demands to have a calculator for one or two standardized tests in high school, you have to figure out a way not to handicap your child from actually learning the subject later on.

 

:confused:

 

Somehow I cannot see how this would be an issue--especially with an advanced student.

 

Are the effects of a couple weeks calculator use before and during a test truly that dire and far-reaching? :001_huh:

 

ETA: Are you saying once a calculator has been used, all is lost? I am truly confused. It's the EXPLORE. Is it truly that much of a concern?

Edited November 9, 2012 by Hilltop Academy

[TCB]

The EXPLORE officially calls for use of a calculator-so I feel it's better for her to have some experience using one rather than not.

 

My dd has done the Explore test twice and has not used a calculator either time. I don't think any of the kids in the group used a calculator. I think the rules may say you can use one, but is by no means essential.

[mathwonk]

no i am not saying that. that would not cause a problem. i tried to say very plainly that there is no problem as long as child can survive well without a calculator.

 

QUOTE: "It is a good idea to know how to use a calculator, but even more important to know how to get along entirely without one."

 

i apologize if i was unclear. the problem is with those children who use a calculator so much in their youth that they do not acquire facility with arithmetic. this is a very big problem. if you have solved it, and figured out how to teach a child to use a calculator, and yet be willing to give it up and practice doing arithmetic by hand and mentally, i think you have solved the problem.

 

i think some of us would benefit by knowing how you do this. It would seem tempting to me, if i were a child, once i find out how to save labor by letting a calculator do all the thinking, not to do any myself ever again. i certainly have suffered with some of my most brilliant students in college, who used them so much they were literally terrified without them. One girl, one of my favorite students, and just as smart as a whip, literally almost cried when her battery ran out on a test. she said "i haven't done math without a calculator since 8th grade!"

 

just as long as you understand that risk, it is all up to you.

[Woodland Mist Academy]

Thanks for clarifying. My daughter has a simple calculator she uses for fun sometimes. I was wondering if I should confiscate it! ;)

 

ETA: She doesn't use it during math. That would defeat the purpose of AoPS.

[kiana]

I see no issue in allowing a child to use a calculator for fun outside of math class or for a specific set of math work. The issue (jmo) is when it's used for most/all work within the class. Some of my college students cannot reliably do calculations such as 29 - 20 or 2(1/2) or .354*10 without a calculator. This is a huge time-handicap on tests.

[mathwonk]

i had a B student in calculus who, when faced with a problem requiring multiplying 13 by 65, added 13 copies of 65. This stuff gives me the willies. I just can't make the problems any easier than that. what if i had asked her to multiply 98 times 65? the problem for me when i try to teach advanced algebra to that student, is that this student does not appreciate that 13 times 65 is the same as 10 times 65 plus 3 times 65. in that form one could do it in ones head (i hope, i.e. 650 + 200 -5 = 845).

Edited November 10, 2012 by mathwonk

[mathwonk]

let me put it another way. i have a friend who likes to go out fishing in his boat. we found out a year or so ago he could not swim, when we had to rescue him from drowning. knowing how to use a calculator but not how to do mental math is like knowing how to use a boat but not how to swim. at some point you will need a serious rescue.

[Crimson Wife]

advanced math classes never use calculators

 

This *REALLY* depends on the specific class. I agree that a calculator should not be necessary for basic arithmetic calculations but certain STEM classes actually specify that the student have a graphing calculator.

[lewelma]

but certain STEM classes actually specify that the student have a graphing calculator.

 

I must be very old fashioned, but my son had to do all his graphs by hand for the regional science fair. I was *very* surprised to see that there were only 2 other students out of 600 that did hand-drawn graphs. But..... my son won 1st place.:001_smile: I saw so many posters with poor choice of graph styles, too much color/glitz, too many graphs (they did not choose the best but displayed them all), etc. If you have to work hard for your graph, you will do it right the first time and really think it through.

 

I even had my sister do her statistics for her master's thesis in biochemistry by hand (they were nonparametric, so pretty easy, but she did have to do about 50 of them). And in the end, she agree with me, that it was a good choice. She *knew* what she was doing. She knew if she had made a mistake. She did not rely on a computer print out. She could explain *why* she used the statistics she did, and what were the potential problems. All this knowledge was because she was down and dirty with the numbers.

 

OP, I do appreciate the need for a calculator when under time constraints. And I think that you could tell your student that she will be doing exam prep for a month with a calculator so that she is very comfortable with using it. But I would warn her, that after the test the calculator is going away.

 

I do speak from experience. In my previous life, I was a high school math teacher and after that, a researcher doing mathematical modelling of population dynamics. IMHO, calculator use promotes lazy thinking.

 

Ruth in NZ

Edited November 10, 2012 by lewelma

[boscopup]

I have no input on the OP (and this comment below is completely tangential to what she was asking ;) ), but...

 

IMHO, calculator use promotes lazy thinking.

 

 

Amen! I say this from experience. I have not had to use my brain in math for so many years, and even as an engineer, I could pull out my handy calculator or use the computer calculator or plug numbers into Excel and do calculations there. I never did stuff in my head anymore. I *could*... back in school. I have good conceptual understanding of math and always did the mental math tricks like Singapore teaches, using the distributive property and things like that. I could figure out a 15% tip in my head very easily. But having calculators available ALL the time has made me lazy, and I find myself using a calculator for VERY simple problems. :tongue_smilie:

 

Going through AoPS myself is helping my brain come back. I do NOT use a calculator for that. :D

 

I do still have my old HP 48G, for when DS needs one for higher level math and tests, but for now, he has a little calculator that smells like chocolate (prize for selling things at school one year) that he plays with sometimes, but he NEVER gets to use it during math time. Mostly, he's figured out how to make it say words when held upsidedown. :lol:

[KAR120C]

i apologize if i was unclear. the problem is with those children who use a calculator so much in their youth that they do not acquire facility with arithmetic. this is a very big problem. if you have solved it, and figured out how to teach a child to use a calculator, and yet be willing to give it up and practice doing arithmetic by hand and mentally, i think you have solved the problem.

(emphasis mine)

Test prep is not "so much". Using it for everything? That would be too much. But the OP is (if I understand correctly) primarily interested in test prep.

 

It really only takes a couple weeks of very light use to teach a child to use a calculator sufficient for test prep. Seven questions a day was perfect for us, just so he knew where all the buttons were. And then the calculator goes away. We don't use it for everything, or anything he could do by hand. Honestly even with higher math, I'm almost always going to want the answer in terms of pi, or in simplified radical form, and not a rounded decimal answer. So he needs to know his standard trig functions, how to factor algebraic equations and complete the square, how find asymptotes and limits and graph by hand... all of that first, and then if at the end I want a (rounded decimal) numeric answer he can plug in the last expression.

 

Is it tempting to use it for everything? I don't know - I never asked him. It's not an option. But every time he has a standardized test we haul out the calculator and send him off. Is it a little ridiculous? Sure. But it hasn't actually hurt his ability to do real math in real life.

 

The one year he had really heavy calculator (and computer) use was the year we did AP Stats. I did make him do one simple linear regression by hand (ten data points isn't going to kill him), but really it's a royal pain in the rear and always doing it by hand isn't giving you any extra insight. Once is plenty. And I'm perfectly happy to let SAS draw scatter plots and boxplots as long as you can reconcile what you know about your data with what shows up on paper, and that when it spits out an equation you can talk about it with good understanding of how your data produced that particular pattern. (and not "the calculator said so")

[mathwonk]

Caveat: These are opinions gained from a lifetime of teaching college and graduate math, but they are still just opinions.

 

forgive me, maybe i should have defined "advanced". I meant primarily pure math classes at the junior/senior math major level in college all the way to graduate courses for PhD's. Abstract algebra, advanced calculus, real analysis, complex analysis, euclidean and non euclidean geometry, differential topology, algebraic topology, algebraic geometry, operator theory, number theory, lie groups, lie algebras, Riemann surfaces, several complex variables, ......

 

Even applied math classes in college, like numerical analysis, tend to use, not calculators, but computers. It is also true that some instructors advocate using computers with programs like Geometer's Sketchpad, in elementary euclidean geometry classes, especially with students with especially weak backgrounds, those who have trouble visualizing circles and lines meeting. Some of my strongest students have also shown me impressive demonstrations on the computer of very complicated geometry constructions such as the Euler line. But these did not involve hand held calculators, and are well beyond what most students will encounter.

 

I don't know exactly what STEM classes are being referred to, but I don't think of them as advanced. The STEM initiative in my state is an introductory government program designed to bring more high school students and undergraduates into math and science, implemented in college mostly through courses in science education departments. (By my definition there are almost no advanced math classes in high school. I did teach advanced vector calculus once to a high school class of about 6 students, but that is extremely rare, and only 2 of them were clearly ready for it and helped by it.)

 

i also should have qualified "never". it is true that anyone who wishes to can introduce the use of a calculator in some way in almost any course (maybe not most of the ones i listed). But in the last decade or two of teaching university math, I taught over 40 different courses from freshman calculus level to 4th and 5th year graduate courses, and none of them used calculators.

 

It is also true that some advanced research in pure math has found a way to use computer programs to make very complicated calculations, but these are far out of reach for any hand held calculator.

 

 

It is undeniably true that students today are well advised to master the use of computers. the problem is, and it is a serious one, that hand held calculators are not of comparable value, and introducing them too soon can be very harmful to learning math ideas. I do not know the best solution to this problem, but i tend to avoid calculators.

 

E.g. many students who use calculators tend to believe the answer on the calculator is correct, rather than an approximation. Since many scientific calculators have at most 7 decimal places this suggests they do not realize that most real numbers cannot be expressed this way. it is very hard to teach the concept of a precise real number to someone who thinks that pi is really 3.1416, or worse 22/7. They may not even realize that these two approximations are different , e.g. that 22/7 has no finite decimal expression.

 

Calculators are used in some college courses, but those courses known to me, are primarily for students who are weak in math and the calculators are an attempt to open up some areas of math even to people who struggle with basic computations by hand. They are not ordinarily used for math majors. Of course they may be used in very restricted circumstances, such as multiplying huge numbers together in an illustration of code cracking and code making, but even here an ordinary hand held calculator does not have the capacity to make a useful demonstration.

 

I can imagine using a graphing calculator to show a student roughly what a graph of a function looks like. Since calculators make approximations and round off error however, it should be also emphasized that the picture shown on the calculator may be significantly incorrect. E.g. two critical points which occur close together may look like only one, or even like no critical point at all. Once a student realizes how poor the resolution is on the calculator screen compared to the information they seek, they may begin to lose interest in these crude approximations.

 

I once enjoyed showing my class briefly how complicated the antiderivative was of some simple looking fraction, (e.g. the integral of 1/(1-x^20), an expression filling a whole page) but that required a computer running Mathematica, and soon lost its shock value.

 

A calculator can have educational value, but mainly in combination with other insights, and works best in the hands of someone who is very strong without it. The danger is using it to try to compensate or replace mastery of basic math skills and concepts. Unfortunately this may be its most appealing aspect to teachers of struggling students.

 

to sum up my recommendation: first learn to get along very well without a calculator, only then is it helpful to use one. Try at all costs to avoid being someone who is lost in doing basic math without a calculator. Learning to program a calculator may be of more value, and learning to use a computer is essential.

Edited November 10, 2012 by mathwonk

[mathwonk]

Another reservation to use even of computers, say the Geometer's Sketchpad program, for teaching geometry is the loss of understanding "how" and "why". When you ask this program to construct a perpendicular from a point off a line to that line, it just drops one, plop, onto the line. This may help an especially inexperienced person to see what the word "perpendicular" means, but it has no value at all in learning how to construct one.

 

The actual construction proceeds by drawing 3 circles, first one centered at the point, and meeting the line twice, then two more circles centered at the two points where the first circle met the line, and finally connecting a point where these two circles meet each other, to the original point.

 

Subtleties then become visible for discussion, such as how do you make sure those circles do meet in the appropriate ways?, that are quite invisible using the computer program.

 

 

Now someone who realizes this could of course rewrite the program to be more revealing of the steps in the construction, but that has not been done to my knowledge. This is a main problem with calculators and computers: they do not reveal how or why, hence are of value mainly to those who either do not care, or who already understand these deeper matters.

 

(One could of course try to have a discussion on why the method used by a carpenter works, of just dropping a weight on a plumb line, more similar to the computer program's procedure. The computer of course uses this procedure even when the line is not horizontal, whereas a plumb line only finds perpendiculars to the theoretical earth's surface, i.e. lines pointing to the earth's gravitational center. Thus a carpenter needs also a "level" to find a parallel to the earth's surface.)

 

In my experience people who teach and use calculators and computers often omit all questions as to what is going on, since they they are content with an "answer". Another problem in that regard seldom appreciated by naive users is that the "answer" on the calculator or computer can be very wrong, and one needs an understanding of the process to recognize when this occurs.

 

i have had students tell me that sqrt(2) is precisely equal to 1.414 (maybe with a few more places), because that is what they read off their calculator. it did not bother them that this number obviously ends in a 6 when it is squared, hence its square cannot equal 2. They even argued the opposite because when they squared their number on their calculator it did come out 2! How do you discuss this with someone having such blind faith in his calculator? One tends to give up.

 

 

Here are some little puzzles especially for the person who uses a calculator to do simple division. Why is it true that a fraction (like 22/7) often becomes an infinite repeating decimal when divided out? Which fractions on the other hand will yield terminating decimals (like 69/125)? Why does the repeating part of the decimal for 22/7 have length 7? What do you think happens in general?

 

 

It is possible that none of these questions would even occur to a calculator user, since the number of places in a calculator answer may not be sufficient even to reveal the repeating behavior. I recall something similar to this as a homework problem in sophomore abstract algebra in college in 1961, and which resisted my attempts to make completely precise.

Edited November 10, 2012 by mathwonk

[Crimson Wife]

forgive me, maybe i should have defined "advanced". I meant primarily pure math classes at the junior/senior math major level in college all the way to graduate courses for PhD's. Abstract algebra, advanced calculus, real analysis, complex analysis, euclidean and non euclidean geometry, differential topology, algebraic topology, algebraic geometry, operator theory, number theory, lie groups, lie algebras, Riemann surfaces, several complex variables...

 

Very few students will ever take these kinds of classes. At my alma mater (Stanford) only 2% of the students major in math. I was talking about the normal math track taken by students majoring in engineering or science, or who are pre-med or pre-MBA. So okay, if your kid is some sort of math genius and is considering a math major, the advice might be different than for the typical STEM student. There are probably at least 10 students taking the regular STEM math sequence for every 1 student taking the math major sequence.

[mathwonk]

You are of course correct. I may be wrong, but I expected to find future math majors on this accelerated learners thread, at least this seemed to be the place to address them. The question that started this thread was in reference to a 7 year old moving into algebra. This seems to me like a potential math genius/math major.

 

My point is still that even students who are asked to use calculators in STEM classes, are possibly being harmed rather than helped. I am constantly giving advice aimed at people who actually want to learn and understand math, not to those who primarily want to pass courses and tests that seem to me misguided. This is my bias. The fact that calculators are not used in advanced math classes is offered as evidence that calculators are not considered of value in understanding math, even for those who take less advanced classes.

 

But I don't think I can say this any clearer than I have done. In my experience people just won't believe such advice until they see the results for themselves. Besides, I could be wrong, and I frequently am. I am more concerned that my advice be clear than that it be slavishly followed. I am just trying to help people make informed decisions.

Edited November 10, 2012 by mathwonk

[Crimson Wife]

Giving advice based on a tiny fraction of an already elite group of students to me seems like giving running advice based on the experience of those who compete in ultramarathons- it simply isn't very relevant to most people.

 

I stand by my original statement that it really depends on the particular math course whether calculators are banned, permitted, or required. The ones I took as a pre-med and the ones my DH took as an engineer required use of a graphing calculator. I personally think it is far more likely that my own kids will be in these types of math courses than math major ones that ban calculator use.

[mathwonk]

"Giving advice based on a tiny fraction of an already elite group of students to me seems like giving running advice based on the experience of those who compete in ultramarathons- it simply isn't very relevant to most people. "

 

Perfectly so, and that was precisely the reason for my last paragraph above. I want my advice to be clear enough for you to decide whether it is relevant to you or not. So many people do not ask this question, which you do ask, but just ask for advice, without weighing it.

 

 

Pardon the philosophy, but this thoughtful group seems as good a place to try it out as any. You seem mostly to be discussing taking courses, and on that score I agree with everything you have said. To me there is a big difference between learning a subject and taking or even passing a class in it, and I am discussing approaches to learning.

 

Indeed some students claim certain classes reduce their understanding of a subject, and I believe that does happen. There are mathematicans who believe that certain tendencies toward abstraction in mathematical writing and teaching (“Bourbaki” style math) have harmed math learning and research throughout the world for decades now.

 

So think about the problem of designing and presenting a class to teach some subject you love and appreciate and perhaps know a great deal about. Gee, how silly of me, all of you are doing that all the time as home schoolers. Ok, how do you go about it? and I mean especially in reference to a subject you actually know well. (For other subjects you may just ask someone else what book to read.)

 

If you are like me, you will immediately be aware that in designing a class it is not clear where to begin. Actually the more experience you have teaching it the less sure you may be that you do know. I.e. even if I think I know exactly what “should” be taught, if after teaching it several times many of my students still exhibit no mastery of the topic, I may rethink whether I chose well either what to present or how to present it. E.g. if you have taught two children, did you teach them both the same way?

 

I confess somewhat to my embarrassment that I never found, in some 50 years of experimenting, an approach that worked with everyone, or even with almost everyone. When I went too fast, some students said they were lost. When I tried to become slower, clearer and more patient, some more advanced students complained they were bored. When I gave too much theory some asked for more examples. (This is probably advice I should have heeded more.) But to me it seemed a completely example oriented approach missed the discussion and understanding of patterns that unified the subject.

 

One piece of advice from an esteemed colleague was to have a goal. I.e. he reminded me it is hard to achieve your goal unless you actually have one. Ideally one should make the goal a clearly defined one, and if possible decide on a way to measure whether it is achieved. If your goal is to have your child pass regular math classes in high school up to calculus, get a 5 on the AP test, and get into a “good” college, you have to be guided accordingly in choice of curriculum. I never think seriously myself about specific goals such as these. I am primarily interested in how to promote understanding of and pleasure in mathematics, especially to those who already find it somewhat interesting.

 

So when someone asks me what calculator to use with a 7 year old, I tend to say none, because my default goal is always for the child to develop a hands on familiarity with numbers that will lead to an understanding of arithmetical operations that will lead later to an appreciation of the structure of algebra.

 

But if your goal is that the child should be able to score well on an SAT test that requires use of calculators, you may have to proceed accordingly. To me those are just survival skills unrelated to learning math. But I want to help you to determine what goal you are choosing and what the consequence may be. Even then, I may be wrong.

 

Indeed I discouraged one of my children from attending college calculus classes while still in high school, because to me it was not as wise a way to understand math, as to spend more time mastering the basics underlying calculus, namely algebra and geometry. As a result when college application time came around, some colleges gave preference for admissions to some of his classmates who had taken such college classes, even though they understood less math than he in my estimation. So in some wordly situations it is an advantage to give an appearance of understanding rather than to have genuine understanding. I struggle with such phenomena still. Some very practical persons may think this another reason to be skeptical of my advice, but I still believe that in the long run, knowledge trumps just having a degree.

Edited November 10, 2012 by mathwonk

[8filltheheart]

 

I stand by my original statement that it really depends on the particular math course whether calculators are banned, permitted, or required. The ones I took as a pre-med and the ones my DH took as an engineer required use of a graphing calculator. I personally think it is far more likely that my own kids will be in these types of math courses than math major ones that ban calculator use.

 

That hasn't been our experience. Calculus at 3 different universities in 2 different states has not allowed calculators. ETA: I just asked our oldest ds. He said he was not allowed to use a calculator for all 3 levels of cal nor for diffEQ. He took these courses at 2 different universities in a different state from where we currently live. Seems similar to what our 11th grader is experiencing here since he isn't allowed a calculator in his multivariable cal class (as well as in his cal-physics.) AP cal bc only allows calculators on parts of the exam. Knowing how to work the problems w/o the calculator allows you to complete them faster w/it for the sections.

 

Section I: Multiple-Choice

The multiple-choice section of the exam has two parts. For Part A, you'll have 55 minutes to complete 28 questions without a calculator. For Part B, you'll have 50 minutes to answer 17 questions using a graphing calculator.

 

Section II: Free-Response

The free-response section tests your ability to solve problems using an extended chain of reasoning. Part A of the free-response section (two problems in 30 minutes) requires the use of a graphing calculator. Part B of the free-response section (four problems in 60 minutes) does not allow the use of a calculator. During the second timed portion of the free-response section (Part B), you are permitted to continue work on problems in Part A, but you are not permitted to use a calculator during this time.

 

Really, learning the math is the difficult part. Learning how to use the calculator is the easy part.

 

:iagree: w/others who have stated that learning to use calculators for standardized testing is prudent. But, for daily classroom math, they really do not need to use one. The advice to not use calculators regularly for high school math is prudent. Enough to be familiar with it, yes. But for all work or even most, no.

Edited November 10, 2012 by 8FillTheHeart

[quark]

Mathwonk, if you keep writing the things you do, you are going to be breaking a lot of hearts here. :001_wub:. Moms of boys will cyber-stalk you (ask me how I know :001_smile:). Moms of girls will be looking for younger versions of you hoping their daughters will hit it off with those boys in the future lol.

 

I think a lot of us are just using the calculator as a tool? Not as a crutch? Just like how I use some curriculum as tools. I am never comfortable relying on any one source 100%. I agree that some classes can lead to kids learning things in a less efficient way too. I am not able to articulate exactly how I know this but my instincts have been very strong about this and I will continue to trust them. Thank you for confirming this suspicion for me. I feel a lot better now about spending so much time pondering over materials and online classes and being so picky about the ones I do choose.

 

I think I have a budding math major and I will continue to role-model doing math by hand myself. Thank you for this discussion!

[Crimson Wife]

Pardon the philosophy, but this thoughtful group seems as good a place to try it out as any. You seem mostly to be discussing taking courses, and on that score I agree with everything you have said. To me there is a big difference between learning a subject and taking or even passing a class in it, and I am discussing approaches to learning.

 

For most students, math is a gatekeeper and/or a tool rather than a passion. The number of math majors is very small, while the number of students who need certain math courses as a prerequisite for other fields is much larger. To make another analogy with running, it's like the difference between those who truly have a passion for the sport vs. those who see it pragmatically as a way to lose weight/improve cardiovascular health/etc.

[lewelma]

I expected to find future math majors on this accelerated learners thread, at least this seemed to be the place to address them. The question that started this thread was in reference to a 7 year old moving into algebra. This seems to me like a potential math genius/math major.

 

Mathwonk,

 

Please continue to post your thought-provoking ideas. You are right, this is the place to do it!

 

Until my ds started AoPS, he wanted to be an engineer. Now, he wants to be a mathematician. I am going to have him read your posts on this thread!

 

Ruth in NZ

[regentrude]

The ones I took as a pre-med and the ones my DH took as an engineer required use of a graphing calculator. I personally think it is far more likely that my own kids will be in these types of math courses than math major ones that ban calculator use.

 

At our engineering school, engineering students are not allowed to use a calculator on their calculus exams. They are also not allowed to use calculators in calculus based physics. So, it's not just math majors.

[mathwonk]

I agree with much you post crimson wife and appreciate that you are providing a different, possibly less narrow perspective. I am glad mine is welcome here as well. I think they complement each other. :)

[Dana]

At our engineering school, engineering students are not allowed to use a calculator on their calculus exams. They are also not allowed to use calculators in calculus based physics. So, it's not just math majors.

 

Is it no calculators at all, or no graphing (programmable) calculators?

[regentrude]

Is it no calculators at all, or no graphing (programmable) calculators?

 

None. Exams are structured so that calculators are not necessary. Students should be able to do the arithmetic involved without a calculator.

 

Aside from testing whether a student has mastered the use of a graphing calculator, I can not think of any concepts in calculus or physics that I could NOT test with a problem designed to be worked without the aid of a calculator.

Edited November 10, 2012 by regentrude

[mathwonk]

I agree there is never any need to require a calculator on a test, at least if the test maker works hard enough to construct the problem.

 

A calculator is most useful in illustrating basic phenomena that are hard to exhibit by hand.

[Caroline]

I agree there is never any need to require a calculator on a test, at least if the test maker works hard enough to construct the problem.

 

A calculator is most useful in illustrating basic phenomena that are hard to exhibit by hand.

 

I totally agree, and am glad that the AP Calculus exam seems to be trending away from calculator questions again.

[regentrude]

I totally agree, and am glad that the AP Calculus exam seems to be trending away from calculator questions again.

 

Is it? Tell me more, please...

I was under the impression that there were problems which could ONLY be solved using a graphing calculator, and that a curriculum can only be approved as AP if the use of the graphing calculator is incorporated. Has that changed? (I had looked through the syllabus aproval this summer...)

[Dana]

None. Exams are structured so that calculators are not necessary. Students should be able to do the arithmetic involved without a calculator.

 

Aside from testing whether a student has mastered the use of a graphing calculator, I can not think of any concepts in calculus or physics that I could NOT test with a problem designed to be worked without the aid of a calculator.

 

Cool. Thanks.

 

When I was in grad school, the graphing calculator wave was just starting. At my cc, the TI-84 is a requirement from Algebra II onward. However, the department doesn't have common agreement on what students should be doing with the graphing calculator. :glare:

 

I've got my son doing most of his work without the calculator, but I do let him use the TI-30XIIS for some of the CWP problems and IP problems if he shows work very clearly. Started doing that last year in prep for the Explore.

[kiana]

I stand by my original statement that it really depends on the particular math course whether calculators are banned, permitted, or required. The ones I took as a pre-med and the ones my DH took as an engineer required use of a graphing calculator. I personally think it is far more likely that my own kids will be in these types of math courses than math major ones that ban calculator use.

 

This will depend more on the school than the course. At my undergraduate university, for example, calculus 1-2 were primarily calculator-free, although a calculator was encouraged in exams in diffeq/linear for some particularly obnoxious problems (such as inverting a 4x4 matrix). At my graduate university, graphing calculators are banned in nearly all lower-division courses due to text storage capabilities. (They are not actually banned in the upper-division mathematics courses -- although I remember the group theory instructor staring blankly at the student who asked if he could use his graphing calculator, before responding "Well, if it makes you feel better to have it with you, I don't see a reason why not.")

 

Given the variety of courses and schools, it'd be unwise to assume anything. Ideally a student should be prepared to use a calculator when needed and to not use it when not needed. I would err more on the side of lower calculator use as learning to use a calculator is imo relatively trivial.

 

However, again, I see absolutely no issue with allowing its use as a fun toy for free-time explorations, as a tool for a (proper) subset of the problems, or providing some practice before a standardized test. (which is really what the OP was asking about).

[Kathy in Richmond]

Is it? Tell me more, please...

I was under the impression that there were problems which could ONLY be solved using a graphing calculator, and that a curriculum can only be approved as AP if the use of the graphing calculator is incorporated. Has that changed? (I had looked through the syllabus aproval this summer...)

 

The exam changed a slight bit last year. Now only 2 of the 6 free response questions require use of a graphing calculator (it used to be 3 of 6). At least it's a step in the right direction...