Math: Why is it so important to "show" your work?

[2GAboys]

I have always taught RightStart which is heavy on mental math, especially in the primary grades because it teaches mental strategies for solving all two digit adding and subtracting problems. There was never written drill, and certainly no kill doing algorithms for adding and subtracting. There was plenty of oral drill using two-three digits numbers and adding or subtracting two digits using mental strategies.

 

Traditional algorithms for all the operations have been taught but there was never pages and pages of drill. Conceptual understanding and problem solving was encouraged.

 

My 5th grader prefers to solve as much as he can mentally and he relies upon his conceptual understanding to figure things out. Lots of times he makes a mistake that I think would have been caught had he done the algorithm. I know traditional American math programs push kids to always show their work. Am I setting him up for problems further down the road because I do not push him to write out his thought process to solve problems??? If he gets a problem wrong, we walk it through and he certainly understands what he did wrong.

[Veritaserum]

It helps to get in the habit of writing it out because when you get into really complex higher math problems, you want to be able to quickly find where you messed up. :)

[Miss Marple]

Because my sons have lost a significant amount of points in their college level math courses because they didn't show the path they took to arrive at an answer. When a calculus test has 5 questions and you answer all 5 right, but didn't show your work, you get a big fat ZERO! In fact, when the answers are all right and no work is shown, the professor jumps to the conclusion that you copied the answer from someone else! While my sons have not been accused of that, they have had points taken off because they omit multiple steps within a sequence. If you don't show your work, your professor has no idea how you got the answer.

 

Yeah, it's a hoop to jump through, but since it makes such a difference on the GPA, it is probably something that should be taught to the point that it becomes habit. And that means teaching it early...we've had quite a few threads on this on the high school board :)

 

It also teaches orderliness in thinking and documentation which are important aspects of life (not to mention the scientific method). When my boys were young, I, too, could sit with them and discuss the path they took, but as they reached high school, there simply wasn't enough time and I needed them to show me their method. Then I could pinpoint the problem and say, "you didn't multiply that fraction correctly" or "you neglected to take the square root to get the final answer". And at that point, what might seem to be an oversight might actually be a lack of remembering HOW to multiply fractions, etc. And one can then see where the gaps, if any, are in the mathematical thinking.

 

So, every day, my boys hear me say, "I don't see your work. Go back and do it again." Someday they will just do it because it is habit...and I'll probably faint away :)

[mom2boys030507]

Lots of times he makes a mistake that I think would have been caught had he done the algorithm.

 

I think this statement says it all. Using mental math to solve the problems is great if you are not getting messed up. As you go higher in math there are more steps to the problem. Showing your work helps you to figure out where there is an error, if made, and also to know what step is next.

 

In your situation, I would have your son write out the steps to the problems he is getting wrong so that he can see where his errors are. Talking about it works some, but writing it out will help me more in the long run. Also, he may realize that it takes little effort to write out more and get it correct the first time then having to go back and do the problem over the second time. This may also help sharpen his mental mat skills if he is making the same type of error as it will give him more practice.

 

Note: my boys are not at this stage yet but I remember all to well losing points on math assignments and tests because I didn't show my work :)

[craftymama]

In many cases it's teacher dependent, even in college. Sometimes, when you show your work you will get partial credit for understanding the material, but making a simple mistake. This was especially true in my higher level college courses. The ones where a single problem goes on for pages. It's a good idea to at least know how to show your work, even if you don't do it often.

[Hedgehogs4]

Lots of times he makes a mistake that I think would have been caught had he done the algorithm.

 

:iagree: that is not to say that mental math is not important. I would say they are equally important.

[stripe]

Math is mostly about the process, not the product.

 

Fermat's Last Theorem was not considered complete until someone (Andrew Wiles) showed why it was true. "This margin is too small" is not enough.

[Trilliums]

In later science classes where they are using math to solve a problem, showing the work is equally important.

 

I do not think a 5th grader needs to be showing their work for every problem though. My son worked on increasing the amount of work shown in pre-alg and alg. In geometry he has to show all his work because of how the problems are structured. He did not have a problem transitioning to showing his work at that stage.

 

As the problems get more involved, showing work is definitely beneficial so they can pinpoint any errors without re-working everything. The student will see this and once it makes sense to them, it will be the route they prefer to take.

[NittanyJen]

In mathematics, in some ways, the answer is the least important part of the problem*

 

The most important thing the teacher or professor wants to see, and grade, is the logic, because logic is what they are teaching. In advanced mathematics, the arithmetic is less important than the higher math processes; it is generally assumed by then that you know how to add and subtract.

 

One must also be able to communicate, clearly and concisely, what you are doing and thinking while problem-solving. Mathematics can be expressed in a clear, elegant symbolic language which the student is learning along with the subject matter, and he is also being tested on his ability to understand and express himself clearly in that language.

 

If the student writes down the answer, 90% of what the instructor is attempting to evaluate is missing.

 

*of course in the long run, we want the students to build a consistently reliable foundation of logic so that in addition to clearly communicating their findings, their findings will, in fact, be accurate. We don't want our bridges falling down or our medications improperly titrated :).

 

In early stages of many courses, it is possible to arrive at correct answers through faulty logic. If the students do not write down their process, the instructor cannot catch and correct the problem at an early stage before large failures occur. There is nothing wrong with menal math; I encourage it. I also, however, encourage strong, clear, bidirectional communication between instructor and student.

[justasque]

Am I setting him up for problems further down the road because I do not push him to write out his thought process to solve problems??? If he gets a problem wrong, we walk it through and he certainly understands what he did wrong.

 

You need to show your work in the lower grades because you're building good habits for when the math gets more complex. Yes, it's good to have the mental math skills to skip writing steps here and there. However, if you're getting things wrong because you skipped writing a step, then you skipped too much. "I just dropped a negative", or "I just forgot the 2x", or whatever, is not good enough. It means your work process doesn't have the checks and balances to help you NOT make these mistakes. (Yes, everyone does it now and again, but a pattern of it is a problem.) You have to actually get the right answer.

 

It's NOT just because you'll lose points on a test. It's because if you don't have, and practice, a work process that proceeds in an orderly way from copying the problem down, to working the equations, to getting the answer, then eventually you'll hit a point where you will have serious trouble with the material. In my experience, the more mathy-smart the kid, the farther they can go without writing things down, so the worse it is when they finally hit the point where they can't keep track of it all in their head. If you hit algebra without good work process skills, then you may have all the conceptual understanding in the world, but you're going to struggle to get the right answer, even if you understand the process in theory. There's just too many places to mess up - to drop a negative or a term or not realize something should be in parenthesis, etc.

 

Fifth grade is the perfect time to get more formal. Different kids need different strategies, but little things can make a big difference. No need to be obsessive, but encourage the development of good habits.

 

~ I encourage them to lay out their problems in an organized way on the page, clearly labeling each one. Don't bunch stuff up, don't put it all over the place, don't write every which way. Part of this is a "classroom skill", yes, but part is a way of keeping their thinking organized.

 

~I have my students circle the problem number; otherwise they will sometimes get confused and use that number in part of their calculations. A simple thing, but it does reduce errors.

 

~ I encourage students to copy the problem over (or at least the "givens" in a word problem) before starting to solve it. This helps reduce errors that may occur if they try to combine doing some calculations while/before writing in the notebook.

 

~ I encourage them to work in a logical way, from top to bottom on their page. Write each step under the previous one, just like they show in most textbooks. Go from left to right. This becomes critical when combining terms in long polynomials in algebra; laying the groundwork now makes algebra much, much easier.

 

If a kid has basically good habits, I wouldn't obsess about it. But if a kid is making a lot of mistakes, or has poor math-writing skills, then I would begin to place more emphasis on it. It will pay off in the long run.

[HejKatt]

Agreeing with PPs on this, basically the advantages are:

 

- student gets practice in laying out the steps for a more complex problem. This includes articulating the algorithm as well as checking details, e.g. units match.

 

- teacher visualizes thought process, and may give partial credit for computation error (vs conceptual error).

 

Also it isn't just the American system. I grew up using Singapore Math and we were encouraged to show our work throughout. This past week, I was working on a word problem from a past-year PSLE paper (given at the end of SM 6) with ratios and without writing down all the details, it would have been frustrating. So it may depend on the problem type - the CWPs from SM lend themselves well to this format of work.

[SorrelZG]

It's possible to be very good at mental tricks for calculating answers without having a true understanding of what is going on. I think that is the case with a friend of mine who said she couldn't show/explain how she got the answers to her teacher/s and also was never good with word problems, just with numbers. Having to show work exposes those kinds of issues so that they can (hopefully) be addressed.

[In the Rain]

 

As the problems get more involved, showing work is definitely beneficial so they can pinpoint any errors without re-working everything. The student will see this and once it makes sense to them, it will be the route they prefer to take.

 

:iagree: My 4th grader is starting to have some multistep problems where it is beneficial to write it out. Once she saw that showing her work was helpful to her (not just me) she became more willing to do it.

 

Until this stage, writing out problems has more to do with me making sure they know the algorithms. Both of my kids preferred to solve problems mentally rather than learning the conventions of crossing out numbers and regrouping.

[LizzyBee]

Personally, I don't think it's important to show all the work for every problem and I have never required that. However, I think it's important to know that a kid CAN show the work if requested. As students progress to more complex work, it's important to know that they know how they got from A to B so that they can apply that knowledge to higher level maths.

[ErinE]

Personally, I don't think it's important to show all the work for every problem and I have never required that. However, I think it's important to know that a kid CAN show the work if requested. As students progress to more complex work, it's important to know that they know how they got from A to B so that they can apply that knowledge to higher level maths.

 

My son and I just had a similar discussion yesterday.

 

If the expression is simple, multi-step arithmetic (2x2x4 or 64-19), I might not require showing the work. But anything more complicated, I require work. Even if it's a fairly simple expression like (6-4)x2, I want to see the thought process. My son has incorrectly answered simple equations because he mentally went through the steps and made a small mistake. If I see the work and the mistake, I know what he needs to work on.

 

Plus, everything people have said above. Showing your work is part of logic and proofs.

[fraidycat]

We have the opposite problem here. In ps, the focus has been on mental math, not showing work. Since I've pulled DD out and I want/need her to show her work, this is a battle that we are engaged in every day, right now. I want all work shown, so I can see what she knows and what she needs help with and where her mistakes, if any, happened.

[chepyl]

In many cases it's teacher dependent, even in college. Sometimes, when you show your work you will get partial credit for understanding the material, but making a simple mistake. This was especially true in my higher level college courses. The ones where a single problem goes on for pages. It's a good idea to at least know how to show your work, even if you don't do it often.

 

I took a calculus in college that was done with calculator programs. I left my calculator at my parent's house, an hour away, before a test. Since I knew how to show all my work from my high school class, I earned partial credit on all the problems and passed the test.

[boscopup]

From a laziness standpoint (which these kids sometimes understand better :D):

 

1) If you mess up and have to rework the problem, it's a LOT easier if you have shown your work and can easily see where you messed up and fix just that part and anything forward, rather than reworking the entire problem from scratch.

 

2) In the real world (science, engineering, etc.), sometimes you do a calculation for one set of parameters, then need to redo that calculation with a different set of parameters. Again, a lot easier to just plugin new numbers and calculate from there rather than figure out the entire process all over again!

 

And as others have said, you can get a right answer using the wrong process, and I'd want to know if that is happening. There is also a big difference between a wrong answer due to wrong process and a wrong answer due to a silly mistake in calculation (adding wrong, using the wrong sign, etc.).

 

I have DS show his work on word problems. Now the level of work I require to be shown will vary. If there is more than one step to the problem, I expect work to be shown for sure. I started gradually requiring work shown in grade 3 math, and now in grade 4 math I'm requiring it even more. By time he gets to algebra and really needs to show work, it will be a habit.