Given a math story problem where you have no idea where to begin, what do you do? Figuring out what to do is mathematical problem solving. These lessons are designed to help teachers teach mathematical problem solving. Each lesson has a handout and accompanying questions that enable the teacher to guide the student toward thinking like a mathematician. The problems are story problems appropriate for upper elementary grades.
Please modify and use the lessons as best works for you. The lessons are divided into sections and a very rough time estimate is given for each section. These time estimates are provided only as an approximate guide.
introduction provides an overview of problem solving and
suggestions for teaching the ideas.
I say to my classes, “A group of your friends are going to Disneyland and invite you to go. Your parents say fine, but you need to earn the money.” Is this a problem?
If your parents say “here’s the money,” is there a problem? No, but by saying you need to earn the money, we now have a problem. What are some components of this problem – things you need to figure out to solve it?
A problem is when you are faced with a situation and you don’t know immediately what to do. If how to proceed is clear, then it is not a problem. If I asked “what is 3 times 4” we know the answer is 12, and that is not a problem!
I tell my
students that when they get a problem, they need to
“Stop-Look-Think.” My goal is to teach what “think” means in
terms of mathematical problem solving. I tell them that math
problems take time to solve, and in fact, most word problems
take me a few minutes to do and some problems take me much
longer than that, so why should they think they can do it
faster? You cannot just read a problem as if it were a story.
Instead, you must analyze each word and sentence to find out
what information is given and how it all fits
Mathematicians think of problem solving in terms of four steps:
Step 1: Unpack the problem
Step 2: Pick a strategy
Step 3: Solve the problem
Step 4: Answer with a few words
Unpacking the problem is critical to solving it. Use wwwww (who, what, why, when, where) to guide you. Investigating a math problem is like a journalist researching a story; you need to ask the questions to uncover the clues.
Pick a strategy refers to all the different ways we can approach a problem such as make a table, draw a picture, work backwards, guess and check, group terms, use a Venn diagram, and so on. There are many strategies and we frequently use more than one in solving any problem. There are many lessons devoted to introducing the various approaches.
Solve the problem is when you actually do some arithmetic. It is the third step, with two very important steps before it.
Answer with a few words to make your answer clear. In more advanced problem solving approaches this fourth step is usually “check your work.” For the children at the elementary level, having them go back and answer the question posed by the problem using a few words is a first step in the checking process.
Getting my students to independently focus on understanding the problem had always been a challenge. Over the past several years, I have developed an approach that enables students to really “get it.” I have a poster with the questions as reference for them:
When we start a problem I have my students put their pencils away – they are to do the first part orally – NO WRITING. I have them go through the problems with their learning partner and DISCUSS the questions. I then have a few people share what they have learned.
I tell my classes that stopping to really understand the problem is a very hard step. Most of us never really do this, it is new for them and takes lots of practice to have it become “normal.” I reinforce that this is an important and necessary part of the time it takes to solve mathematical problems.
Then I have
the students use their pencils to do the problems by
themselves, then review their answers with their learning
partner, and finally we discuss as a class.
As I said in item 3 above, problem solving takes time! Accordingly, be sure to limit the number of problems you expect your students to do at any one time.
I call my students Mathletes and tell them that Mathletes are
To assist them in becoming Arithmetic Champions, I do mathematical sprints to start each class. A sprint is something you run quickly. Basically I have two sheets of related arithmetic problems – drill problems – for my students to complete; for example, 60 multiplication facts. I give them one minute to do as many as they can, read the answers, lead a short one minute break activity, and then give them the second page with one minute. The goal is to see improvement scores between first and second sprint. The total time after the first couple of days is FIVE minutes, and the kids love this form of drill. For the break I use the counting activities described in item 7 below.
I use Bill
Davidson’s sprint material and sometimes develop my
own. Bill also has material to enable you to learn the
methods he has developed to create and administer sprints.
At different points when I feel my students need a break or when I’m transitioning from one part of a lesson to another, I use counting activities. I start these very simply and add complexity over the course of the year. When possible, I like them to cross into the hundreds before stopping. Here’s a partial list:
And for later
in the year: Count by ˝ and reduce to lowest terms (1/2, 1,
3/2, 2, …), by 1/4, etc.
I try to have
a few minutes at the end of class to play “Buzz.” The
traditional game is one where everyone stands up, you go
around the room and each person says the next number in the
series, saying "buzz" instead of the number where appropriate.
If a person makes a mistake, they sit down; the game continues
until there is a winner. I allow people sitting to stand up
again if they correct a mistake of someone standing. In my classes, I call out the numbers
and the students give the response. That way, I can say 12
three times to get the three different factorizations of 12
(1x12, 2x6 and 3x4), or to repeat a number if someone gives an
Traditional Buzz: pick a number, say 7, and students say “buzz” (instead of the number) for all multiples of 7 (7, 14, 21, …) and for any number that has the digit 7 in it (17, 27, 37,…) even though they are not multiples of 7.
Factor Buzz: As you go round the room, students have to give a factorization of the numbers rather than the number. For example, as we go around, the students would say…
1 = 1x1
2 = 1x2
4=2x2 or 1x4
and so on.
I tell the
students that they should ONLY use 1x number only if there is
no other factor. So, for 4, I accept 2x2 and not 1x4. For 12,
I accept 2x6 or 3x4. This activity is also a chance to
reinforce prime and composite numbers, which is an upper
Prime Buzz: Only a slight
change: once they have gotten more experience with Factor
Buzz, they say “buzz” for a prime number (numbers whose only
factors are 1 and itself) and give the factors for composite
numbers. For example, answers would be as follows: 1, buzz
(for 2), buzz (for 3), 4=2x2, buzz (for 5), 6=2x3, buzz (for
7), 8=2x4, 9=3x3, and so on.
Other variations I do later in the school year are:
Remainder Buzz: Take a number, say 5, for which they have to give remainders. For example, if you say 6 they say 1. If you say 28, they say 3, and so on.
Prime Factorization Buzz: if the number is prime, they say “buzz;” if not they have to give the prime factorization. For 12, they would say 2x2x3, etc. I jump around with the numbers to keep it interesting.
Multiples of Ten Buzz: For example, 70=7x10, 7=35x2, 70=14x5, etc. My rule for this version is that I will call each number three times; the first time the student can give me the simple multiple of ten, and the second time the multiple of 2, and the third time the multiple of 5.
Fraction Buzz: Simplest form, do only factors of 2: say 10, response is 5; say 5, response is 2.5; say 1, response is ˝; etc.
MOEMS, Mathematical Olympiads for Elementary and Middle Schools, is the organization I have used to guide my teaching over the past decade. MOEMS provides five monthly contests, given from November to March, which I administer in my classroom. The problems are stimulating, challenging, and provide the structure for teaching mathematical problem solving. I extensively used the MOEMS books listed below as a resource. I strongly recommend every teacher investigate participating in the MOEMS program as an option for some or all of their students.
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