Little blog post on gifted kids showing work in math

[Runningmom80]

Nothing earth shattering, but there are links to other topics in the post as well.

 

http://www.byrdseed.com/to-show-or-not-to-show-work/

[SKL]

You remind me of high school. Apparently I was legendary. Even my niece, 27 years my junior, has heard stories of the girl who refused to show her work because there wasn't a single Algebra (I or II) problem I couldn't do in my head. Made the teacher so mad. LOL. Once he gave me a zero just out of spite. My average in his classes was usually a "B," because of our personality clash. See, I couldn't bear to spend 5 minutes writing out what took 5 seconds to compute. Blah!

 

I like the solution - to make it hard enough for work to be needed. I did show my work in college calculus the following year. :)

[regentrude]

The proposed solution is just common sense. Of course the problem needs to be hard enough to warrant writing down "work".

If you give them more challenging problems, they will start writing down stuff.

But I still think even the gifted student should be made to explain his reasonings - that can be done orally, but knowing how to explain math to another person is an important skill on which I would work even with a kid who "sees" the answer.

[jennynd]

Yep, my son is one of those. But now in algebra, he is much much better writing it down

[EKS]

Well, this is timely.

 

My 11yo got his semester final back from his algebra teacher today. He got a C+. The reason? He didn't show his work on three problems (that he got right). If you recalculate his grade with full credit for those problems, he would have gotten an A-.

[FairProspects]

I've been wondering about this too. Ds never writes anything down if he can avoid it, but he does explain what he is doing to me verbally. He is definitely combining a whole bunch of steps at a time, but I figure if he can get me to understand it, then he knows what he is doing.

[momsuz123]

Hmmmm...this is my 9 y/o - totally!! She is doing more and more in her head. Truth be told, I am not really "great" in math. (I am a physical therapist - didn't need much math). She is already figuring out things in her head, where I am having to say, "hold on" and write it down.

[Bang!Zoom!]

That's a pretty decent blog. How did you happen across that one?

[Mukmuk]

Ds doesn't like to write down his working. AOPS has improved it to the point where he can understand his own writing, although its still sparse. Getting problems wrong and rethinking the process forces you to leave a trail of breadcrumbs for yourself :D. When ds was younger, he'd have some intuitive method in his head, do the problems easily, and then suddenly come to a halt when a new angle surfaced. This was especially true when he did the Zacarro books because there wasn't much teaching. In hindsight, I should have asked for an explanation (written or verbal) on how he did the problem. Regardless of whether the explanation is verbal or written, a good thing to remember is that we want the child to keep re-examining his own understanding as new questions alter their perspective.

[courtney.byrum]

DS8 hated to write problems down - after getting too many wrong because of careless errors he now writes just about everything down (even when he doesn't need to). It took until he was doing 5th grade math before he started doing so and with his current math he always writes it down. He's very happy to only have to explain why he got it wrong when he and I can both see the error.

[Soror]

Oh, my goodness that is my house. We were just in battle this week over it because in a multiple step problem one little mistake throws off the whole problem. I try to have him explain things to me at times but sometimes I cannot follow him as we look at it differently. I haven't cared so much that he didn't write it down before but once he started missing things because of little mistakes and then it taking forever because we had to redo I started making him.

[boscopup]

I've explained to my son WHY it's important to show his work. I did so using an engineer working on a rocket trajectory. Let's say he gets it all done, then his boss comes in and says they need to change one number. If the engineer has shown his work, it's easy to redo the calculations, just changing that one number. If he didn't show his work, he needs to completely redo everything. So which is less work? Plugging in the new number is less work. Also, what if someone ELSE needs to change that number? They can do so easily without reinventing the wheel.

 

Once I explained that to him, he was more willing to show his work. And I told him that he needed to practice showing his work so that it would be automatic later on. He's also seen recently where it's helpful even in his current level of math. ;) We were working a CWP5 problem on the white board. I got the right answer, but his was wrong. Where did he go wrong? When multiplying early in the problem, he'd had one of the numbers off by a factor of 10, which messed up the rest of the problem. So I pointed out his error, and he was able to just plug in new numbers in the rest of the calculations. He had worked the problem correctly, but made a silly mistake early on. He was glad he had shown his work when he got done with that, since there were several steps he would have had to remember. :D

 

Now if the problem is one step and very obvious, I don't make him show his work. He rarely gets such problems now though. Those CWP problems are often multi-step. Sometimes he'll get the answer without writing anything down, but he still knows that he better show me HOW he got there. ;)

[AnIslandGirl]

I have one of these too! On a problem in K when asked in the directions, "How did you get this answer? " Our dd wrote, I used my brain! What do you think I am, stupid?"

 

We had a long talk about respecting authority, not only in speech but in writing as well. She decided that writing, I used automaticity would be a more appropriate answer. She was 5 at the time. Oops!

[SKL]

I've explained to my son WHY it's important to show his work. I did so using an engineer working on a rocket trajectory. Let's say he gets it all done, then his boss comes in and says they need to change one number. If the engineer has shown his work, it's easy to redo the calculations, just changing that one number. If he didn't show his work, he needs to completely redo everything. So which is less work? Plugging in the new number is less work. Also, what if someone ELSE needs to change that number? They can do so easily without reinventing the wheel.

 

...

 

 

See, I would have answered "but since I get the answer right 99 times out of 100 without showing my work, overall it saves more time just to do it over when it's wrong. Besides, doing it twice in my head is probably faster than writing it down once and fixing it."

 

But then, I was a rotten kid.

 

I think there might have also been the element of insecurity over someone seeing written evidence of a mistake I made. Often I could save myself by realizing a wrong answer was way off and just starting over. Other times I just took the hit for a completely wrong answer.

 

FWIW, nearly all of my math mistakes in high school were due to mixing up 3x2 and 3+2, or reversing a sign halfway through the problem. I always figured I'd never make that dumb mistake again . . . .

[Dana]

I teach remedial math at the cc, so I'm coming from a completely different population. However, the number of errors I see where students think they know what they're doing and don't need to show work never fails to depress me. As boscopup mentions, the need to share work with other people is also a reason to show work. We lost one Mars rover due to a unit conversion error (US/metric units). Was someone sloppy and left out an "obvious" step? While my son is learning a new topic, I make him show steps. He can do some problems mentally, but I insist he show me the work for some as well. I've caught some conceptual errors this way that we've been able to correct quickly rather than letting the misunderstanding set.

 

I also use the analogy of writing a paper or essay. You wouldn't use text speak or just do an outline and say that's good enough for your work. I want to see complete sentences with math.

 

I'm grading yet another depressing stack of tests, so I'm frustrated again...In some cases if they'd just show the steps, they'd have gotten the problem correct. I can see some of their thought process but they're insistent on doing things their way & they get the problems wrong or say untrue things (2 + 3 = 5 -4 = 1 for an example when they meant (2+3) - 4. )

 

So just because my son's gifted and can "see" the answer in some cases, doesn't mean he really understands. I don't believe he does until he can prove it by showing the steps at least on some occasions. Then he can skip steps.

 

I also saw this myself when learning inverses. I could "see" the answer for the inverse of a linear equation and just write down the answer. So why did I need to show work? I didn't in high school. However.... I didn't understand what was really going on until much later when taking the inverse of restricted polynomial functions and of rational functions. There was no way I'd have been able to find the inverse of them with my "I just see it" procedure.

[Dmmetler]

I have required my DD, even back at about SM 1 to show me that she knows how to do it "The book way". That is, if they want you to take 6+8 and break it into 4+2+8=4+10=14, therefore 6+6=14, you need to show me that you can do it "the book way" at least a few times, because math books (and math teachers, like your dear old mom) are dumb. They can't read your mind and know that you understand. When she was younger, usually showing me was via telling me what to write on the whiteboard. Now she has to show me by writing it down. Now that she's working on algebra, I think she's starting to see WHY I insisted, because she's getting to problems where she sometimes has to backtrack a few steps when she gets off-and she can't do that if she doesn't write it out.

 

Fred's bridges are good for that, too. Obviously, Stanley Schmidt has had more than his share of students who were able to skip steps, because there have been quite a few times that DD has skipped steps or done it a different way than the one that she was explicitly told to use in the question, only to have the solution say "If you did it this way, you did it wrong, and here's why-and have the author show that, yes, it might work in this case, but it doesn't work in all cases, and there have been more than a few questions that were obviously written for the express purpose of tripping up the reader into making standard mistakes if they jumped too fast. I swear if I ever meet the author, I'm giving him a big hug!!

[boscopup]

See, I would have answered "but since I get the answer right 99 times out of 100 without showing my work, overall it saves more time just to do it over when it's wrong. Besides, doing it twice in my head is probably faster than writing it down once and fixing it."

 

In my story of the engineer, the engineer hasn't gotten the problem wrong. His boss has come in with different data. Maybe the weight of the rocket has changed, or the angle of the launch. Something has been changed where he needs to redo the calculations with just one number changed. So yes, showing his work is quicker. He got the first calculation correct, but he doesn't want to redo it all when that one number changes.

 

(in reality, the engineer has probably written a computer program to do these calculations, so he simply has to change the number and run the program again, but shhhhh... don't tell my son that. ;) )

[FairProspects]

I also saw this myself when learning inverses. I could "see" the answer for the inverse of a linear equation and just write down the answer. So why did I need to show work? I didn't in high school. However.... I didn't understand what was really going on until much later when taking the inverse of restricted polynomial functions and of rational functions. There was no way I'd have been able to find the inverse of them with my "I just see it" procedure.

 

This is exactly what ds does on inverses. But, I guess I'm still confused because if he just sees it, what is there to write down? Clearly, I am not a math person, so I have no idea how to coach him toward what would help him in the future in this area.

[regentrude]

(in reality, the engineer has probably written a computer program to do these calculations, so he simply has to change the number and run the program again, but shhhhh... don't tell my son that. ;) )

 

 

Sure tell him! Because in this case, the engineer will have been required to document his program carefully so that it can be used by some other person as well.

This is how i explain to my Comp Sci students that they must show their work - they all understand documentation of code for other users.

[boscopup]

 

 

Sure tell him! Because in this case, the engineer will have been required to document his program carefully so that it can be used by some other person as well.

This is how i explain to my Comp Sci students that they must show their work - they all understand documentation of code for other users.

 

 

Good point! I've had to work on code that was undocumented. It's awful. My kids will be taught that documentation is their friend, even if it's not as fun as writing the code in the first place. ;)

[Runningmom80]

That's a pretty decent blog. How did you happen across that one?

 

One of the gifted pages I follow on Facebook shared it. I can go back and see which one it was if you'd like!

[Runningmom80]

The proposed solution is just common sense. Of course the problem needs to be hard enough to warrant writing down "work".

If you give them more challenging problems, they will start writing down stuff.

 

Right, which is why I said it was nothing earth shattering. ;)

[Dana]

 

This is exactly what ds does on inverses. But, I guess I'm still confused because if he just sees it, what is there to write down? Clearly, I am not a math person, so I have no idea how to coach him toward what would help him in the future in this area.

 

 

 

Let's say you're trying to find the inverse of f(x) = 3x-7.

I'd "see" the answer as f inv (x) = (x+7) / 3 and I'd be correct.

 

The procedure for finding an inverse (and what I'd expect to see on a test if I gave this as a problem) is:

(1) switch x & y

So x = 3y - 7

(2) Solve for y

giving (x+7) / 3 = y

 

The idea is to "undo" our relation, so that when we compose f and f inverse, we get the identity: y = x.

We can see on a graph that the inverse is the reflection of our graph about the line y = x.

Domain & range also switch.

 

You don't see ANY of this with lines.... but let's look at a parabola...

f(x) = (x+3)^2 -4.

 

The domain is all real numbers, so we'll need to restrict the domain so the function is 1-1 and has an inverse... we'll go from the vertex, so x >= -3. The range is y >= 4. (Our graph is a parabola shifted left 3 and down 4).

 

To find the inverse:

(1) switch x & y

x = (y + 3)^2 - 4

 

(2) Solve for y (still can be done mentally, but getting trickier)

x + 4 = (y + 3)^2

sqrt(x+4) = y + 3

-3 + sqrt(x+4) = y

 

You can check by composing f and f inverse.

 

The domain for our inverse is x >= -4 and the range is y >= -3... the domain & range have flipped from the original function!

 

And rationals are worse... find the inverse of y = 1 / (3x + 8). Not likely to do that mentally! (At least I couldn't when learning them.)

 

Understanding the basics and whys now can help later. You use inverse trig functions later as well.

 

-----

 

It's tough when you don't know where a subject is going. I know I'm going to goof with science a lot.

I've been seeing students screw up math for over a decade now, so I'm going to be very sure my son doesn't make the same mistakes! And he was doing his math while I'm grading & he saw one student's work & apologized to me :) Not going to get complaints about showing work for a while from him!

[Bang!Zoom!]

Ya, Running Mom, if you don't mind, that would be great to know the FB page. It's really unusual to find something that is up the alley I like, and I really did like that blog. :)

[avilma]

The difficulty with showing steps is that it is more an art than a science.

One cannot learn in one day or a week how to show steps, it is a skill that can only

be learned with lifelong practice. Showing too many steps can be as bad as

showing too few steps not to mention that it can become incredibly boring or even

worse it can become a passion in itself (I had a classmate in elementary school

who would be very proud if she could find a way to solve a problem by showing more steps than

other people. She didn't go much further than algebra).

I don't really have a recipe how to go about this, but I always keep in mind while

working with my daughter that I want her to have as much fun as possible so if she doesn't feel

like showing the steps and the answer is right I don't push it. She has already done this in her head

so if I insist that she show me the steps basically I am asking her to do it again. When she gets something

wrong (and really if she doesn't get something wrong I am not challenging her enough) I then

ask her to show the steps in a way that helps her figure out where the error is.

There are cases when I really need her to show the steps because I have in mind a particular algorithm

that I need her to learn. In that case if I feel that she is giving me the result using some other

algorithm, I lay out the steps myself and ask her to follow them for a few cases.

My daughter is only 5 so I am hoping I will be able to find better strategies as she works on more complex

problems and I get more experience in teaching math.

[Runningmom80]

Ya, Running Mom, if you don't mind, that would be great to know the FB page. It's really unusual to find something that is up the alley I like, and I really did like that blog. :)

 

Ok, I'll try to find it!

[kiana]

Ya, Running Mom, if you don't mind, that would be great to know the FB page. It's really unusual to find something that is up the alley I like, and I really did like that blog. :)

 

 

I don't know where she saw it, but I saw it from Hoagies.

[Dmmetler]

Gifted Homeschoolers Forum puts some really good things on their Facebook feed, and I think that's where I saw it.

[shawthorne44]

I see showing your work as part of the work. If I were a math teacher, a correct answer with no work shown would get 50%.

I think part of it is that you have to learn how to show your work. You need to have learned that before the math becomes a struggle.