Problems in Fifth Grade Math
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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.
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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
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Scott.Life, thanks for your comment. You may be right!
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?
Teresa, thanks. I suspect some of the teachers would like to throw away the text books, but dare not for fear of losing their jobs!
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.
Jerilee, yes, today's books weigh about as much as the children, if not more. It's all so wasteful, as they are full of colorful pictures and very few words.
I am getting the math book and the workbooks that go with it, so that I can give Sword extra help -- sometimes just translating what the book says into ordinary language.
She is a good reader and speller, and she is always working on some creative project on her computer, so I am not worried.
In a way, it is good that they are not letting them do much creative writing in school, because that way this outlet will not be ruined as a fun thing to do!
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 :)
Nets, is that it? Is it always the first one that needs to be increased to the nearest multiple of ten? And always increased, never reduced? So this would mean that the process the student goes through will be different if the sum is written 28+23 or 23+28?
I gave Sword the problem 28+23 to do on scratch paper, and I told her that she needs to doctor one of the numbers so that it ends with zero. She said she understood, and she promptly reduced twenty-three to twenty and then added three to twenty-eight and got thirty-one, and added twenty and thirty-one and got fifty-one. That's one way to solve the problem, but if the purpose was avoiding adding the eight and the three, it was not achieved!
What is NCTM?
Storytellersrus, thanks! I'd be interested in learning more about the methods used for teaching your niece.
I don't know why teaching math has to change all the time. It's one thing to find a method that works better for a particular student. It's another to demand that all students use the same method. After all, if the student can come up with the right answer to the problem, does it matter how he did it?
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.
Misha, you look different today! I'm not sure if I know too much or not quite enough. Perhaps if I knew just a little less or just a little more it would make things easier.
Kartika Damon, thanks! I'm not sure it's counter-intuitive to everyone. The problem is that different people have different intuitions.
Opinion Duck, you are right. Following these instructions is a problem in reading, not math. It's all about reading between the lines!
Paraglider, tough problem with hens and eggs! We currently have three hens, and yesterday evening my daughter found five eggs. However, I suspect that the eggs were laid over a period of more than twenty-four hours.
Sufidreamer, I sometimes do the same thing when adding, but only when I feel like it! It comes and goes.
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.
Nets, thanks. I'll take a look!
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.
Mith_moral, thanks! It would be great to get comments from children who are dealing with this issue as well. Sometimes I think that my daughter should just take a survey of the children in her class on how they solve these problems!
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).
EmpressFelicity, 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 gave the example in a comment above that my daughter is capable of increasing or reducing one of the addends to the nearest full decimal and adjusting the other accordingly. She just has no idea which one! The instructions don't say.
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. I think that the step that was missing from the explanation is that you have to determine which of the addends is closer to the nearest whole decimal. Or, I could be mistaken, but some sort of explanation is still forthcoming about what is meant by "easier".
There is nothing wrong with requiring the text book writers to explain what the purpose of the exercise is to the students in clear, rigorous terms that are not subjective.
"Ease" is an emotional word whose meaning varies with the individual. Educators must learn to understand that not everyone has the same subjective experience with numbers.
"...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.
EmpressFelicity, well, in that case, we agree!
Unfortunately, in my daughter's class the students were required to use this particular method and tested on their ability to do so correctly. I don't have a problem with that, as long as the method is defined so that everyone has the opportunity to know what is expected!
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?
Sage Knowles, thanks for your comment.
A comma instead of a decimal point, eh? What country are you located in? I think they used to use a comma instead of a decimal point in Israel, when my parents were children...
I suppose it doesn't matter what you use to mark it, as long as you use it consistently.
There is easier way, and are useful tecniques to learn math , just read my posts http://hubpages.com/hub/HOW-TO-LEARN-MATH-EASILY
Esy Math, thanks for your comment and the link.
I did read a few of your math hubs, and they look like a good way to teach. The problem that this hub addresses is not actually what is the best way to teach math, but how to grapple with the methods being used in my daughter's school system!
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.
Nicomp, thanks! Are your kids still willing to listen?
I'm not in class on daily basis -- or at all, for that matter. However, I do get some idea of what goes on in class, from my daughter's descriptions.
(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.
Nicomp, that's great! Is the high school math curriculum any good where you are? Or is your kid way ahead of what they cover?
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.
Rainier828, I'm so glad this hub helped you and your daughter! Yes, we all do something like this in our heads sometimes, but there's a world of difference between discovering the method for yourself as a shortcut that works sometimes, and being told you have to use it, without a clear explanation of how it works.
BTW, how did you and your daughter solve the issue of which number should be doctored to the nearest multiple of ten? Did you explain that part, or just demonstrate?
Yes, it's true that some things are easier to demonstrate than to explain, like tying our shoelaces. However, I think math really is capable of being explained clearly, and we ought to hold textbook writers to a higher standard.
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.
Moon Daisy, thanks. You write: "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." I totally agree. It would be nice if the teachers encouraged students to find the method that works for them. But as you said, that might be more time-consuming.
i like this too!!Do you write for a News paper??? do you.....you should!
crazy888
Crazy888, thanks. Glad you liked it.
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.
Crazy888, it somehow surprises me that you have a child in 7th grade. I thought you were a college or high school student, from your comments on my other hub. Maybe you should write a short biography on your profile page to introduce yourself to Hubpages readers.
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!
Crazy888, I look forward to that!
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.
Ruby, thanks! It can be quite confusing trying to help with homework, if the textbook or written assignment sheet have not made clear what the goal is to be accomplished. There is more than one correct way to solve a math problem. It's frustrating, though, when we are not sure whether our children are being tested on their ability to solve the problem or on their ability to follow a set of directions in order to solve a problem. It's even worse if the directions are unclear.
Aya Katz...have you read my bio yet???
Yes, I have Crazy888. I think it's a really bad idea for you and your daughter to use the same online ID or handle or avatar. It could potentially lead to a lot of misunderstandings with strangers.
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
Crazy888, I will definitely read your hub!
Kids have the right and their own techniques of demanding things.. they can easily get whatever they want.. nice hub..
Vizey, thanks!
Scott.Life 2 years ago
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.