When to allow a calculator

[Embassy]

I have a almost 12 year old PG visual-spatial learner who is currently working through AOPS Pre-Algebra.  Concept-wise, math comes easy to this kid.  The hardest parts on AOPS so far have been problems that require lots of standard calculations.  He could probably speed through the book if given a calculator to use.

 

Math facts have been quite slow because he figures out the answer in his head each time (6x6 is worked out in his head rather than known immediately).  We worked on math facts for YEARS and if we don't continue to practice, his speed significantly slows down.  

 

Part of the issue is math facts and part of the issue is engagement.  He does not find rote problems engaging - even if they are part of a greater puzzle.  If he is not engaged, his mind wanders.  Memorization has been difficult.

 

I wonder if giving him a calculator to use for rote problems would hurt or help him at this point.  He has a huge interest in a future with STEM and spends his free time pursuing his computer interests (mostly creative like modeling).  

 

It seems like he needs to speed ahead conceptually to be engaged with the content, but I'm hesitant because math facts seem so important.

 

What would you do?

[CadenceSophia]

I might be the wrong person to ask, but I do have a degree in math. I used my calculator for everything. IIRC, we were allowed to start using calculators at some point in algebra and it never left my side. I don't see why he shouldn't be allowed to use a calculator for long calculations as long as he definitely knows how to do it. I would NOT allow him to use (graphing) calculator functions to solve equations yet though. As long as he is responsible or you only give him a 4 function calculator, I don't see it as a problem. 

[EndOfOrdinary]

Normally with AoPS, there are very few calculations. In fact, the whole point is to minimize them as much as possible. The calculator is openly discouraged with this curriculum for that reason - using it often means much less is gained in problem solving. Since huge portions of the program involve factoring out multiples, this might be where the problem lies.

 

I am not a major math facts advocate (much to the horror of the other math teachers I worked with). Often, through repetition of doing computations they begin to stick or through all out frustration of time suck they are memorized down and dirty style. Does your son recognize this as an issue? If he does, and legitimately has tried memorizing but can't, have you considered there might be a bigger issue?

 

More often than an actual disability, I have found kids just don't care. At that point the calculator is enabling. If he is really wanting STEM, he either has to learn quick distribution (my Ds' route) or memorizing. They aren't going away. They will be expected to be performed quickly, recognized quickly, and used to simplify otherwise highly complex situations.

[regentrude]

AoPS (I have used the entire series beginning with Intro to Algebra) is designed to be used without a calculator, except for a very small number of problems per year, mainly in trigonometry, where the book tells you explicitly that you may use a calculator. Very often, the learning objective will be missed completely if a calculator is used. We found that whenever we were tempted to do tedious calculations, we were approaching the problem in an inefficient way - it was pretty much a sign that we were doing something wrong. There were no "rote calculations".

 

I teach physics at a university and see the detrimental effect of calculator dependence in students every week. They have developed no number sense and number familiarity and are not capable of performing simple arithmetic accurately without calculators (I am not talking complex long divisions, but simplifying fractions, seeing perfect squares etc).

I would discourage calculator use as long as possible. With a well designed curriculum like AoPS there is no need.

(ETA: if your DS is interested in STEM, he should be prepared that many university math and science courses prohibit the use of a calculator on exams)

 

My kids were allowed to use calculators for the computations in their science courses, but managed math through multivariable calculus without regular calculator use.

Edited December 12, 2015 by regentrude

[SparklyUnicorn]

If multiplication and division are slow going, you could give him a table.  That is what I did.  That same kid no longer needs it. 

 

 

[mathmarm]

There is minimal need for a calculator in AoPS. If you are needing to perform long, sloggy calculations you are missing the point and not seeing the real problem that is to be solved.

 

If I were you I'd reevaluate why I was using AoPS with this student and try to determine if there was another curriculum that would fit him better.

No matter what math curriculum you go to, I would strongly advise you to instead of a calculator, give the child a hardcopy of the multiplication tables and a print out of powers, etc...

 

Whatever simple calculation he would be performing with a 4-op calculator can be performed by referencing the charts until he no-longer needs them.

[mathmarm]

Also, most STEM schools do not allow calculator use in math classes or even if they are allowed in class, then not on exams. Your son is going to want to learn how to deal with math in a mathematically sound fashion.

 

 

[wapiti]

It's hard to remember, but IIRC, the more (relatively!) tedious calculations in the Prealgebra text are limited to ch 2-3 and perhaps a few other random lessons in the book (e.g. approximating square roots).  Most of the time, if it's getting rather tedious, it's a sign that the student might not be using the concept the "smart" or "easy" way.

 

For most kids, I would say that the small number of lessons with more calculations is worth the calculation practice and practice getting a better feel for the numbers - perhaps the work in the number theory chapter is especially good for that, for a person with fact issues.  For someone with significant issues, I might let him use a multiplication table (of his own making, remaking one every week or something).  I think it's best to avoid the calculator with AoPS in order for the concept work to be practiced as the text intends, unless the text specifically allows it.

 

This perspective does not extend to other math programs.  It has been annoying for my kids to show up to math tests at school without a calculator...

[RoundAbout]

I have a 9 year-old in Chapter 3 of AOPS Pre-Algebra (taking a break for Beast 5a) and the only problems I've been tempted to give him a calculator on have been the few problems where you have to test larger numbers for divisibility by multiple primes where there are no easy divisibility rules - 7, 11, 13, etc.   It gets a bit tedious but figure its good practice for long division even if we get fewer problems done in AOPS. He's also developed enough number sense to test more quickly in most cases (ex. 840 is divisible by 7 because 700+140 is divisible by 7)

Not knowing multiplication tables would be a show stopper for me.  I'd be tempted to spend 15-20 minutes playing multiplication card games daily (RightStart has some really great ones) rather than turn to a calculator.  

Edited December 12, 2015 by RoundAbout

[Embassy]

Thanks for all of your suggestions.  You have given me much to think about.

 

I believe that even if my child does have the multiplication facts down cold, we would still have issues here....I think it is only part of the issue.

 

I thought there might be some learning difficulties in the past, but the educational psychologist felt any issues were related to his very strong visual-spatial learning style.

 

Engagement is a must for this kid.  Let me explain, hoping that you all can give further suggestions or insight.

 

If I give him a problem like 375/13 and have him turn it into a decimal - it will take him a long time.  His mind wanders.  He does calculations quite slowly and I often see him daydreaming.  I sit next to him for math almost all the time.

 

However, if I give him a problem like 37634889342/2432 he is much more engaged and focused.  Calculations may be slow, but no where near as slow as the easier problem.  This kid is a poster child for easy is hard and hard is easy - and this shows up in many subjects.

 

He prefers mental calculations over showing his work and can manipulate numbers quite well.  He doesn't often make errors and seems to have good number sense.   He will often come up with the correct answer by doing a problem in his head in a way that he can't explain - at least I don't understand the way he explains it.   However, multiplication tables are not automatic.  We have worked on them for years.  Addition and subtraction were also issues at one point, but it seemed to go away when I accelerated him through his curriculum.  He was having difficulty with subtraction with regrouping and would make tons of errors - until I accelerated him and he made it to more difficult problems.

 

I don't question AOPS as choice because it really suits him.  The figure it out for yourself (top-down) methods and the abundance of word problems are right up his alley.  He would wilt if given pages of similar problems to do.  

 

Does anyone else have a kid like this?  This kid is a constant puzzle.  Maybe a multiplication chart would be the best move here. 

 

 

 

[Embassy]

I should say that last school year, I focused on helping my kid learn multiplication tables and practice them with mental calculations.  I put conceptual studies to the back burner and it didn't seem to help.

[chocolate-chip chooky]

To get our times tables up to scratch a while back we used the online site called SumDog. Your child plays games against others online and the focus is on being fast.

My daughter hated it for word problems, as it's hard to be fast, but for times tables it was excellent.

 

If you sign up with a parent account, you can select the types of questions your child gets in the games.

So, I'd narrow her questions down to things like:

 

6 times table using division

9 times table using multiplication

 

I knew that every time she played a game, she was working on the specific skills that I'd chosen.

 

[SparklyUnicorn]

To get our times tables up to scratch a while back we used the online site called SumDog. Your child plays games against others online and the focus is on being fast.

My daughter hated it for word problems, as it's hard to be fast, but for times tables it was excellent.

 

If you sign up with a parent account, you can select the types of questions your child gets in the games.

So, I'd narrow her questions down to things like:

 

6 times table using division

9 times table using multiplication

 

I knew that every time she played a game, she was working on the specific skills that I'd chosen.

 

I tried something like that with both of mine.  It freaked them both out. 

It's weird, both of mine understood concepts quickly, but it took a long time to solidify the facts. 

[lewelma]

I have a kid like yours.  Same exact trouble with memorizing math facts.  Way way too boring, but with a hint of something else, almost like some sort of mental block about working with numbers in a memory sort of way.  Not exactly a learning disability, but definitely beyond the realm of normal interaction with a straight forward job that simply needed to get done.  

 

My solution:  No calculators, accelerate until engaged, and continue with the facts.  This meant that my ds had to do the multiplication facts 3 times per day 7 days a week (after every meal) while concurrently working through AoPS intro Algebra.  My logic was that the moment I gave him a calculator, he would lose all motivation to get the facts down, and in the end you have to know them if you are going into STEM.  It is just too important to algebra to see how things factor and cancel to think that using a calculator is even possible.  So AoPS Intro Algebra went *very* slowly, as in it took almost 3 years to complete.  He had the time and the motivation, so I did not really care.  

 

For your situation, you said your kid was PG and 12 doing PreA?  I would suggest that you abandon PreA and move straight into AoPS Intro Algebra.  There are fewer computation problems, and all the PreA material is covered in a faster manner (that is the algebra material, not the geometry or counting material in PreA).  A number of us did Intro Algebra without PreA because PreA had not been written.  Intro Algebra focuses in Algebra and you won't find any decimal problems in the book as far as I can remember.

 

Also, I would suggest that you skip division problems like the one you suggested.  Instead, I would practice estimating the answer with finer and finer detail.  That kind of work will teach your son to break the numbers apart and put them back together.  There is nothing wrong with mental math for calculations as long as you are getting the answer right.  Algebra is where you want to start working towards proper workings.  

 

I'm a maths tutor and work with a lot of different types of kids.  There are many many ways to skin a cat and it sounds like you need to think out of the box with this situation.  I say make the work harder and keep the facts going.

 

Ruth in NZ

[lewelma]

 

This kid is a poster child for easy is hard and hard is easy

 

 I have often described the difficulties my ds had with drill-type math to proof reading a phone book.  It may be easy, and you may have good intentions and the capability, but you just can't do it.  It it not an attitude problem or even a focus problem, it is just so laborious and boring that it cannot be done.  Not just cannot be done with accuracy, but cannot be done at all. 

 

My ds is very similar to yours.  He is his own normal. 

Edited December 12, 2015 by lewelma

[CadenceSophia]

Well I learned something about AoPS! 

 

My son is the same way with the easy is hard, hard is easy. I am in the same situation (to some extent) in that Singapore has a painful amount of repetition for him.

[Embassy]

Thanks all!  Lewelma, it is nice to hear from someone with a similar child.  You all have given me much to think about.  

[PinkyandtheBrains.]

 

 

No calculators here, one is in calculus the other is in pre algebra.

 

My son who struggles with math fact memorization has a multiplication chart to reference as needed. We keep moving forward and working on facts at we go.

 

 

As others have said, AoPS is designed for no caluculator use.

[slackermom]

My dd did AOPS pre-algebra without a calculator. If she got bogged down in a calculation, I would have her re-read the problem, because, as others have said, the problems in that book rarely call for that type of work.

Regarding your problem converting 375/13, I think an aops type problem wouldn't be looking for something to be worked out the tenth place, the way my calculator provided 28.8461538462. It would probably be enough to know it is between 28 and 29, and any further precision would depend on what was being asked.

[daijobu]

Sometimes we use a calculator with AoPS just out of sheer disbelief.  Like, have you ever solved a problem involving the sum of 2 different square roots, where you are taking the square root of the sum of an integer and a square root, but you solve it by squaring both sides, using some factoring techniques and ending up with an integer for an answer?  It's fun to see the same integer result when you do it on a calculator.  

[Embassy]

The problems I referenced weren't from AOPS.  It was my feeble attempt to demonstrate how easy is hard and hard is easy.  I do understand and agree that AOPS can be done without a calculator.  When my older child went through the book, I only had him use a calculator for the statistics chapter.  It could have been done without it though.  

 

I agree that most things require thinking to figure out rather than long calculations.  We don't do long calculations - not unless I want 1 small problem to take an hour!  It is the small calculations required that take a long time.  He would sit there staring at something like 13x6 for 5 minutes (even though he could do it relatively quickly in his head as 60+18, figuring out 6x3 rather than knowing it automatically)...daydreaming because he is not finding it engaging.  Now when there is a word problem, my child finds the solution quickly or if we encounter math in day-to-day life, he is usually the first to answer - quicker than my child who knows his math facts quickly.  This kid is a puzzle!

 

I'm probably going to let him loose on Alcumus for awhile while I attempt to accelerate him through Pre-Algebra.  I'll see how he does with Alcumus to determine the next step and how fast we should go.  As for math facts, I think I'm going to try out accelerating him first.  That worked when he had difficulty with addition/subtraction facts.  Taking time to focus on math facts last year instead of going forward with concepts may have been a mistake.  I think he needs the concept depth to learn the facts.  At least that is my working theory now.  So, no calculator plans here.  His younger sibling has a multiplication chart on the wall that he will look at sometimes.  I guess I won't stop him from doing that for now.

 

Thanks for all your suggestions!