what's the hive position on kids doing math in their funny little heads?

[Nicole M]

My almost 12yo son "prefers" to not "waste graphite" by writing out all those carried carried numbers, and regrouped digits and all that. He does a great deal of his work in his head; working this way, I'd guess he gets an incorrect answer no more often than about once a month.

 

So. If I allow this to continue, will it come back to bite us later, when writing out work is essential? (Does it become essential?) My instinct is that it's good to get into the habit of showing your work. But I'm confused, because he's right nearly always and seems to make more errors when I have him write out the work.

 

Whaddya say?

[nmoira]

Our rule is: If it can be done mentally, it is done mentally. However, this applies only to arithmetic. Equations are written out for word problems (left to right, not up and down), and answers are checked for logical flow. For me, "showing your work" means showing the reasoning behind and being able to explain what is happening at each step. My goal is simple: By the time I permit calculator use, the girls will have had years of experience performing mental calculations.

[Melissa in Australia]

my ds14 does all his math in his head. he is 3/4 way through Saxon algebra 1.

he has dyslexia and has trouble writing all the math down.

I have read that some dyslexic people can do very complicated advanced math in their head, it is just the way their brain can visualise it.

I have told him as long as he is getting 90% plus right he can keep doing it this way, soon as he starts getting mistakes, he will have to show workings out.

I find it fascinating seeing him sitting there staring at nothing for up to 15 minutes and then just wright the answer down.

 

somebody told me that you can get penalized at university if you don't show workings, I don't know if this is true or not.

[Kelli in TN]

I think Myrtle made a good point about this on her blog. And don't forget to watch the little youtube video of her son making his argument for not having to show his work.

 

Even though I use an unamed and perhaps wimpy math curriculum; one that Myrtle would never use, after reading her blog post about showing work I have become pretty strict about writing out the steps of anything that has more than one step.

[8filltheheart]

It is absolutely true. :)

 

My ds is an engineering major and has completed cal 1,2,3 and differential equations. Final answers are almost inconsequential to the entire process. He has gotten marks off for simply not writing down every step.

 

Writing out the work is 90% of the grade in our household. If I can't see the work, I mark it wrong.

[Lolly]

Well, it may make for more arguments later on. Does it completely mess them up forever? no. In algebra, they need to show the steps they took to answer the problems. The scratch paper work (what your son is probably doing now) can actually still be done mentally if they are capable of it. But, writing down the various steps does become necessary.

[snickelfritz]

It is absolutely true.

 

They also usually give partial credit if you work a majority (or even part) of the problem RIGHT and make one error somewhere. In my classes, I usually showed every step I could to get as much credit for the problem as I could.

[PeterPan]

Just as a compromise, did you know RightStart suggests using dots for carrying, underscores for trades, and doing short division (where the carries are written in the division house and nothing long is written out)? But I totally agree, by algebra he has to show it all.

[In The Great White North]

It is totally possible to do all the math in your head, certainly up through pre-algebra.

 

Writing proofs in geometry will teach him to write each step, but it might be painful. Ds had a very hard time doing proofs because he hates writing each step.

 

Some kids get the answer in their head, then figure out the steps and write them down.

[KS_]

My ds would much prefer to not have to write down anything except the answers and is very good at calculating in his head. But because of his very strong personality and resistance to change, I have insisted that he learn how to write the processes down and that he does it for at least some of the problems in each math assignment. I know how critical showing work is for later math and agree with the other posters who said that in most college classes it's not optional and the final answer is only a small part of the total credit for each problem.

 

I know for my ds, trying to learn to show his work at a later age would only make it that more difficult for him, so although I do encourage mental math, I also insist on showing his work.

[Lisa in the UP of MI]

Ditto what many others have said about college classes. In my higher math and science classes showing the work was much more important than getting the right answer. What matters is how we got the answer not the actual number that we came up with in the end. I will allow my children to compute simple mathematics in their heads but once they get to algebra they will need to show their work.

[lovemyboys]

Our rule is: If it can be done mentally, it is done mentally. However, this applies only to arithmetic. Equations are written out for word problems (left to right, not up and down), and answers are checked for logical flow. For me, "showing your work" means showing the reasoning behind and being able to explain what is happening at each step. My goal is simple: By the time I permit calculator use, the girls will have had years of experience performing mental calculations.

 

 

I like this distinction. Dc here are starting to prefer working mostly in their heads which is great. But we have not hit algebra yet. Dh is excellent at this, which I consider a skill, so I'm glad to encourage it. I will keep in mind your comment here.

[forty-two]

As I see it, the main reason for showing your work is so that other people can follow your thought process(es), and (hopefully) understand why you did what you did. The universal, absolute rule is that you need to show all "thinking steps", but you can leave out rote calculations. But determining which step is "thinking" versus "rote" is, alas, more relative than absolute. (However, declaring variables is *always* a thinking step; just try to decipher what an eq means w/out knowing what the variables stand for.) It is dependent on:

 

1) *What*, exactly, you are trying to show: If you are trying to show *why* 1+1=2, a la Myrtle in the linked blog post, you need to show much more detail than if you are merely *using* the established fact that 1+1=2 to show something else.

 

2) The math level of your intended audience: The more math knowledge your audience has, the more steps that you can "leave out", since everyone already takes it for granted. In a geometry proof for high schoolers, if you claim that angle A = 45 deg because its supplement is 135 deg, you don't need to show all the steps involved in 180-135=45. Everyone already understands how the subtraction algorithm works, and can easily check the result obtained. But in a class of 1st graders, you can't assume that each child understands how the subtraction algorithm works, and would need to explain each step in depth.

 

Number two doesn't matter as much to a student showing work for the benefit of their teacher, as the teacher clearly can be assumed, in most cases, to have greater math knowledge than the student. (Although it would come into play if a student was tutoring a younger/weaker student.)

 

Number one, for students, would depend on what knowledge the teacher is "testing" (for lack of a better term). If the teacher is trying to see if the student understands how to use a specific algorithm, like long division, then of course the student would have to show each step in the long division algorithm; however, they could do the subtraction in the long division algorithm without explicitly showing regrouping, as that knowledge is assumed. But if a teacher wanted to see if a student could set up and solve a word problem that involved long division, the focus would be on properly setting up the eq, and the resulting division problem would not need work shown.

 

Basically, the higher up in math you go, all the previous stuff learned can be assumed unless otherwise stated. I believe even in proofy college math classes, the general practice is to consider results proved in previous classes to be "common knowledge", and so you don't need to keep "showing your work" to use them.

 

As far as partial credit goes, the more insight into your thinking you can give is always good, but there is no "thinking" shown by showing each and every arithmetic and algebraic manipulation you do in the course of solving a calculus or physics problem. They assume you already know that stuff. Just write down whatever you need to solve it correctly - even if that is nothing - and move on to the next step that actually involves thinking about calculus or physics.

 

So if your child has demonstrated to your satisfaction that they can do addition with regrouping, then I see no need for them to continually have to "show their work" when they do addition. Same with any other procedure. Do you really expect them to show each and every step of the addition algorithm when they are in pre-calc? Or each algebraic manipulation in physics?

[8filltheheart]

 

1) *What*, exactly, you are trying to show: If you are trying to show *why* 1+1=2, a la Myrtle in the linked blog post, you need to show much more detail than if you are merely *using* the established fact that 1+1=2 to show something else.

 

2) The math level of your intended audience: The more math knowledge your audience has, the more steps that you can "leave out", since everyone already takes it for granted. In a geometry proof for high schoolers, if you claim that angle A = 45 deg because its complement is 135 deg, you don't need to show all the steps involved in 180-135=45.

 

I don't really agree with this. For example, in geometry, while it may be understood that angle A=45, in a proof students do have to write that supplementary angles add up to 180 or the definition of supp. angles or something along those lines to prove where they derived the info.

 

I think that based on the problems involved, it is fairly obvious to both student and teacher which math skills are appropriate to have written out. Just b/c it can be done in their head, does not excuse that is should be.

 

Bright math students typically want to not write down their work. It will be a problem in higher level math courses and their laziness will catch up with them. Typically, the final answer is worth far fewer points than the process that gets you there.

 

It is easier to be in the practice of writing out work from the beginning than having to learn the process later on.

[forty-two]

...I will allow my children to compute simple mathematics in their heads but once they get to algebra they will need to show their work.

I don't think it is necessary to show each and every algebraic manipulation once its been mastered, any more than it is necessary to show each and every arithmetic step, once its been mastered, especially for application problems.

 

Given this problem:

June spent $120 on 10 items of clothing; some were shirts and the rest were pants. If shirts cost $10 and pants cost $15, how many of each did she buy?

Here are the "thinking steps" I would require, once all pre-req skills were mastered:

 

Let s be the number of shirts bought, and p the number of pants bought:

s+p=10

$10s+$15p=$120

[$10(10-p)+$15p=$120] This step is optional; it isn't really needed, per se, once the skill of substituting one eq in another has been mastered, but it is helpful. I was able to solve this problem mentally without writing it down.

s=6, p=4

 

Anything else would be scratch work, and only required if they can't get the correct answer without it.

[forty-two]

In a geometry proof for high schoolers, if you claim that angle A = 45 deg because its complement is 135 deg, you don't need to show all the steps involved in 180-135=45. Everyone already understands how the subtraction algorithm works, and can easily check the result obtained.

 

I don't really agree with this. For example, in geometry, while it may be understood that angle A=45, in a proof students do have to write that supplementary angles add up to 180 or the definition of supp. angles or something along those lines to prove where they derived the info.

 

Quite right. What I was trying to say was that even in a proof, certain things can be assumed, or else you'd have to go back to first principles each and every time. In this case, what I was envisioning was something like this:

Proof of [blah-blah]:

... ... ...

Ang A and Ang C are supplementary (oops on that silly mistake) because [blah, blah, blah]

Therefore Ang A = 45 because the measures of supplementary angles sum to 180.

... ... ...

 

You don't have to make a big production out of showing your work for the subtraction required - the knowledge of how to subtract is assumed.