Teaching Math through Problem Solving Nationellt Centrum för Matematikutbildning vid Göteborgs...

Teaching Math through Problem Solving

Nationellt Centrum för Matematikutbildning vid Göteborgs Universitet

Teaching Mathematics through Problem Solving in

the Primary and Intermediate Grades

• Diana V. Lambdin

Indiana University

USA



Planning and teaching problem-based lessons

• Identify mathematical goals

• Select an appropriate problem

• Plan and carry out the three parts of the problem-based lesson:

• Launch• Explore• Summarize



Effective problems:

• Involve important mathematics

• Consider what the children may know

• Can be approached in more than one way.



Let’s look at some examples of problems



A sample task

Think about the number 5 broken into 2 different amounts. Draw a picture to show ways that 5 things can be in two parts. Make up a story to go with your picture.



A sample task

Provide a collection of coins (real, pictures, or a list) -- for example, 8 nickels (5¢), 9 dimes (10¢), 2 pennies (1¢), and 7 quarters (25¢).

Ask students to find the total in two different ways.

(Can lead to discussion about various ways to make 10s or 100s.)



A sample task

Find ways to measure our Halloween pumpkin.

Write a letter to our pen pals that will help them know how big our pumpkin is.



A sample task

A fifth grader was trying to put numbers in order from smallest to largest. This is what he did:

3.4 3.38 3.45 3.4026What would you tell him?



Now let’s consider a problem AND the lesson launch:

Teacher goals for a 3rd grade class:

• Engage children in exploring multiplication/division (x/÷) concepts

• Assess levels of student understanding of x/÷ -- for planning future lessons



The Doorbell Rang

The Doorbell Rang, by Pat Hutchins, is a story about children sharing cookies among friends.

Copyright laws prevent us from including the complete story in this powerpoint posted on the www.



Suggestions for using children’s literature as a

lesson launch

• above all, enjoy the story first

• read the book aloud

• let children respond personally

• encourage problem responses in a variety of modes (pictures, words, symbols, number sentences)



Benefits of using children’s literature to teach mathematics

• Motivation - stories have universal appeal• Familiarity - most children have

experience with reading or hearing stories• Structure - stories can provide a defined

context where problems can be explored• Language - stories bridge the gap

between informal, oral language and formal symbolic mathematical language



More benefits of using children’s literature in math

• Integration - stories can bring many aspects of the school curriculum together (reading, math, science, social studies, art)

• Shared experience - stories provide children a common starting point for discussing math ideas



Suggestions for using stories as problem launches

• Try to pose problems with more than one answer, or more than one way to get to the answer (or both)

• Encourage multiple approaches and use of multiple representations

• Let children know they will be asked to share their thinking and to react to each other’s ideas



Selecting a problem:

What kinds of multiplication/division problems could you pose from this launch?



The “sharing cookies” problem

Suppose you had 18 cookies. How many different ways could 18 cookies be shared fairly with breaking any?

Using pictures, words, or numbers and math sentences, show all the different ways the cookies could be shared.

Show how many people would get cookies, and how many cookies they would get.



Teaching actions “during” the problem solving

Circulate and listen to studentsAsk questionsEncourage student ideasResist explaining or tellingProvide hints and extensions as needed



Teaching actions “after” the problem solving

• Promote a mathematical community of learnersEncourage student-student questions & dialogueExpect explanations with all answers

• Listen actively. Do not evaluate.

• Summarize main ideas.



Another problem-based lesson:Pattern Block Trains

Teacher goals: perimeter, pattern recognition, use of variables, generalization

Pick a pattern block shape.Build trains of that shape only.

Find the perimeter of each train.Predict the perimeter of any train.



Lesson launch: Let’s try it for triangles!



Hexagon Trains

• Compute the perimeter for each of the first 4 trains.

• Determine the perimeter for the 10th train without constructing it.

• Write instructions for finding the perimeter of any train without constructing it.

• Find as many different ways as you can to compute (and justify) the perimeter of any train.



The “Summarize” phase of the lesson

Ask students to share their formulas -- in words, or in symbols.

Ask for explanations, justifications.

If students have different formulas, compare them. Do they give the same result?



Does the formula make sense with the blocks?

• Sometimes students generate formulas by looking at tables of values.

• Ask them to explain where the numbers in their equations come from.

• Example: P = 4N + 2 Why 4? Why 2?



Comparing different formulas

• P = 4N + 2

• P = 2N + 2N + 2

• P = (2N + 1) + (2N + 1)

• P = (6 - 1) + 4(N - 2) + (6 - 1)

• P = 5N - 1(N - 2)



Discussing Student Ideas

Amanda, a sixth grade girl said:

To find the perimeter of any hexagon train, take the number of blocks in the train, add the next number, then double the result.

Does this always work? Does it make sense?



Next part of the workshop

Return to our original room for wrap-up discussion

Teaching Math through Problem Solving Nationellt Centrum för Matematikutbildning vid Göteborgs...

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