# Problems in Fifth Grade Math

Last year, I bought almost all my daughter's text books, so that she would always have a copy at home. The school provides free copies to the students, but they are very big and very heavy, and the children seldom take them home.

This year, I decided to pursue a different policy. We didn't use most of the books I bought last year, so I decided to focus on what I can teach my daughter independently and let the school teach her what they think she should know.

However, we've been having some problems with math, so I may end up getting the math textbook so that I can keep up with the material they are studying in class.

## Meet the Authors of the Textbooks

The other day my daughter was struggling with some math problems that she had been assigned as homework. Since she was complaining about it, I offered to take a look. One of the problems looked like this:

18 + 26 = _______

"Well, that doesn't look very hard," I said. "What seems to be the problem?"

I rewrote the problem for her on a piece of scratch paper like this:

## 18

## + __ 26__

"Of course, that's easy!" my daughter said. "But I'm not allowed to do it that way. I have to do it sideways."

This had me baffled. "You have to do it sideways? But what difference does it make?"

"I'll get counted off if I don't do it sideways."

"But the answer will be exactly the same!"

"Yeah. But this is supposed to be mental math. I'm supposed to do it mentally."

"Oh. Well, in that case, you do it in your head. Then you write the answer down."

"No! We have to show our work."

I was at a loss. To me "mental math" means that you don't show your work! If you write it down, then in what way is it "mental"? By that reasoning, all math is mental.

But then I looked at the instructions, and it seems my daughter was right. Here is what it said: **"Add or subtract mentally. Use compensation. Show your work!"**

Later in the week, I spoke with my daughter's teacher, and the teacher confirmed that my daughter is capable of solving the problems, but not by the method that the class is currently studying. "Well, if you can send home a description of the algorithm, I'll make sure she learns to do it that way," I said.

"Okay. I'll send some instructions on how to use compensation." She was true to her word. The instructions were essentially copied from page 88 of the textbook, so I'll just cite the textbook here:

**To use compensation, add a number to one addend to make the addition easier. Then adjust by subtracting the same number from the other addend.**

When I finished reading these instructions, I was not very much enlightened. Clearly if you add and subtract the same number from your total, the total will remain unchanged. But what did they mean by "to make the addition easier"?

Fortunately, the textbook also provided an example.Next to the picture of a cute little terrier, there was the following tabulation.

## Category Number of Breeds

## Terrier ................................................ 28

## Toy ..................................................... 23

## Which is easier to add: 28+23 or 30 + 21 ?

My initial response was that they were both equally easy to add. But after a little more thought, I came to realize that this was a rhetorical question. The textbook writers had already decided that the easier sum was "30+21". They wanted the student to add two to twenty-eight and subtract two from twenty-three, so as to get the sum thirty plus twenty-one.

Having determined this much, I was now in a position to ask the all important question: why? Because without answering that quesiton, I would never be able to come up with an algorithm that my daughter could use in order to predict which sum the educators think is "easier."

I'm not a mathematician. I don't know very much about numbers. However, it seemed to me that this is not really a math problem. It's a mind reading problem. "Easier" isn't a rigorous mathematical concept. Easier depends on your point of view. So what was the point of view of the people who wrote the text book?

There must have been some assumptions that they were taking for granted and that they refused to spell out for the students. My job, as a parent, was to ferret out those assumptions.

"This has something to do with the fact that they are using the decimal system to do all their calculations," I mused. "It has nothing to do with the numbers 28 and 23 per se."

For instance, in hex, twenty-eight would be 1C and twenty-three would be 17. Adding two to one addend and subtracting it from the other would not get you a number that ends in zero on either side of the addition sign. If we were doing this problem in base 28, then twenty-eight would be 10 and twenty-three would be N and the sum of 10 and N would be 1N.

The educators who wrote the textbook assumed that it's easier to add two numbers if one of them is divisible by ten. Of course, if they used a different base, say base N, where N stands for any whole number, they would think numbers divisible by N were easier to use in additions.

The rule my daughter needs to learn is that she must determine which of the two addends is closest to a number divisible by ten, then she has to determine which arithmetic process, addition or subtraction will get it to equal the number. In the case of twenty-eight and twenty-three, twenty-three is three away from twenty (by subtraction) and twenty-eight is two away from thirty, (by addition). Since two is smaller than three, then the educators want her to choose two and to add it to both addends, and all so that one of them will end in zero.

That seems like a lot of work. Now all I need to do is find an algorithm for determining which of two numbers is closer to a multiple of ten! Then I can give airtight, rigorous instructions on how to solve an addition problem using compensation.

## Other parents grapple with the problem

Some children just instinctively know what the teacher means, because they are able to put themselves in the teacher's shoes and realize what her assumptions are. For those children, rigorously defined instructions are not necessary.

However, many children do not know what the teacher or the text book writer mean unless given explicit directions. They have no idea why one sum is supposed to be easier than another. They may have a picture of twenty-eight dogs in their head and another of twenty-three dogs, and they do not know why counting them should be easier if they are rearranged into two different groups of thirty and twenty-one. In particular, if one is a group of terriers and the other is a group of toy poodles, the child might even be reluctant to mix the two groups.

Such children can follow instructions, but only if everything is completely spelled out. They will do what you say, not what you mean. The same is true of computers. Perhaps a good exercise in the pedagogy of elementary arithmetic might be to require all text book writers to first write a computer program that performs the calculation the students are expected to make in the manner that they are expected to make it.

(c) 2009 Aya Katz

## Comments

Seems like another example of taking something simple then reworking it around and taking twice as long to get the answer. Honestly as an adult when the day comes that all the math you learned becomes needed do you really want the long way around or the shortest most direct root to your answer. What are the teaching kids to be in school now? Wait I know politicians.

Good grief. That is such a terrible waste of a child's educational years, when learning should be fun and should be adapted to real situations, not hypothetical addends. No wonder this country is in such a dreadful mess. I'm so sorry the curriculum at your daughter's school is so idiotic.

Don't the teachers want to rebel and throw the textbooks away?

Math is the bugga boo of our household. It's the one subject Kaela doesn't get instantly and her self esteem was suffering because one of the boys teasing her. Over the summer, we finally resorted to hiring a math tutor who still comes over 3 days a week.

We buy copies of all the books -- that way there is no excuse if she forgets a book. Also, she can highlight in them and make notes in the margins for studying for tests. As an experiment, we weighed her books this year (7th grade) -- they weigh 62 pounds! She only weighs 102.

On a side note, this household reads all literature books the kid reads and it makes for some lively generational discussions. I highly recommend it -- allows you to ask some probing and relevant questions. It also gives a good guide to knowing what reading is influencing your child.

Back to math? I'm a firm believer that kids need to know practical math that they will actually use as adults before all the compensation, etc. Wondering whatever happened to word problems that applied to real life? Too many children in America never make it out of high school, at least the basics would help them function as adults.

Aya,

Obviously the textbook writers are too stupid or too evil to write a precise definition, but what they mean is that they want her to avoid the carrying by regrouping. Every time carrying is required, they want her to increase the first summand to the nearest multiple of 10 and adjust the second summand accordingly.

Why they want her to do this is another question. It's some kind of NCTM idiocy.

My niece had a stroke and cannot physically add items that line up. It's a miracle she can add at all, but she had lots of help from her parents, who found ways to inspire her thought processes. Parents who are involved, like you, make so much difference in their kid's lives.

It seems every year there is new math. I'm not sure if it is an improvement. I wonder if the point is to teach math or to teach some other thinking process? Doesn't there have to be some reason they play around with seemingly simple concepts? Perhaps they are all like a friend of mine's daughter who is a math whiz and always finding new ways to do math in order to relieve the boredom?

Anyway, I like the way you managed this, Aya. Certainly your kids will grow up exceptional!

He-he, you know too much Aya, that's your problem :D

I had problems with math as a child but if I had had to deal with this I would gone crazy! It seems so counter intuitive!

Aya

The problem is not math but reading skills.

Sometimes reading skills are made more difficult because of the writers technique of presenting information.

I know that looking at your child's textbooks is sometimes a foreign object. I remember looking at my children's homework. When your child has to explain what the teacher wants as opposed to you reading the question and understanding it, that is a problem.

As for math, the problems get more difficult when they become word problems. That is the question is words with numbers asking you for an answer. You have to visualize the 18 plus 26 from the words.

Also, in the math area there are better techniques available than is found in text books. But alas, formal education requires you to follow their rules.

If a hen and a half lays an egg and a half in a day and a half, how long does it take a hen to lay an egg?

(You don't have to answer :)

Must admit, this one made me think. When adding or subtracting large numbers, I do tend to use that method. However, the key is that this is something I instinctively do, and was never taught it.

Everybody has their own unique way of doing things, and forcing one way or another stifles the mind's ability to generate creative solutions and methods for solving problems.

Aya,

I'm not sure (it's so stupid and all.) But I think the point is to get to a summand that is a multiple of 10. Maybe it's all right to do it on the negatived side instead of the positive one. At least it would be interesting to hear the teacher try to explain why not.

The NCTM is the National Council of Teachers of Mathematics,

who are in large part responsible for our present predicament.

p.s. I hope you'll get the chance to look at my Dvorak hub. I linked to two of yours.

I work with children and I encounter this problem all the time! It just gets so frustrating because the kids end up wanting to give up altogether just because a problem is not written according to their usual way of understanding it.

I thought I was the only one that was confused by these methods that teachers and textbooks are teaching the children. Thanks for posting this hub. The kids will feel a lot better knowing that it is not just them.

Interesting hub - I used to teach basic skills maths to adults, and many of them had the same problems as your daughter. For most of them, it wasn't just the maths per se they found hard, it was actually trying to understand what the question was asking!

I don't think it's necessary to require text book writers to compile software that performs a calculation - that's like using a sledgehammer to crack a nut! With the "28 + 23" example you gave, all that the question setter would have had to do was say something along the lines of "If you have to add two numbers like 28 and 23 in your head, sometimes it's helpful to change them to things that are easier to add. If you add 2 to 28, it becomes 30. If you take away 2 from 23, it becomes 21. So your sum changes to 30 + 21 instead of 28 + 23. Which do you find is easier to add in your head, 30 + 21 or 28 + 23? Why?" Cue lots of classroom-based discussion (if the teacher is any good, that is).

"...what you suggest is essentially what the text writers did. The problem is that unless the student personally finds it to be "easier" to add 30+21 rather than 28+23, the student will not have any idea what to do with a similar problem, but slightly different numbers. That was the point of this hub."

I see what you mean, but hopefully a bit of classroom discussion would smooth problems like this out. The crucial thing IMO is for the teacher to make it clear that students don't *have* to use this method if they don't find it helpful, and to encourage the class to come up with other ways of solving problems if that's appropriate.

"She came up with 31+20 by subtracting three from 23 and adding three to 28. However, in what way is this easier?

She still had to add the eight and the three in the sum 28+3."

Because she's still broken the calculation down into steps. Speaking for myself personally, I find it easier to mentally add a single digit number to a 2-digit number rather than to try adding two 2-digit numbers together in one go (unless at least one of them is an "easy" number with a zero on the end LOL). Of course, some people don't have any trouble doing mental addition regardless of what the numbers are, in which case they don't need to use "aids" like this. It is (or should be) down to what suits a person's preference.

math was never a problem for me in school, but i have to say that helping my kids with their grade 5 and 7 homework at times has me stumped! often the question in my mind is 'why would you ever want to do it that way!'... on the other hand sometimes you never know which way of teaching a concept will finally click in someone's brain to have it all make sense...and then they can use that particular strategy in the future...the latest one that has me stumped is why they are teaching my kids to use a comma for a decimal point ...confusing or what?

There is easier way, and are useful tecniques to learn math , just read my posts https://hubpages.com/education/HOW-TO-LEARN-MATH-E

Great exposition. Your subtle observations are absolutely wonderful.

We all need to help our kids with their homework, especially when they are in grade school and they are still willing to listen to us!

I've always been very careful to dance around the issue of how I solve a problem compare to how the teacher might solve the same problem. Sometimes their strategy just makes no sense, but obviously I don't see the big picture. I'm not in the class on a daily basis.

(Bragging Alert)

My kid is in high school and still asks me to make up practice problems and worksheets to review for tests. It's awesome because I love math also.

I agree the methodoligy is confusing at first - but really, it's trying to teach kids a short-cut to what many of us often do instinctively. I personally prefer not to carry or borrow in my head, and will do this very procedure whenever I need to do math in my head. I have a math degree - and yet I still prefer this method to just doing the math outright.

My daughter is in 5th grade, and has the EXACT same text book. I didn't understand what her homework was asking her to do - so I googled "add or subtract mentally use compensation" and found your page. Funny thing is, once I saw the explanation, I knew exactly what she needed to do, and realized it's what I already do naturally. I just wasn't aware of how the teacher had presented it during school.

Try to explain in words how to tie a shoe - it's incredibly difficult to spell out, but yet our young children all learn how to do it. I showed my daughter how to do a few problems and she's off and running.

Nice hub, and an interesting situation. It seems that they're trying to encourage the kids to understand an easy method of adding numbers together in their heads. Rather than having to carry, which is better suited to written calculations. (I know some other people have said this too, but I agree!)

It is frustrating that mathematical methods seem to change so frequently, and they go in and out of vogue, just like the latest fashion styles! It's just confusing if you ask me. And I'm sure that it's not the case that the latest method is always 'better' than the previous one, as it will probably always be the case that some kids find one method easier and others will prefer a different one.

It would seem more practical to give the children a sum and let them work it out the way that they find easiest - but I suppose from a teaching point of view it's simpler and less time-consuming to stick to one method for everybody. And the same goes for marking papers. A load of different methods would also add time to this activity.

It's great that you take so much interest in your daughter's schooling. I'm particularly looking forward to helping with maths homework when my daughter's older.

i like this too!!Do you write for a News paper??? do you.....you should!

crazy888

yes well i remember when i was in fith grade.....and well we didnt realy have textbooks. now a days...schools have fundraisers that support the schools. Trust me. I have a seventh grader and she's bringing home HUGE life science textbooks. i think the seventh grade teachers need to cut her some slack with the rest of the kids. These days you cant go off to school unless you have 2 huge textbooks, your lunch and multiple studing notebooks! good hub...good read...!!!

Also..with the cell phones? My daughter cant get enough f texting all the time to her friends. she will write "OMG" a bunch of time but she does not realize that THAT cost money! teens these days......i wish my daughter was in fith grade.

thats a great idea. when im done with it....i will tell you and then i will have you read it. Its going to exlain a little bit more a bout me!

thanks.....alsmost done!

This is a great hub. Though my kids are grown, I remeber my oldest bringing home some 3rd grade math and having problems with it. I was trying to help him with it and he keep telling me I was doing it wrong! I don't remember exactly what the problem was but the method that the teacher was teaching them to use was way out there and I couldn't understand it. He was getting the wrong sum than what the actual sum was. It just blew my mind that "they" decided to change the way we add and subtract! they should have classes to keep us up to date with their teaching methods.

Aya Katz...have you read my bio yet???

yes you are absolutly right...i have to put more rules to the computer with my family. My number one rule..no chating....guess they never listened to that

Anyway...it just seems like my kids get whatever they want. Read my hub...it will give you an idea of who i am

Kids have the right and their own techniques of demanding things.. they can easily get whatever they want.. nice hub..

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