by: Maggie Martin Connell

8:30 am. The buzz and chatter in the room subsides.
“ Good morning…” says the announcer to the sixty-plus teachers sitting at tables throughout the room, each hopeful that the day will be worthwhile, bringing new light to his/her math teaching.

Introductions are delivered and the guest speaker steps up to the microphone. There is a brief pause.
“ Has anyone here ever eaten a mango?” she asks.
Somewhat perplexed, the audience hesitates briefly before several hands go up.

Maria’s hand is not raised.

“ So then, Maria, you’ve never eaten a mango?”
“ No.” replies a young teacher at the back of the room. The speaker addresses the rest of the group.
“ All right… I would like the rest of you to explain to Maria about mangoes so that she understands. Can you do that?”

There is another pause. Contributions begin slowly, but gain momentum and volume as the exercise continues, several people talking at once.

“… juicy”
“… more slimy than juicy”
“… sort of tastes like a peach”
“… but with an orange sort of flavour”
“… kind of stringy”
“… soft flesh”

“So, Maria, now do you understand about mangoes?” the speaker asks.
“ Sort of… not really…” she replies.
The group is invited to try again.

“…sweet”
“… smells a bit like socks”
“… I agree that it’s more like a peach than an orange”
“… gets between your teeth”
“… sort of rich tasting”

Maria is obviously trying hard to understand but doesn’t look convinced.

 

 

 

 

 

There is a general consensus from the group that Maria has to actually experience a mango to really understand it; that she needs to smell it, peel it and taste it before she can develop a real ‘sense’ of mangoes.

“What about skating?” asks the speaker. After a brief exchange about the set of sub-skills and ideas that make up an understanding about skating, the speaker raises the question again.
“ Could you learn to skate if I explain it to you? What if I show you how?”
“ No,” replies one of the teachers. “I could probably repeat your instructions, but I still wouldn’t really know.”
“ So then, what are you telling me about learning to skate?”
“ You have to put the skates on and feel what it is like to skate.”
“ You have to experience skating?”
“ Yes.”

The speaker continues, “How do you learn to paint or draw?” “What about learning a new language or playing a musical instrument?” “Building a house?” For each scenario, the group agrees that experiencing the idea is critical to ‘knowing’ it.

Consider a time when you have personally experienced a breakthrough with something you were trying to learn, a time when the lights suddenly came on and you felt the rush of clarity and associated excitement that comes with discovery.

Chances are, you were a) interested and b) engaged in an activity that allowed you to have direct personal contact with the idea; you were experiencing it.

This exquisitely simple principle holds true in just about every type of learning situation, even numbers… especially numbers. The straightest path to intuitive understanding in math is through engaging investigations that allow discovery (not telling) to happen. The more complex the idea, the more important this becomes. A good investigation will uncover a variety of possibilities and create the need to record thinking, thereby breathing life into the symbols we ‘teach’.

What might this look like in your classroom? The following lesson outline paints that picture one way. You will undoubtedly bring your own interpretation to it and adjust it to fit the needs of your own students. Keeping the BIG ideas in mind as you do that, will ensure that we all arrive at the same point.

How Will I Share? (connecting fractions and division)

What is the BIG idea?

1. A fraction can represent a division (sharing) situation.
2. The result of the division (sharing) can be recorded as a fraction.
3. There is a relationship between a division. Expression and its associated fraction. 

   

What will I need?

coloured paper squares approximately 3 x 3 inches (a set of 3 per group of 4 students)
scissors

What does this lesson assume?

1. Students understand a denominator as naming the parts (number of equal parts / fair shares in the whole).
2. Students understand the numerator as the count of those equal parts being addressed.
3. Students have had significant experience exploring area, set and linear models for fractions (fraction meanings).

Area Model (part of a whole)

Set Model (part of a set)

Linear Model (part of a set)

What is my problem?

You have invited 3 friends over for a party, and have prepared 3 ‘mystery treats’ to share. The problem is, you forgot to include yourself and you definitely want some too. The guests are arriving and you haven’t time to make more. What will you do?

How do I start?

Stop now and investigate the problem yourself. You may be surprised at what you discover! You will, at the very least, be better able to construct thoughtful questions.

Stop now and investigate the problem yourself. You may be surprised at what you discover! You will, at the very least, be better able to construct thoughtful questions as you lead your students towards discovering.

 

Now let’s say there are 24 people coming and you have ordered 16 pizzas. If you have noticed the pattern, you should be able to know instantly what fraction of a pizza each will get! AND… chances are you won’t forget because you have just had direct personal contact with the idea!

Now you are ready to lead your students to discovery.

1. Set the tone for the investigation by recounting the problem. Tell them you solved your problem successfully and everyone (including you) ended up with fair shares. Challenge students to investigate all possible ways that sharing could have happened.
2. Have students form groups of 4. Give each group a set of 3 treats (squares).
3. As groups work to solve the problem, move about the room, listening for reasoning and providing prompts where necessary. It is important here not to give instructions, but rather to steer students’ thinking and check for understanding. Some helpful questions might be:

• Are these shares fair? How do you know? (When I put them together, they are all the same size)
• Can you explain why you chose to share that way?
• What are the fractional parts called? (Fourths, because there are 4 equal shares)
• How many of those ‘fourths’ did each get? (3 of them)
• Can you find another group who has shared the treats differently?
• What if 1 more person showed up… would you still be able to share fairly? (Yes, but each treat would have to be shared into 5 parts so each part would be smaller)
• How can we record the problem mathematically? (3 ÷ 4)
• How might you read it? (3 things shared by 4 people, 3 divided by 4)
• How much did each person get? (3 fourths of a pizza)
• How can we record that as a fraction? ( )
• How are these recordings the same? How are they different?

The following excerpts from the “IGet It!” Math series are examples of possible strategies students might generate to solve the sharing problem.

Adding Fourths

Taking Fourths

Taking Fourths

NOTE: Groups may have physically divided the squares differently (diagonally, vertically, horizontally,   etc.) You many choose to invite those groups to justify their ‘fourths’. Additionally, you might ask   them to find another group that has divided their treats differently and tell how the fourths are   different and how they are the same.

How do I wrap it up?

Allow sufficient time to bring group back together for a whole class sharing of strategies, discoveries and methods of recording. It is important that all contributions be valued, but not judged as right or wrong.

Have students show and explain their recording methods on the overhead projector or board. Draw attention to the relationship between a division expression and its related fraction:

• the numerator represents the number of things being shared
• the denominator represents the number of people sharing
• either can be used to represent a division situation

Challenge different groups of students to test the pattern with other numbers of treats and people sharing. Does the pattern work every time?

Add variety to the thinking process by turning the question around so that the
fraction is given and students create the division context.

Extend the learning by having students investigate treats of different sizes (large, medium, small) using the same problem of 3 treats shared among 4. Groups should compare their findings and explain why each share was even though the size of the share is quite different. This is an important idea that is often missed. Many students have the misconception that a fraction has a specific value in itself. They need to understand that value is relative to what is being called 1 (the whole).

But don’t stop there!

 

As teachers we want our students to succeed in math. New standards are redefining what that success means, including a vision of a deeper understanding of each idea and its relationships to other ideas. Drawing relationships requires intuitive understanding. Intuitive understanding at its best is born out of rich investigations that give the learner opportunities to discover and construct personal understanding through direct contact with the idea.

n other words… Let them eat mangoes!

Every single one of us (including you) has the right to understand the ideas behind the procedures and reap the rewards and the excitement of that understanding.

 

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