The Ontario Action Researcher - Archive

ACTION RESEARCH
IMPROVING CHILDREN’S KNOWLEDGE
OF MATH FACTS AND PROBLEM SOLVING SKILLS

Margaret Juneja, Vice-Principal, Grand Erie District School Board

The Question: How can I help make my class more efficient problem solvers while increasing their knowledge of the basic math facts?

My concern arose because my class seemed to be entering Grade three with little knowledge of even the basic facts of addition and subtraction. In other years, I felt that I had given the children lots of practice in solving word problems. I had given them daily problems to solve but was not as methodical as I should have been in my approach to helping the children improve. The results on the previous year’s EQAO tests showed that many of my class were not proficient in this regard. In the Math Knowledge/Skills Category, here is where the students placed in last year’s results:

	Level 1	Level 2	Level 3	Level 4
Problem Solving	5	6	2	0
Understanding Concepts	7	3	3	0
Application of Mathematical Procedures	5	6	2	0
Communication of Required Knowledge	7	5	1	0

Table 1.1 – Level Achievement on the EQAO exam by 13 Students in Math Knowledge/Skills Category

I decided to try a two-fold approach to help change the way that I taught Math. I started by making up sets of flash cards in the four fact areas and sent them away to be reproduced on card stock. I made enough so that my class could work in pairs. Every day they would drill each other on the stack of cards and then we had a three-minute quiz. These became harder as time went on. The marks were recorded every day so that improvements could be charted.

My other strategy was to directly teach all of the twelve problem solving strategies (see appendix). I originally hoped to teach two methods a week but soon realized that this was too much. I settled down to teach one a week, each one involving several steps. First, I modeled the method. I went over how it was done and asked questions as I went along. The class copied this first example into a separate problem solving notebook. The second time, the children worked on a similar problem in pairs. All of their work went into the problem solving notebook. The solutions were then reviewed and evaluated. In other words, students were told whether their answer would be considered a level four, three, two or one. At first, the children wrote simple, incomplete solutions. Gradually, however, many of them were able to write quite complex answers. I continually reinforced the ideas of using pictures, numbers and words to show solutions. I also encouraged the use of precise math language to communicate their ideas. Exceptionally good solutions were praised in class. By the third time, the children were working on the problem individually. All solutions were graded and verbal feedback was given in all cases in order to help the students improve their answers. I always tried to point out what the child was doing that was correct, as well as areas in which they needed to improve in order to give a better response. On several occasions the problems were sent home so that families could work on them. Even the parents said that they had difficulty with some of them!

I would ask the children how long they took to work out the problem. Often, an evening would be spent working on them. One parent commented that their child had great difficulty with the problem but that they persisted until the child was able to understand. Another mother, who happened to be in the school volunteering, popped her head in to ask if the solution the family had worked out was correct. I even heard comments from other teachers who had siblings of my students in their classes as to how families had worked on the problems together. One child, in a spontaneous comment, told me that the problems were getting easier. Another child, after learning a few of the strategies, actually used an earlier learned strategy to solve that day’s problem. I made sure to point that out to the class to show them that there were many different ways to solve problems. As long as the solution was correct, the method may differ and still be acceptable.

I made a point to tell parents on Interview Day that I had undertaken this project in an effort to improve the children’s performance in Math. Most parents were very encouraging and said that they were willing to help their children learn the math facts at home in order to increase their competency. However, I did initially receive a few negative comments. One set of parents came in demanding to know if there was a Grade Three Curriculum and if it was being taught. They had viewed the previous year’s test results and expressed great concern over them. I explained what I was going to do about it and they left the interview in a better frame of mind. There was also one negative comment that came back on page three of the first report card. The parent had written that “given the poor results of the school last year” she hoped that her son would be ready to write the Grade Three test come May.

By the end of December, I had taught six of the twelve problem strategies and had drilled the addition facts to eighteen and had almost finished with the subtraction facts. I had kept track of the results of the daily quizzes and most of the children had significantly improved in their recall of the facts taught so far. I had also tried using corrective feedback with my class as the result of giving them a test on subtraction and then going over the questions, one by one, asking individually if they needed more clarification, and then giving them another test a few days later. Most of the children did significantly better the second time around. I had told the children that I would keep the higher mark.

By the end of January, I had started teaching multiplication, of which few of the children had any knowledge. We continued with the same pattern: daily flash card drill with a partner followed by a three minute, fifty question quiz. By examining my formative results to the daily quizzes, I could see that many of the children were improving in their recall of the facts.

By the end of March, I had finished teaching and testing the multiplication facts and was starting division. While in other years, I had found that the children struggled with division, this year I was pleasantly surprised! This class quickly understood the concept, realized that the division is the opposite of multiplication, and continued to do well on the daily quizzes. I am lead to believe that their earlier competency in their knowledge of the facts significantly aided in their understanding of the relationship between multiplication and division, and inevitably, the correct answers. I had also taught most of the problem solving strategies by this time. As I observed the children work out their problems, I was amazed to hear the math language that they used and that they were usually on task. They took great pride in being able to come up with the answer and were eager to show me their written responses. I continually stressed that they must communicate not only how they solved the problem but why they tried a certain procedure.

Before the provincial testing in May of that year, I did not really have any hard data from an outside source to show that my method of teaching math had been effective in increasing the students’ math scores. I did know however, that my class’s attitude towards math had certainly improved. On the whole, they were eager learners, always willing to try and displayed both confidence and competence in their approach to math. For myself, I had become excited about this way of teaching math problem solving skills. I could see the results clearly in the written responses that the children gave. This enabled me to focus more clearly on the process that the children used rather than the mere product. I gave much more corrective feedback, both oral and written. Seeing a child smile when you tell them what a great answer they have given is certainly a wonderful reward. My assessment methods changed in that I made sure to build in a time for the children to have another chance to try an assignment, after corrective feedback had been given.

Even though I had moved to another school this past fall, I contacted my former principal and asked if I was able to access my former class’s results to see if there was concrete data to support the teaching methods that I had adopted. The results, shown in the following table, emphasizes to me how important it is to examine one’s own practice in order to improve student learning – the results can be quite dramatic.

Please note - following the table below are some examples of the problems the students were asked to solve, a list of resources, and some samples of the children’s work.

	Level 1		Level 2		Level 3		Level 4
	1998-1999	1999-2000	1998-1999	1999-2000	1998-1999	1999-2000	1998-1999	1999-2000
Problem Solving Skills*	36%	0%	43%	10%	21%	86%	0%	4%
Understanding of Concepts *	50%	0%	29%	29%	21%	71%	0%	0%
Application of Mathematical Procedures*	43%	0%	43%	14%	14%	81%	0%	5%
Communication of Required Knowledge*	57%	10%	36%	19%	7%	71%	0%	0%
Overall Level of Achievement*	21%	0%	50%	10%	29%	71%	0%	19%
Mean Performance of the Class*	42%	2%	40%	16%	18%	76%	0%	6%

Table 1.2 – Class percentage of Level Achievement in Math Knowledge/Skills Category on the EQAO tests 1998-2000

*1998-1999 = 13 students
1999-2000 = 21 students

APPENDIX

PROBLEM SOLVING STRATEGIES

Look for a pattern
Construct a table or chart
Make an organized list
Act it out
Draw a picture or diagram
Use objects
Guess and check
Work backwards
Write an equation
Make a model
Logical reasoning
Solve a simpler or similar problem

LOOK FOR A PATTERN

1. As you are walking down a street, you notice that the houses are numbered in the following manner: 304, 312, 320.
What would the next five numbers be?
Write two patterns you notice about these five numbers.

2..Explain what is being done in this pattern sequence. Then figure out what the first three numbers would be.
____ _____ _____ 8 16 18 36 38 76 78

CONSTRUCT A TABLE OR CHART

1. There were 12 people at a party when they started to leave. In the first hour, 2 people left. In the second hour, 4 people left. People continued to leave. Every hour 2 more people left. In what hour were there no more people at the party?

2. While walking to school on Monday, Tim saw five birds. He saw ten birds on the way back. The next day, he saw seven birds on the way to school and fourteen on the way back. The following day, he saw nine birds on the way to school and eighteen on the way back. If this pattern continues, how many birds will Tim see on each of the next five days. Use a T-chart to show your answers.

MAKE AN ORGANIZED LIST

1. Matt has a $1.00. He goes to the store where erasers are 10 cents, pencils are 50 cents and pens are 20 cents. How many different combinations of school supplies can he buy?

2. Jane is playing darts. She could land on 10, 20 or 30 points. She has three tries to get a good score. She got a score of 30. How many possible combinations could she have got to get that score? ( 0+0+30 is different from 0+30+0)

ACT IT OUT

1. 23 children form relay teams. Each team must have five children. How many children cannot join a team?

2. Ivan has a secret. He whispers it to 3 classmates. Each classmate tells 3 others. How many know the secret?

DRAW A PICTURE OR DIAGRAM

1. How can 16 members of a marching band form a square?

2. 17 people sit around a campfire. Then 6 people join them. 8 people leave. Later 10 more people leave. How many are left around the campfire?

USE OBJECTS

1. “Attention children,” said the teacher. “Today Harry Hamster is going to talk about his trip.” Sitting in front of Harry was Gertie Goose. She was to the right of Ronnie Rabbit. Freddy Fox was between Gertie and Tommy Turtle. Willie Wallaby was seated beside Freddy and in front of Mickey Mongoose. How are the children arranged in the classroom?

2. Five children catch 28 fish. They share them equally. How many fish are left over to give away?

GUESS AND CHECK

1. 26 27 28 29 30 31 32 33 34 35 36 37
I am thinking of two numbers that are next to each other. Their sum is 73. What are the two numbers?

2. Sharon bought stickers that cost 5 cents, 4 cents and 2 cents. She paid 30 cents for 10 stickers. How many of each did she buy?
12

WORK BACKWARDS

1. Some children left the classroom. In the second minute, 2 more children left than had gone in the first minute. Every minute, 2 more children left than had gone in the minute before. In the fifth minute, 16 children left. How many children in all left the classroom?

2. Campbellville is famous for its soup. One day the recipe disappeared. The Queen ordered the cooks to look for the recipe. They did not return. On the second day she sent out four more cooks than she had sent out the first day. Each day the Queen kept sending out four more cooks that she had sent out the day before. Nineteen cooks went looking on the fifth day. How many cooks in all did the Queen send looking for the recipe?

WRITE AN EQUATION

1. Your teacher has put you in charge of organizing transportation for your field trip to the aquarium. You are going to travel by car. You have 25 students in your class and 5 cars to put them in. How many students will be in each car? Show the answer in four ways: by adding, subtracting, multiplying and dividing.

2. Who has more cards? Tom or Tim?

Tom began with 12, traded 4 away, but got 5 in trade, bought 7 at the store, got 11 for Christmas and gave away 6 to a friend.

Tim started with 45 cards, gave away 24 to a friend, bout 4 more, traded away 6 good cards for 10 plain ones, lost 2 cards and gave 3 to his sister.

MAKE A MODEL

1. Place 8 pawns on a chess board so that no two pawns lie on the same row, column or diagonal.

LOGICAL REASONING

1. George the Dinosaur is 15 years old. His father is 59 years old. How old was George’s father when George was 4?

2. When the brothers Tom and Tim were asked their ages, they replied with a riddle. The riddle had three parts.

Tom is 25 years older than Tim. Tim is 25 years younger than Tom. Tom’s age is even, Tim’s age is odd. Add their ages together and get 63. How old is each brother? 13

SOLVE A SIMPLER OR SIMILAR PROBLEM

1. Three children are on the way to school. They find 50 cents and turn it into the office. After a week nobody claims it so they are told to keep it. How are they going to divide the money? (Because 50 cents is not easily divisible by 3, have the children solve dividing 30 cents by 3 to see the strategy and then doing 50 cents.)

2. How many chickens and how many ducks are there if you have 42 legs altogether? (First have the children try smaller numbers such as 8 or 10 legs. Once they have the idea, then use the larger number. This problem will have more than one solution.)

LIST OF MATH RESOURCES

Text Books

Quest 2000, Exploring Mathematics, Publisher: Addison Wesley Journeys in Mathematics, Publisher: Ginn

Resources

Daily Problems and Weekly Puzzlers, Publisher: Ideal The Problem Solver, Publisher: Creative Publications Linking Assessment and Instruction in Mathematics, Publisher: Ontario Association for Mathematics Education

STUDENT SAMPLE