Why do kids need to show their work in math?

[lamolina]

I hope this doesn't sound totally clueless, but I really wonder why children must show their work in math. It seems to me that if they are able to consistently get the problem correct they have obviously mastered the material right? Could you explain why showing their work is important and/or what skills that teaches?

 

thanks!

[Lily_Grace]

In my experience it is because often, if they make a mistake, they may not know exactly what went wrong. And in higher math if that mistake in a formula gets the right answer some or most of the time it could appear they have grasped the material when they really haven't.

[wy_kid_wrangler04]

I have my kids show their work so when they get to more advanced stuff that they can't do in their head (and the time WILL come) then they will not struggle with the written steps. Plus its insurance for me to know that they actually know the correct steps and not some other way to get the answer correct.

[Beth in SW WA]

I hope this doesn't sound totally clueless, but I really wonder why children must show their work in math. It seems to me that if they are able to consistently get the problem correct they have obviously mastered the material right? Could you explain why showing their work is important and/or what skills that teaches?

 

thanks!

 

What level of math are you working with? For older dc, they show their work because the problems often take up a whole page. It's insightful to see their train of thought as they work through an alg or geo problem. They can go back and see where they may have made a mistake in computation.

 

For my youngers, I don't make dds work out every problem on paper -- EXCEPT the bar models in Singapore. I actually require my dd7 to make bar models in her TT lessons as well -- as she is just learning the Singapore bar model method. It is a brilliant exercise that forces her to 'see' the problem as she processes through the steps. I require dd8 to do bar models on her word problems in all the math programs she uses (Sing, MM, TT).

 

I don't make them write out any problems that are the 'do it in your head' type (estimating, rounding, adding/sub/mult/div numbers ending in multiple zeros, etc).

 

With fractions, decimals, negatives, solving for unknowns, etc -- it is beneficial for dd8 write out every step (even if she can skip steps by doing it in your head). For now this is working well.

 

I would love to read what others do. Great question, OP. :)

[Jim]

We do it both ways. Every math session starts with a math facts sheet and mental math. He does the work in his head. After the new lesson I require he write down each step. For me, it's laying the groundwork for logically walking through a puzzle from start to finish.

 

He may be able to do the problems mentally at his current level but that will not always be the case so I want him to habitually work it through on paper. It will allow us to check every step.

 

 

Jim

[twoforjoy]

I'm having this issue with my son with regrouping. We just started regrouping for addition. He can figure out the problems, right now, in his head, and doesn't get why I'm making him write them out.

 

But, once the problems get more difficult, he won't be able to do them in his head. And if he doesn't understand the process for regrouping, and how to go through it step-by-step, he won't be able to solve the problems.

 

At least this is what I tell him when he complains to me about how it's not fair that he has to show his work.

[chepyl]

Younger children showing work is preparation for more advanced math. Showing work neatly is a skill that must ne learned and practiced. I have my son show work on some problems. He would do them all in his head. Every now and then he hits a word problem that he can't quite grasp in his head, we practice writing word problems out to build the habit of writing them for when he gets to the harder problems. He still does them all in his head and then we write them out to check his answer.

[lionfamily1999]

I have my kids show their work so when they get to more advanced stuff that they can't do in their head (and the time WILL come) then they will not struggle with the written steps. Plus its insurance for me to know that they actually know the correct steps and not some other way to get the answer correct.

:iagree: I learned my lesson when dd started Algebra. Now ds must show his work. He needs to practice writing it out and TAKING HIS TIME.

[Alice]

What everyone else said.

 

My son is pretty good at math but often has difficulty telling how he got an answer. It's usually not the way I would get it or necessarily the way the book tells you. I'm ok with him doing it "his" way in his head but I also want to make sure he knows the more conventional ways. A lot of times this is because the conventional way will help him when it gets more advanced or I think it's just valuable for him to know more than one way to do a problem.

 

I've come to a compromise with him in that I don't make him show the work for every problem but for a sampling so that I can see how he is doing it.

[foxbridgeacademy]

I always hated having to "show my math" a a kid. I figured if I knew the answer why should I have to show how I knew. Now with my DC I see why they should. If they get the answer wrong I can see where. When they get it right it helps to "secure" the correct process in their heads. What I do though is require they only "show" work on the first 1/4 to 1/2 of the problems, or on the ones I think will trip them up. This saves a lot of grumbling from them and I still feel like they are understanding the lesson.

[ktgrok]

so that if they get it wrong you can see WHERE they went wrong. Did they do the problem incorrectly, or did they just make a mistake when carrying a number, or what? My son just yesterday got one wrong, but I was able to see that he had confused a plus sign for a multiplication sign. Also, that way they don't have to redo the whole problem to fix it, just the part they got wrong.

[regentrude]

As others already pointed out:

- so they can see where any mistakes have been made

- so you can make sure that they get the right answer for the correct reason, and not because two mistakes happened to cancel (which is more common than you might think)

- because starting with prealgebra at the latest, many problems can NOT be done without writing them down - careless mistakes, especially sign mistakes, are often a direct result of not being used to writing down every step neatly

[Christie_P]

My 6yo is starting to work some word problems in her math (Math Mammoth) and I'm having her underline the numbers in the problems (i.e. Jack has _six_ cars....) so that when she gets to bigger and longer word problems, she will be in the habit of underlining the important information to make it stand out. (you know how they always love to throw in something that isn't required to solve the problem!)

[boscopup]

If it's a simple problem like "Mary has 5 apples. She eats 2 apples. How many apples does Mary have now?", I'll let him do it without showing work. If the problem is multi-step or needs some bar diagrams drawn to figure out how to solve it, I have him show his work. This seems to be a happy medium where he is learning to show work (so it won't be a new thing when he gets to prealgebra in a couple years), but at the same time, he's able to do simple problems in his head still.

 

Doing Singapore CWP has been helpful for showing work, since they have you drawing the bar diagrams. He enjoys drawing them and will sometimes draw them even if he doesn't HAVE to in order to solve the problem.

 

Now if the problem has multiple steps, like "Mary has 5 apples. She eats 2 apples, then gives 1/3 of the remaining apples to her friend Bob. Bob then invests in the stock market and uses the proceeds to buy more apples. He increases Mary's apple possession by 200%. How many apples does Mary have now?" He better show his work on that one. ;)

[dcjlkplus3]

I let my more mathy kid get by with just giving the answer - BUT.. if it is wrong she has to show her work, so I can see what she is doing - it also helps her to focus and see where she is making the mistake.

[Spy Car]

I'll be the contrarian and say I don't think it is best that a child always has to show his or her work.

 

In fact, I think that demand would limit math skills.

 

Part of being math adept is being able to manipulate numbers in ones mind (mental math). This is a skill that is undermined if a child has to work out solutions on paper.

 

So I am not misunderstood, I think there are times when showing ones work is a valuable endeavor (for many of the reasons stated above).

 

And, critically, I have generally demanded (especially when a topic was new) that solutions derived at "mentally" be justified "orally", that is to say the strategies and mathematical laws and processes need to be explained.

 

But the initial computations are (sometimes) done mentally.

 

Simply writing out a math problem doesn't necessarily show that a child can justify their answer. And solving problems successfully on paper via the standard algorithms is not necessarily a proof a child really understands what they are doing.

 

I like teaching for understanding. Mental math should be part of the equation along with acquiring the skill to be able to "justify" ones work (as opposed to just writing down the procedures).

 

Bill

[crazyforlatin]

I'm with Bill. In fact, I've tried to skip all standard algorithms so that DD can work every problem mentally. We're now working on 2-digit times 2-digit problems and 4-digit minus 4-digit with regrouping problems, and it would be too easy if I allowed her to work them out on paper. It takes time since I have to sit with her and listen to every problem as she calculates them out loud, but I can see vast improvement in how fast she calculates. Whatever mistakes she makes is figured out orally as well. We're also practicing long division orally and it takes a lot of time for both of us, but I'm trying to avoid the education I received in parochial school where almost all problems were taught only procedurally and it would have been difficult to tell if a child understood them conceptually. With word problems, I have her diagram them on paper.

 

But, I do teach the standard algorithms as soon as I feel that the concept has been mastered and the problems can be done mentally and quickly. I don't know how long we can do this mental workout as we progress in math.

 

ETA: Just to clarify, I do have DD show how the problem is worked out, but that is done only orally, and I doubt she will be able to do this once we're multiplying into the thousands.

Edited August 31, 2011 by crazyforlatin

[lamolina]

Thank you all for replying to this! It is interesting to see all the reasoning behind making them show their work. Lots to think about there!

[LucyStoner]

Why? Because my math professors made me do it all through college and if I had to suffer...just kidding.

 

Actually, I agree that it is not always necessary but it does help me see not just where he went wrong but HOW he went wrong...was it an arithmetic error or did he mess up his order of operations or something else. That helps me figure out what to work on or see that he really gets it but had a brain fart and added 1 and 1 to 4 somewhere. It happens to us all. The only problem that a friend of mine missed on the math SAT was that he said a cent equalled a dollar. :D

 

I tutored math in college to primarily high schoolers or remedial college students (though some advanced math as well) and have always mixed mental with written work.

[kiana]

For college classes:

 

If I can do it in my head, I assume that a student may be able to do so also, and won't deduct points for lack of work. However, no partial credit will be awarded for incorrect answers when work is not shown.

 

If I cannot do it in my head, work needs to be shown.

[Haiku]

I hope this doesn't sound totally clueless, but I really wonder why children must show their work in math. It seems to me that if they are able to consistently get the problem correct they have obviously mastered the material right? Could you explain why showing their work is important and/or what skills that teaches?

 

Because when kids get to higher level math, they will not be able to solve math problems in their heads. They will have to work them out on paper, in multiple steps. Might as well get in the habit now.

 

Also, if a child does get an answer wrong, if their work isn't shown, it's hard to know where they went wrong.

 

Tara

[Haiku]

Part of being math adept is being able to manipulate numbers in ones mind (mental math). This is a skill that is undermined if a child has to work out solutions on paper.

 

Not necessarily. My kids can derive their answers however they see fit, mentally. But then, we write them down. As we write them down, we talk through how what they are writing corresponds with what they did in their heads (or differs, if that's the case). I think it's mutually reinforcing.

 

Tara

[txhomemom]

Whatever you do don't neglect having your child right out the steps in a math problem at an early age. I used to let my dd do a lot of mental math because that was her thing, but then it got to a point that we were doing more difficult problems and she was making a lot of mistakes. It took me time to convince her to write them out and she hated having to do borrowing problems, but now she knows them forwards and backwards and she has the ability to manipulate them in a more mental fashion as well. There needs to be a balance.

[ChandlerMom]

IRL people who use math for a living HAVE to show their work -- engineers double-check each other, mathematicians and physicist have to publish and defend their derivations (solutions) sufficiently that someone else can reproduce their work in full. My mom who is an RN says, "if you didn't chart it, you didn't do it." meaning everything should be documented at the time it's done.

 

On the other hand, our children are not professional engineers or mathematicians (yet :D), and math skills often exceed writing skills in elementary years. The point isn't so much getting the right answer, but using the correct process/thinking. So, despite the dire warnings to college frosh "if you get a sign wrong the bridge will fall down!" (actually, that's WHY work is double/triple checked :lol:) if my dc make an arithmetic error but their process was correct, I point out the error but do not *always* make them rework it (or I work it out for them). If they were chronically careless, I'd handle that differently, but I've got a perfectionist on my hands!

 

When I was in 3rd grade my handwriting was horrendous and it was hard to write out my math neatly. We had a 12' chalkboard at home, so I decided to go home, work out the problem on the board (much easier), then I write the answer on the sheet. I had maybe 6 assignments on one side of a piece of paper, just a patchwork of odd shapes blocked off for each assignments, each block full of numbers with boxes around them. I turned it in and my teacher was horrified. :lol: "Where is your work?" "At home on my blackboard," I replied. Since I excelled at math and was the first done with my assignment she couldn't doubt I'd done it myself. "Well, you need to show your work." "Why?" I replied. She paused, and wisely compromised, "Please work out the first problem on each assignment so I can see what you are thinking. And just put one assignment per page." I talked her into TWO assignments per page (she showed me how to just separate them with horizontal line).

 

So....I think it is a balance. It is easy to kill their interest in math by too much writing and busywork, and I agree with Bill that the ability to free think math without paper or pencil is a VERY important skill. Having the time and ability to mentally PLAY with math is critical to enjoying it, like playing an old favorite tune on the piano for fun instead of always working on the next hard piece. However, the process is more important than the answer, and being able to document and explain your solution is also an important skill.

 

For us, the balance is that I rarely require dc to show their work. I choose a couple problems each day for her to "show me how" she solves them IN DETAIL. And I've got a big blackboard. :lol: As they get older I may have them check each other's work, which is a quick way to impress the need for detailed solutions.

 

ETA: math is a subject we mostly do together, so aside form computational practice, concepts are discussed and demonstrated orally by dc, or on the chalkboard or with objects. This means I really do SEE her process and what she understands or doesn't. At least half of math is done in a socratic method discussing the ideas of math (like nature of negative numbers, today).

Edited September 1, 2011 by ChandlerMom

[zenjenn]

IRL people who use math for a living HAVE to show their work -- engineers double-check each other, mathematicians and physicist have to publish and defend their derivations (solutions) sufficiently that someone else can reproduce their work in full. My mom who is an RN says, "if you didn't chart it, you didn't do it." meaning everything should be documented at the time it's done.

 

:iagree:

 

It lays a good foundation for computer programming, too.

 

When you are working collaboratively with other people, everything you've done has to be clear and written down step by step.

[Beth in SW WA]

I'm with Bill. In fact, I've tried to skip all standard algorithms so that DD can work every problem mentally. We're now working on 2-digit times 2-digit problems and 4-digit minus 4-digit with regrouping problems, and it would be too easy if I allowed her to work them out on paper. It takes time since I have to sit with her and listen to every problem as she calculates them out loud, but I can see vast improvement in how fast she calculates. Whatever mistakes she makes is figured out orally as well. We're also practicing long division orally and it takes a lot of time for both of us, but I'm trying to avoid the education I received in parochial school where almost all problems were taught only procedurally and it would have been difficult to tell if a child understood them conceptually. With word problems, I have her diagram them on paper.

 

That is really awesome! All my dc have done it the old fashioned way along with doses of mental math -- but not to the level that you do. Dd8 does tons of math in her head (at random times) when able. It's fun to watch her wheels turn as she's figuring out an equivalent fraction, then either adding, mult or dividing it. I definitely don't have the patience or time to do long division mentally with my kiddos -- but I think that is wonderful that you are doing that. I'm inspired. :)

[Spy Car]

Not necessarily. My kids can derive their answers however they see fit, mentally. But then, we write them down. As we write them down, we talk through how what they are writing corresponds with what they did in their heads (or differs, if that's the case). I think it's mutually reinforcing.

 

Tara

 

Writing answers out after the fact (having done the math mentally) is not the same thing as working solutions out on paper. There are times when it is beneficial to work through the standardard algorithms on paper. And, as I said, I often insist (especially in the pre-mastery stage) ask that mental math be explained, but this (and writing out solutions following mental math work) are very different than only doing procedures on paper.

 

Doing only the standard algorithms on paper is limiting.

 

Bill

[Spy Car]

I'm with Bill. In fact, I've tried to skip all standard algorithms so that DD can work every problem mentally. We're now working on 2-digit times 2-digit problems and 4-digit minus 4-digit with regrouping problems, and it would be too easy if I allowed her to work them out on paper. It takes time since I have to sit with her and listen to every problem as she calculates them out loud, but I can see vast improvement in how fast she calculates. Whatever mistakes she makes is figured out orally as well. We're also practicing long division orally and it takes a lot of time for both of us, but I'm trying to avoid the education I received in parochial school where almost all problems were taught only procedurally and it would have been difficult to tell if a child understood them conceptually. With word problems, I have her diagram them on paper.

 

But, I do teach the standard algorithms as soon as I feel that the concept has been mastered and the problems can be done mentally and quickly. I don't know how long we can do this mental workout as we progress in math.

 

ETA: Just to clarify, I do have DD show how the problem is worked out, but that is done only orally, and I doubt she will be able to do this once we're multiplying into the thousands.

 

I do the same thing. We do learn the standard algorithm but after the work can be done mentally (and explained well by the child).

 

Getting the "right answer" is not my goal, it is making sure the underlying mathematics is well understood.

 

And I'm right there with you on the method for achieving this.

 

Bill

[Sputterduck]

I hated that in school. I thought it was a huge waste of my time. I don't make my son do it. If he can do it all in his head, I want to reward him for that, not make it seem like he shouldn't. In fact, I require him do math mentally quite often.

[nov05mama]

If he is able to do it in his head, I don't require him to write it out. There are plenty of instances where he does have to work out the problems on paper to give him the practice. I am not going to make him suffer (as I remember having to do as a child) by writing out every single math problem he is presented with.

[Ali in OR]

IRL people who use math for a living HAVE to show their work -- engineers double-check each other, mathematicians and physicist have to publish and defend their derivations (solutions) sufficiently that someone else can reproduce their work in full. My mom who is an RN says, "if you didn't chart it, you didn't do it." meaning everything should be documented at the time it's done.

 

 

:iagree: Yep. When you require that your dc show their work, you are helping them develop clear communication skills. I'm not talking about the easy problems where mental calculations are appropriate and should be encouraged. But for more challenging problem-solving type exercises, learning to show your thought processes is a very important skill. My dd's natural inclination is to have different parts of a long problem all over the page. We need to work on a skill that I thought was kind of obvious--work goes top to bottom and/or left to right. Her math skills are coming along nicely, but computation alone isn't enough. She needs to be able to communicate it clearly.

 

My kids are also hearing early about the concept of partial credit. All of my college math and science classes were tough--very few people could have done well without partial credit. You would get a lot of points for having a mostly correct approach and knowing the appropriate formula or processes to use. And if you couldn't quite get the whole problem worked out correctly, you could still do very well if you could demonstrate your thinking and the thinking was along the right track. When I taught high school math, a problem on a test might be worth 5 points. The right answer was only worth 1 of those 5 points--being able to demonstrate how you got there was worth far more.

[Daisy]

I make my daughter show her work because she often intuitively gets the right answer but doesn't even have a clue what computation she used to get it. I want her to KNOW why she is doing what she's doing.

[NittanyJen]

I hope this doesn't sound totally clueless, but I really wonder why children must show their work in math. It seems to me that if they are able to consistently get the problem correct they have obviously mastered the material right? Could you explain why showing their work is important and/or what skills that teaches?

 

thanks!

 

I just asked my mathematics professor husband this question for his opinion. His response:

 

The answer is the least important thing. In math, what we are really teaching them is a way of thinking.

 

You can get the right answer for the wrong reasons. If they don't write down their logic, you cannot catch the error in their thought process-- it's kind of like asking your DC to write an entire essay, and then saying you only need to grade it by reading the last sentence of the essay, because that sentence should be the whole point.

Edited September 1, 2011 by NittanyJen
Added one more thought!

[NittanyJen]

I just asked my mathematics professor husband this question for his opinion. His response:

 

The answer is the least important thing. In math, what we are really teaching them is a way of thinking.

 

You can get the right answer for the wrong reasons. If they don't write down their logic, you cannot catch the error in their thought process-- it's kind of like asking your DC to write an entire essay, and then saying you only need to grade it by reading the last sentence of the essay, because that sentence should be the whole point.

 

This is why, even when doing the mental math in Singapore, DS8 has to be able to tell me in words how he arrived at his answer. I don't make him write them all out, because with his dysgraphia, that would be torture. But on a spot-check basis, I make him talk me through what he is thinking (or think out loud) so that I can make sure he is really doing what he is supposed to be doing.

[stripe]

Sometimes it's the journey, not the destination.

 

And I like to know how my kids think. Sometimes they come up with an interesting way to solve a problem on their own.

[tammyw]

I have started making DD8 show her work in math. I find it's much easier for me to see where she made the mistake (when she does) and we can figure out together how/why she made the mistake (often it was just a silly error, and when discovered, she doesn't have to go back and start from the beginning).