Lesson 7: Problem-Solving

from Teaching math in the intermediate grades

Lesson 2: Mathematics Curriculum in the Intermediate Grades

O b j e c t i v e

Plan a lesson that uses a problem-solving strategy

Introduction

Not all word problems promote problem-solving skills. In this lesson, you will let the characteristics of a good word problem, when it is best to give a word problem promote problem-solving skills, and how to process the students' varied solutions.

Thin k

The problem-solving strategy involves students being challenged to collaborative solve real-world math problems that they have not yet previously encountered. It student-centered and promotes critical and creative-thinking skills, problem-solvin abilities, and communication skills. The integral part of this strategy is the time givei to the students to struggle with the problem, and its beauty is in the varied solution that the students would produce.

There are three main elements of problem-solving that you should take note of; (1) the word problem, (2) the time given for the students to struggle with the problem, and (3) the mathematical discourse that happens during the struggle and during the processing of the student-generated solutions.

The Word Problem

In many Filipino classrooms, word problems are given at the end of the lesson and the students are expected to answer them by applying the concept or skills that have just been taught to them. In most cases, the teacher first demonstrates how to solve a problem, and then the students would independently answer a similarly structured problem. In this practice, the students are not doing problem-solving— they already know how to solve the problem! They know that the just-taught lesson is the key to solve the problem and they pattern their solutions to what the teacher has demonstrated. In using the problem-solving strategy, the problem serves as the starting point of the learning experience. Therefore, it is given at the beginning of the lesson. The challenge for you, the teacher, is to choose or create a problem that can be

How you present the problem also matters especially for the elementary grades. : is not always helpful to introduce the problem by posting it on the board; doing s may intimidate some of the students and reading and comprehension skills may itervene. Instead, it is suggested to narrate the problem in a storytelling manner to e-igage the learners. Encourage the students to imagine the scenario and allow them :o clarify information if they find some details confusing. Showing drawings or real -bjects might help.

The Time Given to Struggle With the Problem

The goal is for the students to collaborate—share their ideas with each other— to come up with a solution. Encourage the students to use their previously learned knowledge and skills to solve the problem and to communicate their ideas with their classmates through words, equations, and/or illustrations. It is natural for the students to find this phase burdensome especially when it is their first time to engage in such an activity; critical thinking and communicating ideas are not easy tasks after all. So,

t it is the task of the teacher to encourage the students to think out of the box. Tell the students that there is more than one way to solve the problem, so they do not need to worry about their solution being wrong as long as every step they did is meaningful to solving the problem.

The Mathematical Discourse

This is the most exciting element of the problem-solving strategy. While the students are working in small groups to solve the problem, you get to move around and enjoy the mathematical talk that the students are engaging in. Of course, you may intervene in the students' discussions when corrections and clarifications are needed but be careful not to give hints. It may be tempting to do so especially when the students are struggling but don't. As you encourage your students to think, believe that they actually can. Allow yourself to be amazed at how the students would defend their thinking, correct each other's ideas, and figure things out on their own.

Remember that all the student-generated solutions, as long as correct, can be directed to the concept or skill that is the objective of the lesson. The challenge is how you would process those various solutions, make sense of each of them, and use them to generalize or come up with a solution that makes use of the knowledge/skill that is the objective of the lesson. In this phase comes the importance of the teacher's fluency of the subject matter.

Study the following lesson plan. Take note that the plan only shows development of the lesson that involves the problem-solving strategy; other parts are not included.

Topic: Division of fractions

Grade level: 5

Target learning competency: By the end of the lesson, the learners will be able to divide a whole number by a fraction.

Prerequisite knowledge and skills:

1. Fraction as part of a whole

2. Fraction as repeated subtraction

3. Division of decimals

4. Multiplication of fractions

5. Reciprocal

Presentation of the problem:

I have 6 liters o f milk. I will transfer the milk in to glasses. Each glass can hold | liter. How many glasses can I fill?

Present the problem above in a narrative approach that will engage the students. See example below.

"Have you heard of feeding programs? What do you know about feeding programs? Who benefits in feeding programs? Have you participated in such? How many children usually participate? What kind of food is usually served in feeding programs?

Last weekend, I volunteered in a small feeding program. I got to meet some Grade 5 children like you. I was in charge of distributing milk. We brought with us 6 liters of milk. My task was to pour the milk into glasses. If my estimate is right, each glass approximately contained | liter of milk. The milk we brought was just enough for all the children in the area. How many children do you think we served during the feeding program?"

Generation of solutions:

The students will work in pairs or triads. Encourage the students to think about the problem and share their thoughts with their classmates. Assure them that there is

Watch out:

The students should realize that the given is a division problem. Their discussions with their partner/groupmates should revolve around figuring out how many | s would "fit in" 6. Give guide questions to pairs/groups who may have not realized this. In giving guide questions, determine first what the students know and then build on that.

Processing of solutions:

Have some pair/group write their solutions on the board and explain.

Possible solutions:

(1) Illustration

s a a!

Use of decimals

— = 0.4 5 6 t 0.4 II II

60 4 15 4160 4_

20 20

N=15 glasses

4 _ _ 2 _ _ _ 2 _ 5 5 “ 5

Guide the students to realize that the different solutions all lead to the same answer, which is 15—there are 15 children who were served during the feeding program. Ask the students who among them did the same as the ones presented in class.

Tip:

Name a solution after the students who shared it. For example, the illustration is "Paolo and Jose's technique." Doing this would (1) deepen the students' sense of ownership of their learning and (2) motivate them to think of unique solutions when given the same task in the future. Lead the students to realize that 6 -s-1 is equal to 15.

(See the continuation o f this plan in Lesson 10)

The next part of the lesson involves the students discovering the algorithm for dividing a whole number by a fraction. This applies another teaching strategy that will be discussed in Lesson 8.

Assess

Answer the following questions to solidify your understanding of the problemsolving strategy.

1. How are the three possible solutions shown in Experience different from each other? What goes in the minds of the students who would possibly give those solutions?

2. As a teacher, how would you ensure that the word problems you will give genuinely promotes problem solving?

Challenge

As you may have noticed in the sample lesson plan, the problem is basically a usual division problem that you may see in common mathematics assessments. It is not even a nonroutine word problem. However, the students' thinking and creativity are challenged because (always remember) it is the first time that they will encounter such a problem. So, the task of the teacher is to present the problem in an engaging way. Do the following to practice this important task.

1. Browse the DepEd mathematics curriculum guide. Choose a topic from Grades 4 to 6. Write the topic and grade level below.

Browse the DepEd mathematics teaching manual for the grade level you chose. Find a word problem from your chosen topic. Write the problem below like how it is written in the teaching manual.

How would you present your chosen problem in an engaging way? Put it in a situation where the students can relate to. You may use your own experience, too (many times children are interested in what's happening to their teachers' lives ©)! Imagine you are talking to your students. Write your script below. If you plan to use materials, write a note about it.

2. The possible solutions in the sample lesson plan are written by real Grade 5 students. So be convinced that Filipino students are actually capable of thinking! As would-be teachers, your task is to anticipate such possible solutions. How? Consult with experienced teachers! Show your problem to some teachers and ask them how they think the students would answer if they are given the problem for the first time. You may also ask the children themselves. Talk to your nephew/ niece, godchild, neighbor, etc. Share with them your problem and have them explain to you how they think they can solve it. Write in the boxes three of the possible solutions that you have gathered.

Possible Solution 2 i

Possible Solution 3

Write a lesson plan that makes use of the problem-solving strategy. Use the topic, problem, and possible solutions that you have answered in Challenge. This activity will be part of the learning portfolio that you will compile at the end of this module.

T o p ic :________________________________________________________