kitchen table math, the sequel: Top Ten List

Thursday, May 24, 2007

Top Ten List

I found a packet in my 5th grader's math binder about a week ago.

Remember, this is math class. 5th Grade Math Class.

Top Ten List of Problem Solving Strategies:
1. Act out or use objects
2. Make a picture or diagram
3. Use or make a table
4. Make an organized list
5. Guess and Check
6. Use or look for a pattern
7. Work backwards
8. Use logical reasoning
9. Make it simpler
10. Brainstorm

Kids use these "strategies" to solve word problems.

I don't even know where to begin with this. This is a Top Ten List? My first thought is, what strategy is left that is so lame brained it couldn't make this list?

Act it out? Presumably, this will help when you have to solve the question on 1/3 of a football team wears glasses, 1/2 of the glasses wearers are blond, if 4 kids are blond glasses wearers, how many kids are on the football team? Or something like that. Get all your blond classmates together and line them up.

No mention of using an algorithm. How about, decide if you are using multiplication/division/addition/or subtraction? That seems like a good starting point.

Brainstorm? guess and check?

This is not from EM. I have no idea where the teacher found this gem.


Catherine here, parachuting into Lynn's post.

I thought that list looked familiar. (hit refresh a couple of times if necessary)


Anonymous said...

8. Use logical reasoning


GoogleMaster said...

These might actually be useful:
2. Make a picture or diagram
3. Use or make a table
9. Make it simpler

But the list as a whole is ridiculous.

LynnG said...

I gotta think that a 5th grader looking at that would not find "use logical reasoning" to be particularly helpful.

If you are giving kids problems so difficult, or so beyond what they know how to solve, that they are reduced to acting out as their best hope, you are doing something wrong as a teacher.

SusanS said...

Yeah, where's "Apply any operations you've mastered?"
Oh, right. Well, maybe, "Use your calculator and apply any operations you've been somewhat exposed to."

This is where Singapore and Saxon differ mightily. After the operation is mastered, they don't look back to diagrams. Diagrams were used as a bridge to help with the logic early on. After that the push is for abstraction.

Saxon explains it out more, but Singapore does the same thing, starting with easy 2-steppers around book 2, I think.

None of the teachers I've dealt with ever treated word problems as a teachable thing. My SPED son just appeared with a huge 4-5 step problem (huge for him), but no real instructions or practice before on how to analyze it.

Luckily, we had practiced enough in Saxon that I could at least walk him through it.

If I had not been there and someone had bothered to check his answer, they would have concluded that he just couldn't do it.

Catherine Johnson said...

None of the teachers I've dealt with ever treated word problems as a teachable thing

It is just STAGGERING.

Two years of "accelerated" math & Christopher has barely done any word problems (except for the ones on tests, of course), let alone received instruction in how to set them up and solve.

Catherine Johnson said...

Several of these strategies come from Polya.

Anonymous said...


I'm just now learning that logic itself is a teachable thing. Who knew?

And actually I was planning on using illogical reasoning, so glad they left me a reminder on the list to use LOGICAL reasoning. Phew.

They should have thrown in a #11

11. Stare at the page until the correct answer comes to mind.

or these would be just as useful...

12. Resort to ESP

13. Use "the force"

14. Look it up online

BeckyC said...

Googlemaster correctly sums it up - there are only a few usefully named strategies.

For problems where you need to determine the order of a set of values whose locations are given relative to each other but not in absolute terms, or to follow a given walk through time and find the beginning or the ending point, it helps to draw a picture of the spatial situation (also known as a number line).

For classic rate problems, it helps to make a table of values going forward in time (by extending a given pattern and testing for a given end condition).

For classic mixture problems, it helps to make an organized list of all possible pairs of values that fit one given condition (then test to see which pair fits the other given condition).

Of course, there are times when a child can "do it in his head" or "see through to the answer." But lines and tables and lists are a trail of bread crumbs that will reliably lead him out of the woods, no matter what.

Of course, the word problems I see my kids do in K-5 are just classic rate and mixture problems from algebra, but with carefully chosen smallish positive integer values so that the table or the list does not get too big.

Guess and check is not useful - kids can learn to solve these junior rate and mixture problems by systematically developing tables and lists, and without using formal symbolic algebraic techniques. And the skill of developing a table of values will serve a child well when he progresses in mathematics and needs to plot the equation of a line (or two) by hand.

Singapore Primary teaches kids how to use bar models - one bar model for each linear equation - to do the linear transformations and substitutions and combinations. To solve two linear equations in two unknowns with smallish positive integers.

Independent George said...

Becky - I completely agree about the tables, charts, and graphs. The thing is, though, as you've explained they're really useful in specific circumstances. You can't just randomly chart a bunch of numbers and expect the answer magically appears in your head.

In other words, a student would need to already have an idea of how to solve the problem before he could adequately chart it. That list isn't strategy; it's desperate flailing.

LynnG said...

"work backwards" is a great strategy in an appropriate situation. I recall Catherine having a longish post on KTM on working backwards and how to teach it. Learning how to work backwards on anything other than the most simplistic of problems is not intuitive and can result in slamming into a few trees in the proverbial woods.

It isn't enough to tell a kid to "work backwards" and then consider the task taught.

Singapore Math has been a godsend for my kids. She is the only kid in the class that has been taught how to problem solve in a systematic way.

Doug Sundseth said...

Rather than piling more ridicule on that absurd list (it's not that it doesn't deserve it, but I'm just not in the mood), here are some strategies that might actually be useful:

1) Identify what sort of answer the problem needs. (If the question is of the form "What percentage of the students are blond", an answer of "7 students" cannot be correct.)

2) Identify the units in which a correct answer will be expressed.

3) Write down the equations that you know from the stated problem.

4) Compare what you now have with other problems you have done and work the problem.

5) Make sure the form of the answer you got matches what you identified about the form of a correct answer in steps 1 and 2.

6) Think about whether the answer makes sense. (If you've calculated that the sales tax on a $10,000 car is $6.5 million, you might have made a mistake somewhere.)

7) Recheck your answer by working the problem backwards.

Catherine Johnson said...

Hi Googlemaster!

Anonymous said...

Saxon explains it out more, but Singapore does the same thing, starting with easy 2-steppers around book 2, I think.

Singapore starts two-step word problems in the 2B. An example is this:
358 adults took part in a parade.
169 of them were women.
(a) How many men were there?
(b) How many more men than women were there?

Notice that they are building up to a "true" 2-parter by having the child explicitly solve the intermediate step.

Russian 1st grade math has "true" 2-parters, but Russian 1st graders are 7 years old.

-Mark R.

BeckyC said...

The specific circumstances I refer to are the problems in the 5th grade WASL practice set from Port Angeles Washington, that our school district uses to teach problem-solving in K-5. Based on LynnG's example, I think it's a common approach around the country.

It's giving kids algebra problems that will be later solved by symbolic means, before the kids have been taught the symbolic means.

I was interested to find that my boys aren't ready for basic symbolic manipulations of linear equations.

Although my boys understood the genesis of an expression like 3x + 6y = 18, directly translating from a given condition like "3 cars and 6 balls cost 18 dollars," when I tried to step them through a transformation like dividing both sides by 3... they couldn't fathom it. We can't skip the next year of pre-algebra. So writing down the equations is not a starting point for my boys, yet.

And when the homework is a mixture problem, and we can find the solution by making an organized list of all possible pairs of positive integer values that fit 3x + 6y = 18, and then test to see which pair fits the second given condition... it's not desperate flailing. It's teaching them to approach a specific set of problems in a systematic way.

I don't get to decide whether these WASL problems are the best use of instructional time at our school - I do have to get my kids through their homework in the most valuable way I can find.

Catherine Johnson said...

Singapore Math has been a godsend for my kids. She is the only kid in the class that has been taught how to problem solve in a systematic way.

The accelerated math kids have not been taught how to solve problems.


The first two years of the course are procedural.

I'm repeating myself, I realize.

I'm thinking that this summer I'm going to teach C. the classic algebra 1 word problems via direct instruction.

I'll probably pull from Saxon & the Mildred Johnson book.

I also have to teach C. percent, which doesn't seem to have been "covered" particularly.

Definitely have to teach him percent increase & percent decrease.

Saxon says you MUST use visuals to teach percent increase & decrease, which I assume is true for the bulk of students. So I'll use his oval drawings.

PaulaV said...

I am in the same boat regarding word problems or I should say the lack of word problems.

My third grader's teacher has said that the kids can't seem to get word problems. She mentioned to me that the third grade teachers are noticing that the kids want to get to the answer without trying to figure it out on their own. According to her, "the third grade team believes the kids did too many computational worksheets in second grade."

I have seen only a handful of word problems come home this year, and the year is almost over.

I know what we will be doing over the summer.

SusanS said...

This is why I love the Singapore word problem book with the bar models. I just wish they went further into the upper grades (although maybe they do. I haven't checked.)

You would think all of these curriculums would be crawling with word problems. Isn't that the "real world" application they keep talking about?