Working out of a math textbook: showing work, habits to build, etc.

[TheAttachedMama]

My kids will be moving out of a "workbook" and into a more standard textbook approach for the first time next year.   (Previously, we used Singapore workbooks.  This year we will be working out of the AOPS textbook for pre-algebra.  However, I think this question applies to any math textbook.)   

My two older kids tend to be really sloppy in their work.   I think this leads them to make a lot of careless mistakes, so this is something I want to work on this school year.    

What type of paper or composition book do your children use for math?  (I would love some links!)   

What organizational strategies have you taught your kids for setting up their math practice/homework?   (Boxing answers, re-writing the original problem, margins, spacing, etc. etc.)

 

[HomeAgain]

I have a kid with sloppy handwriting.  For the past two years he's worked out of this centimeter graph paper notebook. I don't always make him rewrite the original problem (I help with that still, and word problems he's only required to write the initial set up, not copy the word problem), but he is required to box his answers.  For the first part of his notebook he drew colored lines to separate the problems and their work space to make it easier on my eyes.  I require all equal signs to line up, which for some reason was the hardest task to learn. ?

As he gets older we'll switch to 1/4in graph paper, but the centimeter one is just right for now.

[Angie in VA]

Yep. Graph paper.  Usually graph paper has one side w/ small squares and the other side w/ larger squares. That should accommodate a child who has large handwriting.

[Mshokie]

We do graph paper and fold the paper in half longways, so that he writes the problems in two columns. Otherwise, he was squeezing many problems next to each other. He also circles/boxes the answers. 

[El...]

What an excellent question.  I'm following for ideas, and am off to order graph paper!

[ScoutTN]

When my dd was struggling with math neatness or showing her work, I sometimes made her re-do the work. If it's hard to read or incomplete, I do not accept it. Only takes a little of this to get results. 

It helped to explain to her, by asking questions, that if she doesn't show her work, I cannot see where her mistake was, cannot see if it was a careless error or one of understanding.

We too often use graph paper. My kids like a 3 squares per inch one rather than the standard 4. 

[Arcadia]

3 hours ago, elroisees said:

 I'm following for ideas, and am off to order graph paper!

 

You can print the 1/4” grid graph paper from below link for math. We used the 1/8” grid graph paper in same link for science lab graphs. https://www.waterproofpaper.com/graph-paper/grid-paper.shtml

OP,

My kids typed for their AoPS online classes and DS13 prefers blank paper for math instead of college ruled paper. Ironically it did help because for AP Calculus exam there are no lines and he just have to write neatly in the space provided. He boxed his answers for math, physics and chemistry. 

If I can’t read their work, I can’t help or grade their work. Neither can their public school teachers. So my kids were forced to be neat from the time they were in public charter school.

Another thing was showing their work. No working no credit and their grades go from A to B for public school tests. By the time DS13 took his AP exam, he was so familiar with grading rubrics that he shows all relevant working just not to lose points. 

DS13 typed most of his work because his hands get sore writing to the extent he switched to writing with his left hand when his right hand is tired. DS12 doesn’t tire from writing as easily and type Math slower than he writes so he does mainly writing. 

[Monica_in_Switzerland]

We did like you, SM to AOPS PA.  So far, my oldest is just finishing up the last chapter of AOPS PA, here is our experience:

I started with him using a notebook (graph paper) for SM 5.  This is the last level of SM he did, and I wanted to use it as a transition year to get him used to correctly transcribing info from the workbook into his notebook.  This helped to separate out skills a bit and let him develop them more slowly.  Because our particular  notebook does not have margins, he starts by drawing a ruler-width margin across the top and down the left, then splits the remaining work area into two columns with a line.  He then writes his full name, date, and book section in the upper margin.  From there, left margin is for problem numbers.  I did not necessarily require that he copy out the original problem.  We spend a lot of time discussing what work to show and how, and it's still a learning process even after wrapping up AOPS PA.  He always boxes his answer.  Once he finishes going down the page, he goes to the second column and starts from there and works down.  Then he goes on to the next page.  

Hope that helps!

 

[daijobu]

I like these green engineering pads which the gridlines printed on the back of each page.  That way the grid lines are faint and easy to ignore when not needed and don't obscure your writing, especially with pencil.  (And you get to pretend to be an engineer!)

Teach your kids to read actively.  That is, they should be doing a lot of writing as they are reading.  In fact, I would model this behavior for your students because the first few chapters of pre-algebra in particular are very dense.  Here's an example that I posted on another thread:

"Our explanation in Problem 3.28 not only tells us how to find the prime factorization of the least common multiple of 24 and 90.  It also tells us that this prime factorization must be included in any common multiple of 24 and 90.  That is, any positive common multiple's prime factorization must have 2 raised to at least the 3rd power, 3 raised to at least the 2nd power and 5 raised to at least the 1st power.  Also, any multiple of lcm[24,90] must also be a common multiple of 24 and 90."  

You really need to have the numbers on a pad of paper in front of you with the prime factorizations and multiples listed out so you can see what Richard is saying here.  I would read aloud to your student and write out everything.  

Also:

• line up your equal signs
• no stream of consciousness equations
• try to do the problems on your own first (but don't beat yourself up if you can't figure it out in a few minutes, just go straight to the solution)
• If you didn't find the solution to the problem on your own (see above) write out the solution they provide.  Same with all exercises.
• My students are allowed to do any calculations in their head...as long as they get the correct answer.  Incorrect answers must be written out.  

[daijobu]

This is a link to someone doing a live solve of an AMC.  I thought the structure of his scratch paper was interesting: dividing each sheet into 4 boxes, one box per problem.  This is not what I do,  (What if you need more space?) but maybe something to consider.  

 

[Lori D.]

Graph paper -- has lines for each step of the process, and the boxes help with lining up digits. If the student is a little more coordinated, s/he can turn regular lined notebook paper 90˚ and use the lines as columns for keeping digits lined up.

Be prepared for a lot of griping and whining of not only having to copy the problem out of the textbook onto the paper, but also having to show all the steps.

re: copying problems
Like above poster, I did a lot of the copying of problems onto the graph paper for DS#2 who struggled with the act of writing (his hand was exhausted by the time he had the problems copied before he even got to solving), and who has stealth dyslexia, and had troubles correctly copying from book to paper. Another option is to scribe some problems for your student (esp. in the beginning), while they talk through the steps, and you explain (over and over) how you to line up and write each step below the previous step. Another option is to do some of the problems on the whiteboard.

Another thing to drill into the student from early on is as soon as the problem is copied, put the pencil down and use the index finger of one hand to move under each element of the problem in the textbook while simultaneously using the index finger of the other hand to move under each element of the problem of the copied version to double check that all the signs are correct, and that no element was left out or reversed. Model this step. Then you use your fingers and the student says each element, looking back and forth. Then eventually move to the student using his/her fingers and saying it while you watch. Finally have the student fly solo. It is VERY worth it to take 2-3 weeks of drilling this to make it a habit.

re: showing your work
 Starting with Pre-Algebra, it is critical to get students into the mindset that it is not enough to just have the "right answer" -- that showing the process is equally important -- for them to see where they got off-track, and also because Algebra topics are all about showing *process*, and not just about answers (like math facts). So I strongly recommend starting from day 1 with the policy that a right answer is worth 1/2 point, and writing out the steps is worth the other 1/2 point. They can have 100% correct answers, but the work is an F (50%) if they failed to write out the steps, because showing the steps is just as important as the final answer. 

Also be sure to explain how this works in their *favor* -- they can get a wrong answer, but showing the steps of solving gets them partial credit, which can result in a higher grade even for a number of incorrect answers.

[Monica_in_Switzerland]

15 hours ago, daijobu said:

 

• My students are allowed to do any calculations in their head...as long as they get the correct answer.  Incorrect answers must be written out.  

 

BWAHAHA!  Are we twins???  I'm ??? over here!

[Tanaqui]

Quote

What organizational strategies have you taught your kids for setting up their math practice/homework?   (Boxing answers, re-writing the original problem, margins, spacing, etc. etc.)

 

1. Every single step is labeled, in order, and the steps are put near to each other in a sensible way - you don't have step a on the left side, step b somewhere on the right, and step c in the middle.

2. You have to write out any formulas you're using at the top of the problem. If you're trying to find the time spent going from point a to point b at x speed, then before you even start you write out t = d/v. I don't make them rewrite the words in word problems, but they do have to put down the equation they're working with, which means that if they're using a formula they have to plug in all values into that formula before they start.

3. Even if you're using a calculator or mental math, the equation has to be written out. It's not enough to simply think "Okay, obviously I used a calculator, let's put the answer in the right spot", you need to somewhere indicate what you did and that this is the answer. Sure, okay, we all know that your next step in this problem is combining +7 and +1, and that means +8, but ffs, just write it somewhere, 7+1=8. Now it's stupid, because it's self-evident, but one day it won't be and you'll make an error and never catch it.

4. The equals sign does not mean "look out! here comes the answer!" It means "this side is equal to that side". If you have two or three or four equations to do, they must get written separately! You cannot simply say 7+3 = 10-2 = 8x5 = 40 because, in the end, 7+3 != 40. You might think you'll never get confused, but seriously, it's a tiny thing to just rewrite it properly. That is my hill, and I will die on it.

It's a lot of reminders and a lot of work, but I feel it really does pay off when they start doing more complex math. Sooner or later you hit a point where you need to have neat work habits to keep track of what you're doing, and building those habits now is important. Plus it's a kindness for their math teachers once they have teachers besides moi.

[daijobu]

3 minutes ago, Tanaqui said:

 

4. The equals sign does not mean "look out! here comes the answer!" It means "this side is equal to that side". If you have two or three or four equations to do, they must get written separately! You cannot simply say 7+3 = 10-2 = 8x5 = 40 because, in the end, 7+3 != 40. You might think you'll never get confused, but seriously, it's a tiny thing to just rewrite it properly. That is my hill, and I will die on it.

.

 

You will not die alone.  I will join you on that same hill.  Vive les mathematiques!  ?? ?? ??

[nixpix5]

Graph paper

Box answers 

And I give less work incentives for neat, careful and precise work. I will have a student do odds for example and then I grade the work. If everything is correct and looks neat I shave off some of the work. If answers are neat but incorrect I shave off a few but have them do some more for practice. If it is sloppy all works is completed and corrections are written out in full on new paper. 

Kids take the path of least resistance. If they know they will have to write out their work again if they rush or do all problems if messy, they will suddenly be more careful and neat. It worked with my first two and I do the same with my little ones. It is the only way I have found to create the habit of caring about neat and precise work. 

[Zoo Keeper]

This paper is helpful for one of mine ...

here it is again, only with graph lines.

 

 

 

[daijobu]

Here are couple of nice documentation tricks that I use when I'm in the thick of it.  

(1) Every time you perform some operation to both sides of an equation, you can preface the change with a little extra notation

     x + 3 = 6

S3: x + 3 - 3 = 6 - 3 

Where  S3  stands for "Subtract 3 from both sides."  D4 stands for "Divide both sides by 4."  

I don't do this all the time, unless I'm demonstrating for other students and I want them to be able to recreate what I did.  Or I'm working on something complicated and I want to remind myself what steps I've taken.  

(2)  If you are solving 3 (or more) linear equations with 3 variables, I find it useful to number the equations with lower case Roman numbers:

(i) 3x + 7y -z =23

(ii) 8x - 7y + 2z =99

(iii) 3y + 17z = 33

Now if I am eliminating one variable from equations (i) and (ii) I'll label my work:

(i) and (ii): 23 + z - 3x = 8x + 2z - 99

and so on, so I can keep track of which equations I've used already.  

And as others have stated it's important to model the behavior you want to see.  

[Jackie]

We’ve just recently been making the transition from workbook (Beast Academy) to AOPS, with a few chapters of Jousting Armadillos in between. 

I’m another vote for graph paper. My kid generally writes large, so we use a large-grid graph paper. 

I’ve been writing the problems for her onto the graph paper during this transition. At this stage, I can usually look at a problem and figure out approximately how many lines it should take to do the work, and space the problems accordingly. 

Yes, box the answers.

We’ve watched all of the AOPS Prealgebra videos. He models writing out the steps in the problems as he does the videos. The videos are well done and worth watching regardless, but this is a good way for a student to see what is expected to be written in each type of problem.

[4KookieKids]

Ahh! This thread pushed me over the edge and I just ordered our first AOPS book! ?

DS isn't quite done with Singapore or BA yet, but will finish them up this coming year and passed the placement for aops prealg, so I think we're just going to start going through the prealg super slowly together as something special when his little sisters aren't around (he does his other math independently at this point, and it's a good thing, because three little sisters are very distracting! lol). He's a messy, disorganized kid, as well, so I want to give him lots of time doing it *with* me before I let him loose with it. 

I've been eyeing AOPS ever since I heard about them but I couldn't justify getting them... until now. I'm pretty pumped that this day has come! lol. 

[daijobu]

15 hours ago, 4KookieKids said:

Ahh! This thread pushed me over the edge and I just ordered our first AOPS book! ? 

I've been eyeing AOPS ever since I heard about them but I couldn't justify getting them... until now. I'm pretty pumped that this day has come! lol. 

 

I'm excited for you!  FYI, the first chapter is deceptively "easy" but in reality is quite dense, I think, because you are dealing with "easy" material, that is: addition, subtraction, multiplication, and division.  You'll be applying a lot of rigor to what may be "obvious" though it really isn't.  Addition and multiplication aren't so bad.  But sit up straight when you go through negation, subtraction, and division.  You may even want to keep a notebook of definitions and properties to refer to in your proofs.  Reciprocals and division are fairly tricky, but important to understand.  But that's the beauty of AoPS: you'll get a deep understanding of the why of these basic operations.  And you won't be taking calculus learning negation by rote. 

[4KookieKids]

1 hour ago, daijobu said:

 

I'm excited for you!  FYI, the first chapter is deceptively "easy" but in reality is quite dense, I think, because you are dealing with "easy" material, that is: addition, subtraction, multiplication, and division.  You'll be applying a lot of rigor to what may be "obvious" though it really isn't.  Addition and multiplication aren't so bad.  But sit up straight when you go through negation, subtraction, and division.  You may even want to keep a notebook of definitions and properties to refer to in your proofs.  Reciprocals and division are fairly tricky, but important to understand.  But that's the beauty of AoPS: you'll get a deep understanding of the why of these basic operations.  And you won't be taking calculus learning negation by rote. 

 

 I routinely teach proof based courses at the local University, both at undergraduate and graduate levels, so my main concern is actually requiring too much rigor from a youngster… ? I don’t want to ask more of him than is age-appropriate, especially given his organizational challenges, but it’s hard to know where the balance is! 

[wendyroo]

3 hours ago, 4KookieKids said:

 I routinely teach proof based courses at the local University, both at undergraduate and graduate levels, so my main concern is actually requiring too much rigor from a youngster… ? I don’t want to ask more of him than is age-appropriate, especially given his organizational challenges, but it’s hard to know where the balance is! 

For what it is worth, I am choosing to slowly require more rigor and organization from DS, rather than jumping in with both feet and risking souring him to AOPS or math in general.

DS started AOPS prealgebra back in November when he was about 8.5 years old.  He has autism, ADD and anxiety, and is organizationally challenged to say the least.  He had no problem with the math, but he wrote almost nothing down and what he did write was often huge, illegible and scattered all over the page despite working in a graph paper notebook.

At first all I required was that he number his problems and box his answers.  Period.  Since I could barely read anything he wrote, when we corrected his answers he would read them to me one by one (and circle and redo problems he got wrong).  Lo and behold, he had a hard time reading his writing too!!  And frequently he would have to redo problems simply because he couldn't figure out what his answer said.  Also, when he got problems wrong he would have to start completely from scratch because he hadn't written any of his work.  Once in a while I was even super busy when he finished his math and we would have to wait hours or even until the next day to correct his work.  By that point he had forgotten what the problems were about and was completely lost in his quagmire of a notebook.

Slowly, slowly, slowly I have nonchalantly offered math organizational tips, and even more slowly his notebook is becoming more comprehensible.  At this point he has made a complete 180 and now instead of cramming 20 problems onto one cramped page, he is writing bigger than ever and leaving HUGE amounts of space around all his problems.  It is a good thing notebooks are cheap this time of year, because he is going through one for almost every chapter.

My plan is to slowly require more and more neat, precise write ups of the problems.  For now, he is thriving, so I am loath to rock the boat.  He is almost done with chapter 7, and is consistently getting ~95% of the chapter problems correct on his first try (~100% on his second try) and about 85% of the challenge questions correct on the first try.  Clearly his method is working for him, so I'm going to let him run with it while nudging him to make small changes along the way.

Wendy

[4KookieKids]

2 hours ago, wendyroo said:

For what it is worth, I am choosing to slowly require more rigor and organization from DS, rather than jumping in with both feet and risking souring him to AOPS or math in general.

DS started AOPS prealgebra back in November when he was about 8.5 years old.  He has autism, ADD and anxiety, and is organizationally challenged to say the least.  He had no problem with the math, but he wrote almost nothing down and what he did write was often huge, illegible and scattered all over the page despite working in a graph paper notebook.

 

Ha ha. Our sons sound very similar. ? No worries about jumping in too quickly. Initially, I'll just scribe for him, and then I can both model what and how I'd like him to write and also pause to talk about conventions and expectations. That way, by the time he's doing it on his own, hopefully he'll have some clearer pictures in his head of what it should look like. Or that's my hope, at least! lol.