Saturday, February 14, 2009

Ideas on helping children with hard word problems

As I work with my daughter through Singapore Math word problems, I try to keep track of the more successful strategies I've used to help her when she's stumped. Of course, I'm not a certified teacher, so I wonder if any of these are ever covered in Education School math methods classes?

In case they aren't, I thought I'd share a couple here.

1. In some cases, a combination of large numbers and multiple steps may intimidate the child enough that he or she doesn't know where to start. If this is the case, have your child begin with a parallel problem that involves small, easy numbers.

2. Sometime so much is going on in a problem that the child doesn't know which operation to perform when. In this case, charts may be helpful.

Let's see how this works with a given exercise, namely, a word problem from the end of Primary Mathematics 3B:

Isabella cut a rope 414 m long into pieces. Each piece was 9 m long. She divided each piece into four equal parts. How many parts are there?
Using Strategy 1, we might have our child first do the following problem:
Isabella cut a rope 20 inches long into pieces. Each piece was 4 inches long. She divided each piece into two equal parts. How many parts are there?
Using Strategy 2, we first ask:

What are the units (denominations) in this problem?
Answer: a 414-meter rope; 9-meter rope pieces; "parts" of the 9-meter rope.

We then have our child make a chart, heading each column with one of the units:

414-meter rope | 9-meter rope pieces | parts of 9-meter rope pieces|
------------------------------------------------------------------------------------
________________________________________________________

We then ask what quantities go in each column. We have our child fill in the known quantities, leaving the unknown quantities blank or "?":

414-meter rope | 9-meter rope pieces | parts of 9-meter rope pieces|
------------------------------------------------------------------------------------
____1______________?__________________?__________________

It should then be transparent that the first step is to fill in column 2, and that this involves dividing 9 into 414. We then get:

414-meter rope | 9-meter rope pieces | parts of 9-meter rope pieces|
------------------------------------------------------------------------------------
____1_____________46_____________________________________

Going from the middle column to the right-hand column is less transparent, so we reread the word problem to see how "parts" and "pieces" are related, and if necessary, ask hypothetical questions involving low numbers (if there were one "piece," how many "parts" would there be?) and record the answers in a second chart, until the pattern is established and we return to the unknown:

pieces | parts
----------------------
__1_____4_______
__2_____8_______
...

We then ask which unit we know the value for, and have our child fill in this value in the appropriate column:

pieces | parts
--------------------
__1_____4________
__2______8________
...
__46______________

It should then be transparent how to find the remaining unknown and solve the problem.

9 comments:

Jackie Ballarini said...

You may want to read PĆ³lya. His problem solving strategies are listed here.

Sadly, I don't think it was a math methods course in which I first learned about them.

Lori said...

These two approaches are taught and reinforced in Everyday Math. I have to agree with Jackie, I did not learn them in a math methods course.

EM starts with the use of manipulatives, adding and reinforcing the use of diagrams and using key words, breaking down the problem into manageable parts, weeding out any unnecessary information, and concludes with checking to see if the answer makes sense. While I am disappointed in some aspects of EM, such as lack of repetition of basic skills, I am pleased with the number story component.

By the way, your child is lucky to have a dad like you!

Anonymous said...

Looking for key words is not a good idea. Things are not necessarily going to be written with key words meaning the same thing.

There are 15 candies altogether. Two are outside the jar. How many are in the jar?

Students taught to look for the "key" word altogether might add. What makes them key words anyway? the textbook writer?

bky said...

Anonymous has a point (although I didn't see any reference to key words in the original post). Someone in math ed once gave me this example: There were 5 apples on the table. Bob brought in 3 more. How many were left on the table? Some curricula teach kids that "how many left" means subtract, so they are going to say "2."

The strategy of working with simpler numbers to figure out the structure of the problem is a very important idea. With my 4th and 5th graders in homeschool, they still are often stumped by problems only because of numbers like 21.75, 47.83 and so on ... what is going on is that they still do not abstract a problem the way an adult does. A more experienced problem solver reads "Joe has a certain amount more than Bob ..." rather than "47.83 more ..." until he thinks he needs the particular numbers. Kids get hung up on how hard or easy the actual computations are before they even know what computations they need to do. If I ask them to think about the same problem with easier numbers (20, 6, etc) they often see what to do right away.

Another thing that helps with this is to take simple problems, replace the numbers with "a certain amount" and "how much he has" and have the child walk through the steps. It is helpful even to have very simple problems (fruit on table) and have them identify what operation they would need to do on the given pair of numbers to get the answer.

lgm said...

I would use 'draw a picture' on those.
Here's a short summary of the elementary aged strategies used in the text our district uses (from Polya):

http://www.eduplace.com/state/pdf/hmm/05/efficacy/g23552_hmm05_p30-34.pdf

Lori said...

I have been thinking about the key work thing. I don't think it can be discounted completely. We use it as one tool in our toolbox. This is also where the "checking to see if you answer makes sense" strategy kicks in.

Home schooling has always intrigued me, and I am not sure if home school students are required to take standardized tests, but the key words so show up in many of the word problems on the PA State Tests.

Anonymous said...

If a student understands the problem and what it is asking, "key" words are irrelevant. If a student is dependent on key words, and cannot solve the problem correctly without them, then any time in life when a problem is worded without those so-called key words, or the words are used differently, as in the examples above, that student will not be able to solve the problem. It is an ineffective "tool". Better to give students tools they can use in all circumstances.

bky said...

Lori -- For almost any question about homeschooling,the answer depends on which state you are in. I live in Michigan, where there is practically no oversight on homeschooling. In some states parents have to get some kind of school-district approval of their curriculum, so I hear from folks who have lived in, say, California. At one point I thought about having my kids take the state NCLB-related test but then decided not to, since I knew that our scope and sequence was strongly out of sync with the public schools, so there seemed to be little point.

I think that Anonymous' last remark about keywords is on the money: if kids know to read a problem and get the mathematical content, keywords are irrelevant. Therefore the goal should be to get kids to read for understanding -- it is as much about literacy as about arithmetic. It is also not something that can be done in one lesson, it needs to be the ongoing framework in which word problems are addressed.

By the way, one thing I love about Singapore math is that early on, I think from the first grade, they include word problems that invite misreading, along the lines of the "how many bananas were left" example from above. These aren't trick questions, unless the student approaches them like tricks (which is what the keyword approach leads to), but they are questions that you simply have to take at face value and not load in other assumptions about what words mean.

Lori said...

Wow! I have to say, I am really liking what I am hearing about Singapore Math! Thank you for the fantastic conversation!