Writing out math problems

[ShepCarlin]

OK, so this is our fifth year homeschooling. I really thought by now I'd sort of have this all figured out. The last 4 years I used Life of Fred for math for both boys. It was great up until last spring when nothing seemed to work. We tried out Teaching Textbooks 3.0 in August and TT got 3 thumbs up - both boys and I liked what we saw. So we are about 4 weeks in TT. My 14 yo is using the Algebra I and my 11 yo is using Pre Algebra. 

I am literally losing hair, sleep and of course my sanity over the fact that they both refuse to write out their work on paper. They insist on doing it in their heads. I'd be sort of ok with this if they got the work done CORRECTLY. However, they are not. Of course, I can't see where they made a mistake in a problem as it is in that thick skull of theirs. I knew this would be a risk using an online program as they much prefer just typing an answer in the box and moving on. However, I'm a firm believer in you need to write this stuff out...when it's there line by line, if a mistake is made you can catch it and work on why the mistake was made. I know I'm preaching to the choir here but I just want to lay out my math beliefs ? I've made them redo entire lessons when they've not written out the problems....doesn't seem to bother them. 

Where have I gone wrong? When we did LoF, I made them write everything out so this is certainly nothing new. Granted there is more work now as TT is on average 20 problems where LoF was never that many unless it was a bridge (a bridge is a test for those of you unfamiliar with Fred). How can I make them realize they need to write this stuff out? My 14 yo is taking a physics class at our co-op this year. There is more math in it than anticipated but he is doing well with it. Except for the fact that he isn't showing his work! His teacher said the same thing I have said to him. All I get from him is the "I know, I know". I will say this week he has been a little better at it but the 11 yo has me literally at my wit's end. The 11 yo has never wanted to write out his math. Ever. It's been a struggle for a while now but I seem to have come to my breaking point. I just went over today's work and lost it on both of them. The 14 yo's main offense was he didn't date the paper or write down which lesson it was - I know not the biggest deal but I had just finished looking at the 11 yo work that didn't have much work to look it. Or if he did write it out, he just stopped when he got the general idea of what the answer would be (ex. if the answer was 23.45...he stopped at 23.4)

I don't think this is a dysgraphia issue. Both will write for history, Latin and English. Many thanks in advance for any advice you all may have. I hope I'm not the only one with obstinate boys.

 

[ghcostafamily]

Maybe just print the ebook lessons and that way they are writing less than if they’d done it from scratch?

[RootAnn]

If they get it wrong, don't they have to go back and redo it? Do you want to make this a hill to die on? (Metaphorically)

I would sit with each one of them for a week and first model how to write things down. Then, I'd scribe for them as they tell you what to write down. Once they are verbalizing it well, I'd sit and watch them write everything. (Might take two or three weeks!) After that, I'd have them show you the correct written lesson before they can move on. Every time. For a month or more.

Lots of work on your part.

It is possible your younger doesn't know how to write his work out.

[Sneezyone]

We’re not using TT but DS is my second child in a row who’s been resistant to show work in math. DD resisted until the middle of 6th which was way too long. This year it *is* a hill to die on for us. If he gets the answer wrong, I mark it wrong and make him redo it and show me the work. Little by little, he is starting to figure out that showing work helps him just as much as it pleases me. I also go out of my way to praise him when he shows his work the first time, even if the answer is wrong. For ex) “Wow, thanks for showing your work! That makes it so much easier for me to find the snag...see, here, where you copied the problem incorrectly?” I anticipate this being a year-long process.

Edited October 3, 2018 by Sneezyone

[mom31257]

I tutor math and have dealt with this among my students. I'm finally getting them to show most of their work. I think they simply do better for me than their parents because I'm not mom. 

Perhaps you could negotiate to show half of the work, and if it's all correct, he can do the rest of the lesson without showing it? 

[RenaInTexas]

Stick to your guns. I think RootAnn gave good advice. 

Showing work should be taught in Arithematic and it should be a 'hill to die on' in PreAlgebra. I have yet to see a student be successful in Algebra without having developed this skill. This skill goes beyond math. I tell my students that when they start their career and tell their boss that the answer to his question is 12 ... the boss will want to see how you got that answer. I worked in industry for 15 years, there was no profession where you produce a number that didn't require showing how you arrived at that number. 

But the skill will also be needed in higher levels of math: working with polynomials, geometry proofs, systems of equations, etc...

Keep the faith, the struggle is worth it.

[8filltheheart]

I agree it is a hill to die on. I would not scribe anything for kids that age. I would sit right beside them though while they are doing their math and make sure they are understanding and writing out every step. If they refused "bc they already know," I would take away privileges and buy an additional math curriculum and assign double math every day for insolence.

Fwiw, I don't accept no for an answer for an answer from my kids. Period. No way they would be allowed to refuse. If they don't understand how, that is one thing. That is why I would sit with them while they worked and wrote out the answers. Understanding requires teaching.  Refusing to do what is asked is an obedience issue.

[EKS]

I think that the format of TT (and other programs like it--ALEKS, Thinkwell, etc) encourages shoddy habits.  When kids just have to input an answer into the computer, it makes it seem like that's the only important part.  Does TT still have solution manuals?  If so, I recommend that you grade the work yourself.  Just remember that, especially as things get more complicated, there will frequently be more than one way to do things, and if you're going to do the grading, you're going to need to be able to recognize when the student has used a legitimate alternative solution (even if the answer is wrong).

That said, sometimes the problems are really, really easy, and it really is a slog to write out the work.  I've found that it also helps to ensure that the work is difficult enough so that the student sees the utility of writing things out.  I'm well aware of the arguments for learning to write work out while it's easy so that when it gets hard, they don't have the double whammy of difficult math combined with the difficulty of writing it out, but sometimes something has to give.

Edited February 21, 2019 by EKS

[theelfqueen]

On the writing materials -- sometimes my kids do better if they can do work on a white board. (When I'm sititng next to them) something about the impermanence takes off the pressure of writing it out on paper -- the "I wrote the wrong sign, I'll just erase it" part. I've spent a lot of time making my kids write stuff out but this strategy is one we still use in Geometry and Pre-Calc... when a day is particularly frustrating, out come the white boards. 

 

[Andromeda]

My daughter is like this. Her mind works quickly, and she can actually do a lot of the math correctly in her head in Algebra and physics.

I have been working with her on writing out answers for 3 years now, and we are starting to make some progress. The way I can explain is that for many people, in order to solve a problem, their mind goes from A to B to C to..... My daughter's mind often just goes from A to X, and it is the right answer. It is difficult for her to write down the steps because she doesn't think that way at all.

Having said that, if she does not write out the problem with the steps.... she gets to do it all over again, later, instead of screen time, properly, even if the answers were correct. There is much groaning, complaining, and sometimes there are tears - hers and mine, but we get it done. I model the problem solving process, many times, probably every day. For physics I make her set up the problem by writing out what is given, with the units, then writing the formula that best fits the situation, reworking the formula (with all the steps) so that the only part left is to plug in numbers, writing the formula with the numbers but not yet calculating anything, then calculating each piece, circling the answer, checking the units, and then reading the problem in the text to make sure we answered the question that was being asked.

That equals about 2/3 page of written work per problem, with diagrams as needed (neat and labeled) as opposed to an one line answer. It is still work in progress, but the important part is that there is progress.

 

ETA

She writes with Frixxon pens, which are erasable.

Edited October 3, 2018 by RosemaryAndThyme

[RenaInTexas]

One thing that I have found to help some students is doing their math work on graph paper. 

Edited October 3, 2018 by RenaInTexas

[Targhee]

1 hour ago, HeighHo said:

What you do is tell them you don't want answers, you must have solutions as it is your responsibility as the math teacher to teach them how to communicate their findings appropriately, just as in English you teach the format and content of an essay.  They must present solutions to their problem sets on paper, and anything that is incorrect needs to be corrected and shown to you on the whiteboard as they explain their thinking.  Of course you won't just make a verbal demand, you will provide training and a standard of excellence.  You will also have the student placed correctly, so they aren't showing solutions for things that have been mastered and internalized years ago.  Once placed correctly, the usual rule is that material from this year needs to be justified, material mastered from years ago doesn't unless it is necessary for the road map used to show the grader what you have learned  (your one liner in the future is 'gotta show what you know').  My compromise with my lad was write out solutions to problems, but not exercises.

Beyond that, make sure you have proper writing materials and use unlined paper.  Show them how to organize the paper, how to write in straight lines, what to do if a graph is to be shown, and encourage a certain height, big enough for the bifocals the Calc teacher will probably be using.  Writing materials...some experimenting is needed. Sometimes a fatter barrel is easier.  Sometimes a 4 color pen is helpful. 

Lots of good stuff here.  It took getting my kids into challenging math that required them to write things (can’t do it all in their heads) to make it evident they needed to write.  AOPS prealgebra did this für my oldest (even though AOPS uses very “neat” Problems as far as computation goes), and Beast Academy did it for my next two.  

The only thing different from this post I would suggest is trying graph paper - it helps my kids keep their work organized better (the oldest two have very sloppy writing/issues with fine motor).

[Targhee]

20 minutes ago, theelfqueen said:

On the writing materials -- sometimes my kids do better if they can do work on a white board. (When I'm sititng next to them) something about the impermanence takes off the pressure of writing it out on paper -- the "I wrote the wrong sign, I'll just erase it" part. I've spent a lot of time making my kids write stuff out but this strategy is one we still use in Geometry and Pre-Calc... when a day is particularly frustrating, out come the white boards. 

 

White board helped a lot when we were working together. 

[ShepCarlin]

Thank you so much everyone! Lots of good suggestions here and I'm so glad to hear I'm not alone. I figured I wasn't but it's good to hear it. I do some of this already...the graph paper has been a big help and I used to do the white board thing but kind of let that fall to the wayside this year. All the white board suggestions were a good reminder as the 11 yo does seem to like it better than paper. I'm also learning he just doesn't like to be on his own - I can't just say watch the lecture for lesson 5, then do the problems while I go off to do laundry or such. I need to be in the room with him, guess I jumped the gun trying to make him more independent with his work. I've printed out the lessons before but he prefers to see them on the screen. One thing I'm learning about my boys' generation is they really don't mind reading lots and lots of info on a computer screen whereas I prefer books over laptop/Kindle/iPad any day. Guess I'm just old fashioned but books don't have a battery life!

[lewelma]

I am a math tutor for grades 7 to 12 pre-A to Calc. I haven't read all of the responses, but here are my thoughts.

First and most importantly, I don't want to see *your* work, I want to see *the* work.  Math has a very set way of being expressed once you hit algebra, and a student needs to know what this looks like.  This is *the* work that I'm looking for and which will be graded on.  I really don't want to see crap out of your head.  Math at this level is logical and linear-- *the* workings progress in a certain way, and anyone who knows anything about math expects to see the workings written in that special way.  

So for Trig, I want to see

the formula on the first line

the numbers filled in (without doing any calculations)

the calculator line (what you actually put in the calculator)

the answer appropriated rounded with units and a note as to how it was round (1 dp, for example).

If you can get it right in the calculator, you can skip the calculator line, but only if you are getting the answer correct 99% of the time. You also can skip the formula line if you can plug in correctly. So *the* workings look like this (but with actual fractions written on two lines, not the diagonal slash), but only the 2nd and 4th line are actually required:

Tan@=O/A

Tan 40 = 15/x

x=15/Tan 40

x=17.9 cm (1dp)

I don't want to see :

Tan@=O/A=15/x  Tan40=0.839  x=15(0.839)=17.9

or some such thing.

The same goes for Algebra or prealgebra. You must work DOWN.  You must make mathematical statements (explained below). As long as you show the required workings (I'll tell you the minimum expected) you are allow to skip steps as long as you get it right.  If you get it wrong regularly, you have to do the skipped steps until you are getting it right at which point you can skip it.  In algebra, there are some crutches we use, like subtracting 8 from both sides, where you actually write this. This is a crutch, and eventually it will be lost.  Students only need to do the crutch until they can get it right without the crutch, then you simply stop using it.  The age at which this happens can vary dramatically from within a week of being taught, to Calculus (I do have a calculus student who still uses algebra crutches).

Mathematical statements: Because you are always working down, it is expected that each line is equivalent to the one before.  This is true for expressions and equations.  If you need to do some working on only one piece of a long equation and don't want to be writing the whole thing down over and over again, then you make a little box to the side an do your side workings there, and then once you have finished the desired manipulation, you fill it into the main body of your work, keeping your mathematical statements going down. In my example above with the trig, these are not mathematical statements and if written down each line would not be identical.  This is very confusing to read, and represents what a student will write if told to show *his* work, rather than being shown what the workings are supposed to look like.

As for the minimum steps required for algebra, if I as the teacher can do it in my head and wouldn't write it down, then the missing step is not actually required.  So

5x+1=3x-2

2x=-3

x=-3/2

Middle step is not required if a student is consistently getting it right. If they get it wrong, they must add in the step, plain and simple. No arguments. However,

5(x+3)=4(x-2)

5x+15=4x-8

x=-23

Middle step is required.  You could do it in your head, however you are likely going to take longer than just writing out one step, and you are more likely to get it wrong.  The middle step is not one that a mathematician would expect to see skipped. 

The goal is to get students to own it -- own the workings, own the process, own the learning.  So they need to be empowered to drop steps that are obvious to them, as long as they are writing the *required* steps.  If they get it wrong, they need to be taught to acknowledge that they need to add in the intermediate step until they are always getting it right, then they can drop it again.  No one wants to be told to show *your* workings when you actually did not have any because you did it in your head.  To be required to show *the* workings that all mathematicians expect in a written answer, is more understandable to most students.

Calculations: Actual computational work required for algebra on up can be done with official algorithms (long division), mental maths, jotted scribbles, or a calculator.  However, this work is NOT apart of the algebraic mathematical statements so should be done in a calculation box to the side and then placed into the mathematical statements.  I don't really care how you do computation as long as you do it right.  If you are getting it wrong, then we will figure out why and find a fix. 

Ok, that is about all the brain dump I have.  Happy to answer questions.

Ruth in NZ

[Sneezyone]

3 hours ago, lewelma said:

I am a math tutor for grades 7 to 12 pre-A to Calc. I haven't read all of the responses, but here are my thoughts.

First and most importantly, I don't want to see *your* work, I want to see *the* work.  Math has a very set way of being expressed once you hit algebra, and a student needs to know what this looks like.  This is *the* work that I'm looking for and which will be graded on.  I really don't want to see crap out of your head.  Math at this level is logical and linear-- *the* workings progress in a certain way, and anyone who knows anything about math expects to see the workings written in that special way.  

So for Trig, I want to see

the formula on the first line

the numbers filled in (without doing any calculations)

the calculator line (what you actually put in the calculator)

the answer appropriated rounded with units and a note as to how it was round (1 dp, for example).

If you can get it right in the calculator, you can skip the calculator line, but only if you are getting the answer correct 99% of the time. You also can skip the formula line if you can plug in correctly. So *the* workings look like this (but with actual fractions written on two lines, not the diagonal slash), but only the 2nd and 4th line are actually required:

Tan@=O/A

Tan 40 = 15/x

x=15/Tan 40

x=17.9 cm (1dp)

I don't want to see :

Tan@=O/A=15/x  Tan40=0.839  x=15(0.839)=17.9

or some such thing.

The same goes for Algebra or prealgebra. You must work DOWN.  You must make mathematical statements (explained below). As long as you show the required workings (I'll tell you the minimum expected) you are allow to skip steps as long as you get it right.  If you get it wrong regularly, you have to do the skipped steps until you are getting it right at which point you can skip it.  In algebra, there are some crutches we use, like subtracting 8 from both sides, where you actually write this. This is a crutch, and eventually it will be lost.  Students only need to do the crutch until they can get it right without the crutch, then you simply stop using it.  The age at which this happens can vary dramatically from within a week of being taught, to Calculus (I do have a calculus student who still uses algebra crutches).

Mathematical statements: Because you are always working down, it is expected that each line is equivalent to the one before.  This is true for expressions and equations.  If you need to do some working on only one piece of a long equation and don't want to be writing the whole thing down over and over again, then you make a little box to the side an do your side workings there, and then once you have finished the desired manipulation, you fill it into the main body of your work, keeping your mathematical statements going down. In my example above with the trig, these are not mathematical statements and if written down each line would not be identical.  This is very confusing to read, and represents what a student will write if told to show *his* work, rather than being shown what the workings are supposed to look like.

As for the minimum steps required for algebra, if I as the teacher can do it in my head and wouldn't write it down, then the missing step is not actually required.  So

5x+1=3x-2

2x=-3

x=-3/2

Middle step is not required if a student is consistently getting it right. If they get it wrong, they must add in the step, plain and simple. No arguments. However,

5(x+3)=4(x-2)

5x+15=4x-8

x=-23

Middle step is required.  You could do it in your head, however you are likely going to take longer than just writing out one step, and you are more likely to get it wrong.  The middle step is not one that a mathematician would expect to see skipped. 

The goal is to get students to own it -- own the workings, own the process, own the learning.  So they need to be empowered to drop steps that are obvious to them, as long as they are writing the *required* steps.  If they get it wrong, they need to be taught to acknowledge that they need to add in the intermediate step until they are always getting it right, then they can drop it again.  No one wants to be told to show *your* workings when you actually did not have any because you did it in your head.  To be required to show *the* workings that all mathematicians expect in a written answer, is more understandable to most students.

Calculations: Actual computational work required for algebra on up can be done with official algorithms (long division), mental maths, jotted scribbles, or a calculator.  However, this work is NOT apart of the algebraic mathematical statements so should be done in a calculation box to the side and then placed into the mathematical statements.  I don't really care how you do computation as long as you do it right.  If you are getting it wrong, then we will figure out why and find a fix. 

Ok, that is about all the brain dump I have.  Happy to answer questions.

Ruth in NZ

 

Yeah, I would have failed Trig under those rules. My answers always looked something like your "bad" example. I'd start with the formula filled in (I never wrote it out) and solved the dang thing. Done. Calculator line? Is this a thing now? You actually have to write what you input in your calculator?

Edited October 6, 2018 by Sneezyone

[regentrude]

9 minutes ago, Sneezyone said:

Calculator line? Is this a thing now? You actually have to write what you input in your calculator?

I greatly encourage my students to do that in their physics classes. First, when there are many terms, they are very likely to make mistakes if they don't write down what they intend to enter. Second, when I evaluate their work, this lets me see why their final answer is incorrect (as is often the case).

[Sneezyone]

4 minutes ago, regentrude said:

I greatly encourage my students to do that in their physics classes. First, when there are many terms, they are very likely to make mistakes if they don't write down what they intend to enter. Second, when I evaluate their work, this lets me see why their final answer is incorrect (as is often the case).

 

Interesting. I never used my calculator much as we still had reference tables in our textbooks. I will keep this in mind for my kids once they reach the calculator use stage. So far, no need.

[regentrude]

I strongly recommend graph paper for math.

And teach the student to write one equation per line. Run-on equations on one line often morph into expressions that are not actually equal, because students will start performing calculations and write stuff like 5x-2=3x+4=2x=6=x=3 which of course is complete nonsense. This needs to be written as:

5x-2=3x+4

2x=6

x=3

One suggestion I have for algebra learners: write the operation you intend to perform to both sides of the equation behind a vertical line at the end. That is how algebra is taught back home and better than the haphazard way many of my students indicate their operations. For the above example, a beginner would be required to write:

5x-2=3x+4       | -3x

2x -2 =4           |+2

2x=6                 | :2

x=3

Once proficient, the student can omit those extra steps and can skip steps. But for the learner, this is a valuable reinforcement.

[regentrude]

2 minutes ago, Sneezyone said:

Interesting. I never used my calculator much as we still had reference tables in our textbooks. I will keep this in mind for my kids once they reach the calculator use stage. So far, no need.

I never let my kids use calculators in math, all the way through calculus( except perhaps 2-3 problems per year). But they did use calculators in their high school physics and chemistry courses.

Edited October 6, 2018 by regentrude

[Sneezyone]

2 minutes ago, HeighHo said:

 

In my day this all had to be justified if a beginner, using properties.

5x-2=3x+4    Given

5x-3x-2=3x-3x+4  Subtraction Property of Equality (shortened to Subtr after we had that internalized)

5x-2=4  Simplify

and so forth

 

Oh, Lord, when was this? I'd have yanked out my eyelashes one by one. I did that for geometry proofs but that's it.

[Sneezyone]

Just now, HeighHo said:

 

This was when Dolciani was used for Algebra in public high school, late 70's for me.  Yes, I was well prepared for studying engineering.

 

I used Dolciani circa 1985. None of that was required. I think I was equally well prepared for upper level math, no probs at all through pre-cal but I had zero interest in engineering despite scoring higher in math. No accounting for taste.

[Sneezyone]

9 hours ago, HeighHo said:

 

Most likely depended on teacher. My first high school did not require it.  My second did.  Same text, but when I switched at end of first quarter my first high school was behind the second by two chapters, so I did the independent study section at the new high school.  I would not have been well prepared if I had stayed at that first high school...the class I was in omitted too much, clearly the minimum per the teacher's guide suggestion. Independent study was excellent, I was able to do the all the Dolciani high school books.

 

Probably. We followed the accelerated/honors sequence but I don't think (looking at the TG for the version I used) that was called for/required in the years I used the texts. 

Edited October 6, 2018 by Sneezyone

[lewelma]

2 hours ago, Sneezyone said:

Calculator line? Is this a thing now? You actually have to write what you input in your calculator?

Well it depends.  If a student is always getting it right, then no.  But if a student is not, then yes.  It really depends on the level of the student, and I teach students who struggle.  I have spent many minutes trying to find errors in the workings when actually it is the use of the calculator that is the problem.

[chocolate-chip chooky]

3 hours ago, texasmom33 said:

Have you ever considered writing a book? Something along the lines of Everything a Homeschooling Parent Needs to Know about Executive Function and Various and Sundry Other Study Related Things. And Math. ? I would totally buy it. 

Sorry to derail. I'm just all sorts of fan-girling about your posts. 

Put my name on the pre-order list.

[8filltheheart]

9 hours ago, regentrude said:

I strongly recommend graph paper for math.

And teach the student to write one equation per line. Run-on equations on one line often morph into expressions that are not actually equal, because students will start performing calculations and write stuff like 5x-2=3x+4=2x=6=x=3 which of course is complete nonsense. This needs to be written as:

5x-2=3x+4

2x=6

x=3

One suggestion I have for algebra learners: write the operation you intend to perform to both sides of the equation behind a vertical line at the end. That is how algebra is taught back home and better than the haphazard way many of my students indicate their operations. For the above example, a beginner would be required to write:

5x-2=3x+4       | -3x

2x -2 =4           |+2

2x=6                 | :2

x=3

Once proficient, the student can omit those extra steps and can skip steps. But for the learner, this is a valuable reinforcement.

My 3rd grader is doing this right now. Instead of putting it at the end, she writes it directly underneath the the corresponding part of the equation on both sides (our focus is on if it is done on one side of the equation it has to be done on both to keep the equation balanced.) I tried typing it out on here but with the automatic double space on return it looks clumsy.  But -3x would be written under the 5x and the 3x; the next step would be writing down the equation after that subtraction and then under -2 she would write + 2 as well as +2  under the 4, and so on. But a 3rd grader writing out exactly what they are doing step by step like that makes sense.

My older kids never write down that level of detail. It seems obvious what they have done. They always return to the margin on a new line when they are moving onto a new operation, so looking at their work you can determine what was done.  They rarely use calculators except for occasional complicated division so there is nothing to write down in terms of calculator entries.

[wendyroo]

11 minutes ago, 8FillTheHeart said:

My 3rd grader is doing this right now. Instead of putting it at the end, she writes it directly underneath the the corresponding part of the equation on both sides (our focus is on if it is done on one side of the equation it has to be done on both to keep the equation balanced.) I tried typing it out on here but with the automatic double space on return it looks clumsy.  But -3x would be written under the 5x and the 3x; the next step would be writing down the equation after that subtraction and then under -2 she would write + 2 as well as +2  under the 4, and so on. But a 3rd grader writing out exactly what they are doing step by step like that makes sense.

My older kids never write down that level of detail. It seems obvious what they have done. They always return to the margin on a new line when they are moving onto a new operation, so looking at their work you can determine what was done.  They rarely use calculators except for occasional complicated division so there is nothing to write down in terms of calculator entries.

This is how I have my algebra learners do it as well.  I emphasize that they are actually adding two equations, the second of which is simply -3x = -3x or 2 = 2.  Setting it up this way is a good foundation when they start to solve systems of equations with combination.

Wendy

Edited October 6, 2018 by wendyroo

[Skippy]

On 10/2/2018 at 9:26 PM, ShepCarlin said:

Where have I gone wrong?

You have not gone wrong. This was my experience as well. It was so difficult to get my boys to show work. One was great at figuring out complicated equations in his head. This was the one thing that I had to tell them over and over again. So my advice is: Just keep reminding them. You don't need to change math programs if you otherwise like what you are using.

[Kuovonne]

To encourage my kids to show their work, I agree to help them figure out what went wrong only if and only if they have written down their work. If their work is written down in an orderly fashion, I will painstakingly go through their work to find the error. Then I will point out the type and location of the error. (If it is really obvious, I might just say the type or the location.) On the other hand, if there is no written work, I provide no assistance at all. The problem is simply wrong and it is up to them to fix it or produce enough written work for me to examine.

For my kids, providing graph paper has not helped. One kid has great handwriting and the other has lousy handwriting. The kid with great handwriting didn’t need graph paper for lining things up. The kid with lousy handwriting would rather write on a blank piece of scratch paper. I tried switching her to a math program where she had to write out the problems, but that didn’t work out. So, she is back to a worktext and scratch paper.

[lewelma]

An event 3 years ago really impacted how I perceive of showing your mathematical workings. My younger son was struggling to write, so we took him in to get tested for dysgraphia. They worked him through a battery of tests that took 2 days and about 5 hours. I was in the room because he wanted me to be. He was 11 at the time. For the math section, the final question was something like you have 5 oranges and 8 apples costing $20, and 8 bananas and 6 oranges cost $18, and 9 applies and 3 bananas cost $21. How much does each fruit cost? (this is not the question, just something like it). I got out a piece of paper and simply coded it as three equations and three unknowns, but then realized I was going to get fractional answers.  Yuck!  Well, my ds had not started algebra certainly had never done simultaneous equations, had never seen a problem remotely like this, plus he could not write. Although he was allowed to use paper, he did not touch it. It took him 15 minutes to get the answer. He did it in his head.  To say that the examiner and I were flabbergasted, would be to undersell our response.  Neither of us could figure out how he did it. It was an amazing display of both raw intelligence and memory. When we got home, I was really curious about how he did it.  So we talked. I pulled out a piece of paper so I could actually write down what he did since he could not write, and what he explained made no sense.  Clearly, he was using ratios in some way. But we had not yet covered ratios, so he had no words to describe his intuition.  His 15 minutes of insight could not be coded into standard mathematical language. At least not by me. I was at a loss.

Because my ds could not write, he did all of his math in his head, and had for years.  I often scribed for him, but it was more me showing him what to write down rather than just writing verbatum what he told me to write.  So that week during math, I tried to scribe for him by just writing exactly what he told me to write, and it became very clear that he had no idea. None.  He could get the answer because of his mathematical insight, but he could not code it.  Over the next year I came to understand that this was a piece of his dyslexia.  He could not *code* his thinking into mathematical language of expressions and equations. He thinking was web-like and based on intuition, it was not linear or really logical, and certainly not structured in a standard way.  And I came to believe that this was going to be a bigger and bigger problem as he advanced in math.  Given his amazing mathematical intuition, it would be sad for him to be limited in math because he could not write it down. His mathematical insight needed a strong linear, logical foundation of writing to be put to great use in higher math.

This was the beginning of my journey to *teach* him *how* to show his work.  It was absolutely not about showing *his* work because *his* work was a jumble of insight that could not be written down.  It was about rewiring a piece of his brain so that he could take that jumble and code in into linear logical steps.  This took 3 years. But this process showed me that there is more than one reason why students don't show *their* work. My son had to be trained not just which steps to write, but how to *think* like a mathematician. Intuition is a wonderful ability to have, but it simply won't get you far in math without proper mathematical thinking.  And writing is thinking made clear.  If you cannot write it, you are not thinking it.

My point is, to ask a student to show *her* work, is the wrong approach in my opinion.  You need to train a student to write the workings in a certain way, and that certain way when repeated day after day, year after year, will train a student to see math differently.  It is no different than practicing scales in violin, over many years you train the ear to hear if notes are out of tune. Drill is what is required.  So for my son, he had to drill proper workings to be able to train his brain to think linearly and logically. To do it the other way -- show your jumbled workings so I can see what you are thinking -- is to miss half of what teaching kids math is all about.

Ruth in NZ

 

Edited October 6, 2018 by lewelma

[regentrude]

11 hours ago, 8FillTheHeart said:

. Instead of putting it at the end, she writes it directly underneath the the corresponding part of the equation on both sides (our focus is on if it is done on one side of the equation it has to be done on both to keep the equation balanced.) 

 

11 hours ago, wendyroo said:

This is how I have my algebra learners do it as well.  I emphasize that they are actually adding two equations, the second of which is simply -3x = -3x or 2 = 2. 

The problem with writing the operation underneath comes when the operation that is performed on both sides is not simply an addition or subtraction. If you do that, how would the student show that she is multiplying both sides with a factor? Or taking the square root on both sides, or raising both sides to a power? Or taking the natural logarithm? In all of these cases, you are NOT merely adding two equations.

Edited October 6, 2018 by regentrude

[8filltheheart]

Just now, regentrude said:

 

The problem with writing the operation underneath comes when the operation that is performed on both sides is not simply an addition or subtraction. If you do that, how would the student show that she is multiplying both sides with a factor? Or taking the square root on both sides? Or taking a logarithm?

This is for 3rd grade. The most complicated it gets is combining like terms and dividing both sides by a variable's coeffiecent.

My older kids aren't writing out what they are doing for those sorts of steps. They have been doing them since 3rd grade. 

[wendyroo]

7 minutes ago, 8FillTheHeart said:

This is for 3rd grade. The most complicated it gets is combining like terms and dividing both sides by a variable's coeffiecent.

My older kids aren't writing out what they are doing for those sorts of steps. They have been doing them since 3rd grade. 

I agree.

We do sometimes have to multiply or divide both sides.  We just write something like (x3) under both sides of the equation.  The parentheses remind the student that they have to multiply the whole side of the equation by the factor, and that that might require adding parentheses.  So the write up would look like:

(1/3)x + 2 = 3x
     (x3)         (x3)
3((1/3)x + 2) = 3(3x)
(3/3)x + 6 = 9x
x + 6 = 9x
 -x         -x 
   6   =   8x
 /8        /8  
(6/8) = x
3/4 = x

[serendipitous journey]

On 10/5/2018 at 6:35 PM, regentrude said:

...

One suggestion I have for algebra learners: write the operation you intend to perform to both sides of the equation behind a vertical line at the end. That is how algebra is taught back home and better than the haphazard way many of my students indicate their operations. For the above example, a beginner would be required to write:

5x-2=3x+4       | -3x

2x -2 =4           |+2

2x=6                 | :2

x=3

Once proficient, the student can omit those extra steps and can skip steps. But for the learner, this is a valuable reinforcement. 

Oh sweet: that is a much more elegant, straightforward way to write this (for the student who still needs to):  thanks so very much.  Just very elegant. 

[serendipitous journey]

4 hours ago, 8FillTheHeart said:

This is for 3rd grade. The most complicated it gets is combining like terms and dividing both sides by a variable's coeffiecent.

My older kids aren't writing out what they are doing for those sorts of steps. They have been doing them since 3rd grade. 

Mine is.  ?  Well, he is when he starts making tons of errors.  I let him skip this step when, as lewelma & regentrude suggested, he doesn't need it.  But when errors start becoming common, we go back to it until he cleans things up again.  That said, he is nobody's model math student (though he is a total sweetie). 

[forty-two]

23 hours ago, lewelma said:

 It was about rewiring a piece of his brain so that he could take that jumble and code in into linear logical steps....My son had to be trained not just which steps to write, but how to *think* like a mathematician. Intuition is a wonderful ability to have, but it simply won't get you far in math without proper mathematical thinking.  And writing is thinking made clear.  If you cannot write it, you are not thinking it.

My point is, to ask a student to show *her* work, is the wrong approach in my opinion.  You need to train a student to write the workings in a certain way, and that certain way when repeated day after day, year after year, will train a student to see math differently.  It is no different than practicing scales in violin, over many years you train the ear to hear if notes are out of tune. Drill is what is required.  So for my son, he had to drill proper workings to be able to train his brain to think linearly and logically. To do it the other way -- show your jumbled workings so I can see what you are thinking -- is to miss half of what teaching kids math is all about.

I had (and am having) a similar experience with my oldest dd.  She could get the answer, but she couldn't explain it, had no idea how to express it in equations.  (I had thought about it in terms of her not being able to verbalize her reasoning; it hadn't occurred to me that maybe her intuitive reasoning itself couldn't be effectively verbalized in the first place.)  And also, she had no idea what to do if she couldn't intuit the answer - no idea that, even when you have no initial idea how to solve the problem, you can still take whatever you do know and get started and work your way toward an answer.  Either she saw it or she didn't, and as far as she was concerned, there was nothing she could do to move from "can't see it" to "see it".  I used a lot of growth mindset concepts with her - that just because she can't see it *now* doesn't mean she'll never see it ever.

Anyway, at least half of 3rd-5th grade math was me teaching her to show her work.  And it was very much teaching her the *thinking* behind it, teaching her how tell what she knows logically and linearly.  It was very late in the process before she was willing/able to use equations to attempt a problem that was not immediately apparent.  I'm noticing similar issues with writing - she knows what a passage means, but setting it down in words is hard, because spelling it out, explaining it, does not come easily.  She's having to *learn* how to arrange her thoughts into a logical, linear order.  It's an interesting thought to me, that maybe part of the issue is that *her* thoughts simply *can't* be set down in a logical, linear order.

[lewelma]

You can really see how a student is thinking when they have to show the workings for a word problem.  Students who have been taught mental math tricks strategies (e.g., stepping stones, shared numbers, etc)  are very good at break numbers apart and put them back together, but they are very bad at coding a word problem into equations.  This becomes a huge problem when they hit algebra.  I have seen students take a question like "You have 75 pies and 15 students, how many pies does each student get to eat?" Tell me it is an addition problem.  The first time I saw this, I was like what??  Basically there are 75 pies, and each student eats a pie, so 15 are eaten, then they eat another set, so you now have 30 eaten. etc. They are doing repeated addition instead of division.  This is a mental math trick, that won't work with algebra. Another example, "You have 116 pies and 75 are eaten, how many do you have left?" The mental math student takes 75 and adds 25 to get 100, then add 16 more to get 41.  So it is an addition problem. Students who have been trained only with mental math and no traditional computational algorithms, have been *trained* to think in a certain way, and it is *very* hard to retrain their brain to actually recognize what operation is being done, because the mental math strategies obscure the true operation.  This is why many of my students struggle with moving to algebra, because NZ is all about mental maths in primary school.  

 

[8filltheheart]

7 minutes ago, lewelma said:

You can really see how a student is thinking when they have to show the workings for a word problem.  Students who have been taught mental math tricks strategies (e.g., stepping stones, shared numbers, etc)  are very good at break numbers apart and put them back together, but they are very bad at coding a word problem into equations.  This becomes a huge problem when they hit algebra.  I have seen students take a question like "You have 75 pies and 15 students, how many pies does each student get to eat?" Tell me it is an addition problem.  The first time I saw this, I was like what??  Basically there are 75 pies, and each student eats a pie, so 15 are eaten, then they eat another set, so you now have 30 eaten. etc. They are doing repeated addition instead of division.  This is a mental math trick, that won't work with algebra. Another example, "You have 116 pies and 75 are eaten, how many do you have left?" The mental math student takes 75 and adds 25 to get 100, then add 16 more to get 41.  So it is an addition problem. Students who have been trained only with mental math and no traditional computational algorithms, have been *trained* to think in a certain way, and it is *very* hard to retrain their brain to actually recognize what operation is being done, because the mental math strategies obscure the true operation.  This is why many of my students struggle with moving to algebra, because NZ is all about mental maths in primary school.  

 

For young kids, HOE is a great program for getting kids to visualize how to set up simple algebraic word problems and what they are doing mathematically.

[aanes821]

I would keep a notebook next to the computer. If they do a lesson and nothing is written down/no work is shown, I would make them redo the lesson. No moving on until the work is shown. At least make them go back and repeat the problems they had wrong and show their work on those. You are still the teacher and the mom. If they were in school and the teacher said she needed to see their work and they didn't show their work, it would be marked wrong. My kids are not quite teenagers though, so maybe this is just wishful thinking.

[lewelma]

22 hours ago, square_25 said:

 

Sorry to revive an old thread, but this is fascinating to read about. Can you give me an example of an algebra problem that they'd have trouble with as a result? 

Sure.  Basically, any word problem that asks you to form an equation.  You can get the answer using primary school methods, but when being tested for an algebra exam, the goal is to code it into algebra so that you develop the skills required as the problems get harder.

For example

Amy bought two seedlings for her garden. One variety was 8cm tall when she bought it and grew at a rate of 2.5 cm per week. The other variety was 5 cm tall and grew at a rate of 3 cm each week. After a certain number of weeks the two seedlings were the same height. Form an equation to work out when the two seedlings were the same height.

My students would not know that you *multiply* 2.5 by the number of weeks.  So they don't know to write that term as 2.5w.  So can't write the equation 8+2.5w=5+3w.  Their approach would be guess and check, and if required to show their work, they would put it into a table if you were lucky.  Basically, they would see it as repeated addition 8+2.5=10.5; 10.5+2.5=13, and they would just go up on each side until the two answers matched.  If the word problem required 2 decimal accuracy, they would be in trouble. 

All of my students are particularly bad at recognizing division, as they see it as repeated addition to get up to an answer because that is easier to do in your head. 

 

[lewelma]

In NZ, the national assessment is called "applying algebraic procedures in solving problems" so if you solve the problem without using algebraic procedures you can't pass the test.  The goal is not to get a correct answer, the goal is to learn how to apply algebraic procedures in solving problems. 

As to how to teach this, I'll try to write more about that later.

Edited February 21, 2019 by lewelma

[wendyroo]

2 hours ago, square_25 said:

To be fair, that’s a correct solution. I have a lot of trouble with kids who think there’s only one correct way to do things. They got trained out of their mathematical confidence.

How would you encourage the algebraic way without making kids think this is wrong? 

Not lewelma, but I thought I would throw in my two cents.

I would always validate that the guess and check (or other correct method they came up with) was correct and good thinking on their part.  But then I would challenge their method by tweaking the problem a little: How tall will the first sapling be after 20 weeks?  Which sapling will reach 1 meter tall first?  How long will that take?  How many more weeks will it take the other sapling to reach 1 meter tall?  etc.

Using the guess and check/chart method, each of those problems would have to be approached more of less independently.  I would try to lead the student to the idea that the algebraic equation is useful because it provides easy answers to all those questions and countless more.  NOT that the guess and check method is bad, but simply that it has its strengths and the algebraic equation has its strengths and it is important to start to learn when each method makes the most sense.  You could compare this to finding the area of a shape via dividing into a grid versus multiplying the side lengths.  There certainly are some convoluted shapes that might be better approached with the grid method, but for rectangles whose areas can easily be found by multiplication, the grid method would be inefficient and prone to errors.

I would also talk about the goals of math.  Certainly right answers are one goal, but perhaps even more importantly, math is a method of communicating.  For example, if you were to buy a sampling from the garden supply store, which would be more convenient, them giving you a huge chart that showed how tall the sapling would be at 1 week, and 2 weeks, and 3 weeks, and ....  OR would it be more convenient for them to let you know that to find the height you just had to multiply the number of weeks by 2.5 and then add 8?  And what if you were communicating with a computer?  More and more often math is being used to tell a computer what to do.  So what if you were programming a video game and needed to tell the computer how fast to let a character travel based on how much weight they were carrying.  It would be a waste of time and space to make a chart of every possibility, but with one equation you could tell the computer exactly what you wanted it to do.

Wendy

[wendyroo]

6 minutes ago, square_25 said:

Wendy, would you come back to guess and check once in a while to verify that their intuitions match the algebra? Or once you get to algebra, do you stay with algebraic techniques? 

Oh, I am all about double checking with guess and check...or just any method other than the one they first used to see if they get the same answer.  Not every problem, but certainly any they aren't completely sure of how to solve...and I make sure that my kids are consistently challenged with problems that stretch their abilities.

This is coming up a lot right now with my 9 year old because I am having him work through some old SAT tests.  Some of the math is just beyond what he truly knows how to elegantly solve, but I am still encouraging him to play with the numbers any way he knows how to see if he can at least eliminate a couple wrong answer choices.

For example, one problem that came up recently was:
Which of the following is an equation of a circle in the xy-plane with center at (0 , 4) and a radius with endpoint (4/3 , 5)?
It then listed 4 answer choices in standard (x – h)^2 + (y – k)^2 = r^2 form.

He is well versed in linear equations, but he had never encountered the equation of a circle, so did not know the significance of the r squared term.  The goal wasn't to teach any new math, but rather to see how creatively he could use the math he already knows, so the first thing I had him do was graph it as accurately as possible.  From there he could clearly see that he could find the length of the radius via Pythagoras, which could lead to the y intercepts.  At that point it was just a matter of guessing and checking his way through the answer choices until he found the one that produced the correct y intercepts.

In my experience, with my kids, math concepts go through a fairly standard learning progression. 
First, the child starts spontaneously asking about and experimenting with a concept.  My 5 year old is currently thinking and talking about multiplication a lot.  His official math curriculum hasn't gotten there yet, and he isn't ready to formally learn Multiplication with a capital M, but he still thinks it is wildly funny to "trick" me with "really hard" questions like 1 * 3 billion!!! 
Second, as their understanding of the concept develops, they frequently double check themselves with older, more comfortable methods.  When Prodigy recently asked the 5 year old how many legs 5 dogs had, he quickly counted by 5s up to 20, and then he proceeded to draw the 5 dogs and their legs and count to check himself. 
Third, their math curriculum eventually teaches the topic and their knowledge starts to be formalized.  They learn the vocabulary and algorithms that describe the concepts they have been developing in their heads all along.  Now they are firmly nudged into practicing the concept over and over until their arithmetic as well as their intuition is solid.  (Of course, by this point their challenging topics of interest are much more complex, so this drill is just wrapping up loose ends on a topic that conceptually is old hat.) 
Fourth, finally, they are masters of the topic and they feel indignant that anyone would even suggest they write that step down in a math problem or double check their answer.  It is at this point that I start looking for unique and challenging applications of the topic so that their skills and facility continue to grow.  At that point with multiplication I might introduce exponents or factorials or combinations or calculating approximate tax, because all of those seem very grown up, but sneak in extra multiplication practice.

Wendy

[wendyroo]

16 minutes ago, square_25 said:

That's interesting! I haven't seen that progression myself, but I've also been doing something fairly non-standard, so maybe that's why. For this specific thing, I just defined multiplication without doing any skip-counting, so probably my daughter didn't have the chance to play around with it herself.

Yeah, I don't tend to define much for them.  My kids seem to thrive on the discovery method, so most of what I do is label discoveries ("What you are doing is called multiplication.") and ask interesting questions that challenge them to grapple with topics more deeply.  I've never taught skip counting, but all of my kids have stumbled upon the idea naturally.

She also doesn't seem to have any patience with calculational drills at all: I got her to practice adding out loud for a while, but she moans so much about simple calculations on a worksheet that I haven't been doing it at all. So even though she's not solid on her times tables, we've kind of moved on conceptually. We'll probably come back and drill a bit more at some point, though. 

My kids don't have a choice about drill.  I think arithmetic fluency is important, so I have them steadily work on it.  I use the method that Susan Wise Bauer describes as pecked to death by ducks.  Just a little bit of work, consistently done every day.  The rest of the kids' math time is spent exploring more challenging concepts and puzzles.

Oh, question for you: do you introduce the standard algorithms at the usual time? (Like, adding with carrying, borrowing, etc.) I've been avoiding them entirely for now, but I imagine I'll go back and do them at some point. 

My kids work through Math Mammoth for arithmetic (along with many other materials for deeper, conceptual learning).  So far, all of my boys have started their kindergarten year somewhere in the middle of level 1.  I start them there not because that is the level of math they are working at (they already have a firm grasp on place value and are mentally adding and subtracting two digit numbers and exploring multiplication), but because that is the level of "math writing" they can do.  They work on it for 5-10 minutes a day, and it acts as review and reinforcement as well as introducing algorithms, vocabulary and notations they might not be familiar with.  It also provides them with practice reading and working carefully and independently.  Math Mammoth introduces two digit addition and subtraction in columns along with regrouping in addition at the end of level 1, so my boys learn it sometimes during kindergarten.  Regrouping subtraction isn't covered until 2b, so my kids learn how to formally write that algorithm toward the beginning of first grade.

Since it is mostly review for my kids, we move pretty quickly and add a lot of supplements.  My second grader just finished MM4, so learned the multi-digit multiplication and long division algorithms.  He was, of course, already doing both of those operations, but only mentally or conceptually.  He often calculated them via a hodge-podge partial product method he had figured out or through logical reasoning.  Now that he knows the algorithms, he seems to use a nice balance of those and mental math and his old partial product method depending what best suits the problem.

 

[wendyroo]

14 minutes ago, square_25 said:

Do you think doing calculational drills that are specifically calculation and nothing else is important? For example, if she's solving the system

x+y = 15, xy = 56, (well, in triangles and squares instead of x and y),

then she's practicing quite a lot of multiplication along the way, since currently we solve those by trial and error (and number sense, I suppose). But last time I gave her things like 7*8 to practice, she took longer to do it than she does a question like the one above (which generally takes quite a few multiplications for her before she gets it.) 

Actually, I had a bit of trouble when we did a lot of Beast Academy questions of form "35 + 56 = ", for some reason, because she started misusing an equals sign and I found that troubling. So that probably also dissuades me currently. I might be overly paranoid about this issue, though: I see misunderstanding of equals signs so often in older kids that I'm sensitive to it. 

Thanks for the detailed responses! I like hearing what other people are doing for their math with mathy kids. 

I have four kiddos, so around here I need to foster independence. 

Something like "x+y = 15, xy = 56" is what we would work on together during problem solving time.  It might come from Beast Academy or Singapore Challenging Word Problems.  During problem solving time, we do a lot of parallel solving, so we both work on the same problem independently.  Then, when we both reach an answer (or at least make a go of it), we compare notes.  Did we get the same answer?  What methods did we each use?  If we got different answers, what support/defense can we provide for our calculation?  Can we come up with a third solving method to act as a "tie breaker"?

I try very hard not to be seen as all-knowing during problem solving time...and with Beast Academy puzzles that is often pretty easy.  I model good problem solving techniques (like correct use of the equal sign), but I also try to model resilience, perseverance, not being frustrated by wrong answers, a willingness to try one solving method, realize it is not panning out, and start down another path.

I would never leave that kind of high-level, challenging math to be done independently.  I think deeper, conceptual understanding it built through modeling, interaction, Socratic Dialogue, etc.  However, I do think there is value in my kids then spending some time working on lower level math independently.  It is not pure, tedious drill and kill, Math Mammoth is actually a very conceptual curriculum, but I do have the kids working independently through a much lower level than they are conceptually.  When they sit and tackle it all by themselves, it isn't meant as a huge arithmetic challenge, but rather as a challenge in close reading, organized thinking, diligence, focus, and self-control.  The fact that it often really cements their mathematical foundations is just icing on the cake.

[moonflower]

This board is such a good education.