Chapter 15

Finding the Child-learning Methods that Lead

Students toward Critical Thinking

If students don’t understand how to solve problems that require critical thinking, should we give up? Or, can we discover or create methods that lead students to the answers? Can we find child-appropriate learning methods?

Child-learning Methods are the topic of much academic discussion. But let me share with you my own life experience. When I was eight years old, my teacher asked me to answer a well-known critical thinking problem involving chickens and rabbits. I didn’t know how to solve the problem.

Perhaps I was not smart enough to solve the problem at that time. After I found a better problem-solving method to teach my daughter, Gina (when she was 5), I proved that I was wrong.

Problem

After attending Gina’s birthday party, Sonia’s mom asked Sonia what she saw at Gina’s house. Sonia said that there are some chickens and rabbits in Gina’s backyard. Mom asked Sonia how many chickens there are and how many rabbits there are. Sonia forgot the number of chickens and rabbits, but she remembered that there were 9 heads and 28 feet. How can Mom help Sonia to find out how many chickens and rabbits there are?

This is the way I learned to solve the problem:

The number of rabbits: (28 – 9 x 2) ¸ 2 = 5

The number of chickens: 9 – 5 = 4

179

The above method and answer are both correct. But even if the expression, (28 – 9 x 2) ¸ 2, is broken down into three separate steps (as below), it is still hard for students to understand.

To find the number of rabbits:

1^st step: 9 x 2 = 18

2^nd step: 28 – 18 = 10

3^rd step: 10 ¸ 2 = 5 (rabbits)

To find the number of chickens:

4^th step: 9 – 5 = 4 (chickens)

The problem is not an easy problem for 4^th, 5^th, or 6^th graders to solve. So I asked myself, “Can I teach my 5 year old daughter, Gina, to solve it, because it is an interesting problem?”

Diagnostics

The method I learned was correct, as noted above. But the problem was too hard to understand when I was 8. The problem has only two clues. The first clue is 9 heads. The second clue is 28 feet. The problem didn’t tell me how many chicken heads, rabbit heads, chicken feet or rabbit feet there were. That made the problem more challenging.

To solve the problem, we must know that a rabbit has four feet and a chicken has two feet. Based on that knowledge and our clues, let’s evaluate the three steps that I learned. I will explain the procedure in detail.

In the 1^st step, since each animal has at least two feet, we can write 9 x 2 (9 being the total number of animals, since each animal obviously has one head).

In the 2^nd step, since a chicken has two feet, the remaining feet are rabbits’ feet. To find the number of remaining feet, we solve 28 – 9 x 2. The result is 10.

In the 3^rd step, since each rabbit has two feet not yet accounted for, we divide the remaining feet by 2, to obtain the number of rabbits. We can write (28 – 9 x 2) ¸ 2, or 10 ¸ 2. The result is 5.

In the 4^th step, since we know the number of rabbits and the total number of animals, the number of chickens is 9 – 5, which equals 4.

After reading my explanation, how do you feel? Is it complicated to you, an adult? Do you think that 3^rd, 4^th, 5^th, or 6^th graders will be able to understand the method? Because it is a very challenging problem to many adults, it may be too challenging for 3^rd to 6^th grade students. Is there a more child-appropriate way to solve this problem?

Solution

Of course, there is always more than one way to solve any problem, but our goal is to find a method useful at the elementary school level. I think that I found the best way to teach this problem six years ago.

When people ask me how I found a Child-learning Method to solve this problem, my answer is two words: “by heart.” And this is a lesson to all parents: as an outgrowth of your love for your child, you should challenge yourself to teach your child as much as you can, and find creative ways to teach abstract concepts.

To teach my 5 year-old daughter, I developed a six-step method to solve the problem, one that is easier (that is, more child-appropriate) than the adult-oriented method above.

1^st step: I asked Gina to draw 9 circles, since there are 9 heads (animals). Each circle represents a head.

2^nd step: I asked Gina to draw two “feet” on each circle, since each animal has at least two feet.

3^rd step: I asked her to count the number of feet she drew. Since she drew two feet for each head, she told me that she used 18 feet (that is, 9 x 2 = 18).

4^th step: I asked her how many feet remained. She answered that there were 10 feet left (that is, 28 – 18 = 10).

5^th step: This is a very interesting part. I asked Gina whose feet are the remaining feet. She thought about it for a second and told me, “The remaining feet belong to rabbits.”

I continued to question her, to see if she could distribute the remaining feet to rabbits. Guess what her response was? Instead of answering my question, she asked me which of the heads were rabbits’.

I thought that I should give her a hint. “Can we assume that any of the heads could be rabbits’ heads?” I was surprised when she said yes. I realized that we must be very careful not to miss small detail links while teaching.

After she drew two feet on one of the heads, I asked her what she should do next. She said that she should draw feet on more heads.

I asked her how many more feet she needed to draw. She said that she needed to draw 8 more feet. (I was thinking to myself, “How would I answer her if she said, ‘I don’t know’?” I probably would have given her a hint, such as, “How many feet are remaining?”) She drew 8 feet, two each on four of the remaining heads.

6^th step: I asked Gina to count how many heads have four feet and asked what type of animal that is. She counted and told me, “There are 5 heads with 4 feet and they are rabbits.” Then I asked her how many chickens there are. She answered, “4 chickens.” What would I have done, if she hadn’t known what type of animal had 4 feet? I would have given her a hint by asking, “Which animal has 4 feet?”

Let’s compare the way that I learned and the way that Gina learned. It should be easy to tell which is the Child-learning Method.

The way I learned, at age 8, not fully comprehending:

1^st step: 9 x 2 = 18

2^nd step: 28 – 18 = 10

3^rd step: 10 ¸ 2 = 5 (rabbits)

4^th step: 9 – 5 = 4 (chickens)

The way Gina learned, at age 5, with full comprehension:

1^st step: Draw 9 circles to represent 9 heads.

2^nd step: Draw two feet on each circle.

3^rd step: Count how many feet have been on the

circles: 9 x 2 = 18

4^th step: Find the remaining feet. 28 – 18 = 10

5^th step: Distribute the remaining feet on some of the

circles (two feet per circle): 10 ¸ 2 = 5

6^th step: Count how many heads have 4 feet, (the

rabbits), and how many heads have 2 feet (the

chickens).

A very complicated problem is solved using a method that a child of 5 can understand. Isn’t it wonderful to have a method that reaches the elementary school-age child? Isn’t it fun to challenge ourselves?

Questions to Ask Ourselves: the diagnostic list of questions below will provide us with direction to help our students or children to solve the word problems.

(1) Do we give them problems that are interesting and appropriate for the age group we are teaching?

(2) Do they like the problems?

(3) Why do they like (or not like) the problems?

(4) Does the problem challenge them?

(5) Why are the problems challenging?

(6) Do they have difficulty solving the problems?

(7) Why do they have difficulty solving the word problems?

(8) What types of word problems do they have trouble with?

(9) Why are certain types of word problems difficult for them to solve?

(10) Do they understand the problems?

(11) What strategy do they use to solve the problem?

(12) Do they have difficulty developing or implementing the strategy?

(13) What steps in the problem-solving procedure do the students have trouble with?

(14) When children have difficulties, do they lose their interest?

(15) Can we provide necessary hints and helps to foster their interest?

(16) Do we give them enough time to think?

(17) Do we foster the logical thinking students should use to solve the problems?

(18) Do we provide necessary tools for them to solve the problems?

(19) Do we ask them to use or find different methods to solve a problem?

(20) Do we ask them to compare different ways to solve the problem?

(21) Do we ask if they can use the same strategy to solve different problems?

(22) Once they master certain problems, what level of problems should they be given to solve next?