teaching problem solving to second graders, part 2

(Part 1 is here.)

Now the students are ready to start solving simple comparison problems with variables.

First the students are taught how to translate a phrase like "T is less than H" or "R is more than W" onto a number family.

The lesson might be taught like this:

Sometimes we refer to a number without telling which number it is. We can call that number J or B or any other letter. Here is a sentence that tells about two numbers: J is less than M.

We don't know which numbers J and M are, but we can put those numbers in a number family. J is less than M. So J is the small number. M is the bug number.

The big number goes at the end of the arrow. The small number goes close to the big number. Here's how you write it.


Then an example using "more," like "R is more than W," is taught.

After the students are firm on this skill, they are ready to tackle a translation like "J is 18 more/larger than K."

This must be a difficult skill because in CMC they scaffold the instruction by circling the number which tells how much more. Like this:


Eventually the scaffolding is faded.

To translate the problem, the students are instructed to ignore the circled number. This makes the problem identical to one they know how to translate -- "J is larger than K." They know that this can be translated into:


Then they are taught to place the circled number, 18, into the only available spot in the number family. Like this:


Once the students are firm on this skill, they can be given a problem that they know how to solve like "F is 12 more than 56." Now they should be able to translate this to a number family and solve for F.

Then the problem can be made more difficult by specifying two variables and the value of one of the variables, such as "R is 250 more than P. R is 881. What number is P?"

Finally, the students are ready to start solving real word problems like "Fran was 14 years older than Ann. Ann was 13 years old. How many years old was Fran?"

Here's how they are taught how to solve these kinds of problems:


This is a good stopping point. This represents a month worth of instructional time for the lessons and the practice. That's for lower performers, higher performers can probably learn this in about a week. Bear in mind that the students are learning and practicing about 10 other strands of material while all this is going on.

The value of this problem solving technique (like Singapore Math's bar graphs) is twofold. First, it reduces solving math problems to a systematic process; this will clarify the student's thought process. Second, the use of the written number families frees up the novice student's working memory which is taxed heavily in solving word problems. Given enough practice, these skills will become automatic for the student and lodged in long term memory. When this occurs, the burden on the student's working memory is becomes much less and the need for the number family prompt is diminished.

Teaser for next lesson: The students learn how to solve word problems like "Jerry weighed 72 pounds. Terry weighed 94 pounds. How much heavier is Terry than Jerry?" To solve problems like this, the students are taught the concept of moving forward and backward along the number family line.

teaching problem solving to second graders

Jill ran 2/3 of a mile farther than Steve. If Steve ran 7/3 miles, how far did Jill run?

If the NAEP is any indication, this is a simple problem that many students can't reliably solve by the 11th grade. Which is a real shame because if a student can't solve a simple problem like this, he can't do basic algebra. The student's math education has effectively come to an end.

The biggest stumbling block is translating the word problem into a mathematical expression. (Calculators are no assistance here.) This kind of mathematical reasoning eludes many students. Fortunately, it can be systematically taught. For example, in Singapore Math this skill is taught using bar graphs starting in third grade. A fair amount of digital ink has been spilled on bar graphs on KTM, so I'm going to show you a diferent way of teaching problem solving.

I'm going to show you how the technique is taught in Connecting Math Concepts (CMC) beginning in the second grade. By the end of the second grade, students should be able to solve a problem, like the one above, correctly at a high rate. Problem solving is taught the entire 2nd grade year in CMC, so it's going to take quite a few posts to cover it all. So let's intoduce the technique in this post and I'll periodically write new posts until we've covered it all.

In CMC, simple problem solving is taught via the concept of number families. Here's a number family:



Beginning in this first grade, the student is taught that number families show three numbers that always go together in addition and subtraction facts. In the example, the three numbers in the family are 2, 3 and 5. You can derive four problems from each number family, two addition and one subtraction:

• 2 + 3 = 5
• 3 + 2 = 5
• 5 - 3=2
• 5 - 2 = 3
  

The students are taught that the "big number" always goes at the end of the arrow and the "small numbers" always go above the line. (In DI courses skills are taught using terminology that the students are familiar with. Proper math terminology like subtrahend, minuend, and addend aren't taught, if at all, until after the students are firm on the skill. This reduces the liklihood of student confusion.)

Next the student is taught how to derive the addition and subtraction problems from the number families. Here's an example of each:



An addition problem can be written for each family that has a missing big number, like the bottom family in the picture. Students are taught that if the big number is missing, they are to write an addition problem that ends with the "how many" box (4 + 19 = []). For subtraction problems, students are taught that if one of the small numbers is missing, they are to write the big number first and subtract the small number from it to find the missing number (57 - 12 = []).

Once the students are firm on this skill, they are given some math puzzles to solve. For example, the students are directed to complete the number family, write the addition or subtraction problem, and the answer to the following set of facts: The big number is a box, the first small number is 38, and the second small number is 39.

the student should be able to derive the problem: 38 + 39 = 77.

Now the student is ready to learn about the concept of variables.

The student is told that sometimes a "letter" is used instead of a box in a number family. The letter works just like a box. It's the missing number.

Here's a problem:

The first small number is 14. The second small number is 56. The big number is P.

The student should be able to construct the proper number family using the skills he's been taught so far.


The student should also know that in order to solve for P, he has to add. The student should also be able to write the correct addition problem 14 + 56 = P and determine that P = 70.

The student is then instructed to cross out the P in the number family and write 70 like this:



This seems like a good enough place to stop for this post. Don't want to overload your second grade heads. This sequence takes about five weeks to go through--the first five weeks of second grade, including practice. I'd estimate that this sequence represents about an hour or two of instruction time and another few hours of guided and independent practice.

In the next post we start to get out of the puzzles and into the good stuff -- real problem solving.

Many of you can probably see where we're going with this already.

Here's a teaser.

A student should be able to set up simple comparison like "A is less than B" or "G is more than H" just by using the number families and rules for placing the "big number" and the "small number." Once that skill is firm it's just a hop, skip, and a jump away from setting up a problem like: J is 5 less than K. Solve for K if J = 3.

(Go to Part 2)