by: Maggie Martin Connell

 

If 3 isn’t 'three', then what is it? That isn’t a trick question. We make that mark to communicate a very specific idea about quantity. But, if we lived in China, we might name the idea something else and make an entirely different mark to represent it, even though the idea itself hasn’t changed. So, if ‘three’ is neither the mark nor the word, then what exactly is it? Consider the following scenarios:

fig.1

All of these have something in common... 'three'-ness. We want to be able to communicate this quantity to others in a way that is efficient and eliminates ambiguity, so we assign a word and special mark to it that distinguishes it from 'two'-ness or 'four'-ness or any other -ness. Since that word and its corresponding mark can vary across cultures, we have to conclude that both the mark and the word simply represent an idea, but they are not the idea itself.

Over time, brilliant mathematicians have developed progressively clever ways of using the mark to communicate other profound ideas related to quantity. Each new discovery allowed us to better cope with an increasingly complex world.
Consider...

fig.2

To bring us any one of these powerful tools, they had to have understood that this science we call 'Mathematics' follows patterns and that, if the same steps are applied to other situations sharing a similar idea, it ‘works’ every time!

What does this have to do with your classroom tomorrow? Everything!

The problem is… somewhere between the birth of this symbol for 'three'-ness and today, we forgot our history. In the name of efficiency, Mathematics became the art of computing, using established procedures that represent someone else’s understanding. For the general population, learning those procedures became more important than the original ideas they represented; the medium became the message! A math culture evolved where the symbols that were supposed to represent the math became the math itself. Precision ‘digit-dancing’ became the key to success... even if you didn’t know where that dance was leading you.

But this story isn’t just about 3. It’s about the entire collection of marks representing mathematical ideas. Figure 3 shows a few common ones:

fig.3

We have learned to record them, say the words associated with them and manipulate them to arrive at ‘answers’. But do we know why we are doing it, other than because the teacher said so? Have we considered the BIG ideas those digits represent? Can we tell someone why each step in the ‘digit-dance’ makes sense?

Consider a symbol we have seen since the very beginning of our school math experiences: the lowly, often taken-for-granted = sign.

When 74 students from 6 different middle-level schools were asked what it means, 68 of those students replied, “equals”. When asked to explain what ‘equals’ means, some of the students simply repeated the word in a different tone and could go no further. Other responses included:

• “That’s what it makes.”
• “It means two numbers combine to make a bigger one.”
• It’s sort of like a question mark.”
• “It’s when you know the problem is over.”
• “When 2 or more numbers are added, divided, multiplied or subtracted, you add the = sign to indicate the answer will be next.”
• “If you multiplied two numbers, they would be equal.”
• “The equivalent of a problem.”
• “Equals means 'amounts up to'. For instance, 2 plus 2 equals 4.”
• “The sign before the answer to a question showing that the answer is coming prior to that.”
• “It means the adds.”
• “Two things combine to make… AN ANSWER!"
• “It’s what you have to say before you give the answer.”
• “It means a certain thing (most likely a number) is = to the next possible number.”
• “The answer is...”

In the vast majority of cases, students interpreted the = symbol as a ‘command’ to give an answer!

Out of the 74 students, only 8 expressed the idea that it means both sides are the same:

• “Both sides have to be the same or it’s wrong”
• “It only works if each side has the same amount.”
• “It signifies the same amount.”
• “… is the same as.”
• “To have the same value. For example, $1 equals 100 cents.”

In the same survey, 9 teachers were asked the same question. 8 of the teachers gave a response that indicated ‘the same as’ and 1 teacher gave the following response:

• “I think equals means the amount if you are saying ‘blank and blank equals blank’.”

No responses in this survey included the word ‘balance’. It is clear that the idea behind the mark has been mostly missed for these students. Additionally, many of them have adopted the misconception that the purpose of the = sign has to do with the physical placement of the answer.

But what if we taught this idea inside out? Instead of ‘teaching’ what to do with the symbol, what if we set up investigations where kids could explore what balances a number sentence and what does not? A good investigation would uncover a variety of possibilities and create the need to record trials. Students would then be using the symbol as a way to record the idea of balance rather than to simply arrive at an answer.

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. Although this investigation is presented through basic addition and subtraction facts, it can be adapted to any grade level, 1-8.

 

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Does It Balance? (making sense of = and ≠)

What is the BIG idea?

1. We can use the = symbol to communicate balance in an equation and the ? symbol to communicate imbalance.
2. The number of ways to create balance in an equation is limited only by how creative you allow yourself to be.

What will I need?

a pan balance
colour tiles (cubes or other multi-coloured counters)

What does this lesson assume?

1. Students are familiar with the symbols + (when recording addition) and – (when recording subtraction).
2. For this lesson, students understand addition as the joining of sets and subtraction as take away.

3. Students have had exposure to both horizontal and vertical formats for recording addition and subtraction.

 

How do I start?

Set the tone for the investigation by reinforcing the idea that the purpose of any symbol in math is to represent an action or thought (idea):

1. Place 4 red tiles in the left pan. Then place 3 blue tiles in the same pan. Ask the students how they might record what you just did. (4 + 3 is one way) Pointing to each part of the recording, ask students what it represents or what it is telling us. (4 is the number of red tiles – that is how many we started with, 3 is the number of blue tiles, the + symbol tells us we put those groups of tiles together.) You might also reverse the questioning to help students see it from different perspectives. For example:
   
   
   • Which part tells us about the number of blue tiles we started with?
   • Can you show me the part of our recording that tells us what action we took?
   • If someone came into our room right now and saw what we recorded, would they be able to guess what we did? How would they know?”
     
     
2. Ask students what they notice about the pan balance (It’s tipped to the left.) Begin adding yellow tiles to the right pan until it balances the left pan (7 yellow tiles). Have students suggest where that number could be recorded. Most times, children will associate the recording spot with the corresponding part of the physical model (right pan = right side).
   
   

   Leave enough space to insert the = symbol:

3. 
   Ask students what they notice about the pan balance now. (It’s balanced.) Invite suggestions for recording that piece of information. It is likely that some students will suggest the = symbol, since they will have had prior exposure to recording basic addition facts (see assumptions above). Invite a volunteer to insert the symbol for balance where it makes sense in the recording. (4 + 3 = 7) As you read the recorded equation together, use the word ‘balances’ instead of ‘equals’. In time, when the idea is firmly rooted, you will use these words interchangeably.
   
   
4. Ask students to predict what will happen if a blue tile is removed, then invite a volunteer to test the prediction. Take suggestions from students for recording what just happened. Introduce the symbol for imbalance (≠) by drawing analogies to common signs in our world.
   
   
   The point here is to establish that we commonly use a line drawn through something to indicate what it is NOT. Invite suggestions from students as to how we might use this idea to record another idea that something does NOT balance. Suggested ways to record might include:
   
   
   
   Either way is acceptable provided students can explain the reasoning.
   
   Author’s note: The symbol for imbalance (≠) is not traditionally taught at this level, but I have yet to discover a good reason for that. If it is taught as an idea and not a symbol in isolation, it makes perfect sense to introduce it along with the = symbol. Think about it… your back must have a front; to know ‘up’ you have to understand ‘down’; for the inside of a cup to be useful there has to be an outside. Reason gives us the logical sequel… to know balance you must understand imbalance, AND you have to have a way to record that distinction.
   
   
5. Invite students to work in small groups to predict other ways to balance 7 tiles, then use the pan balance to test their predictions. All trials should be recorded, whether they balanced or not. As students are investigating encourage some groups to consider placing the 7 tiles on the left and adjusting the tiles in the right pan. Ask them how their recording might change. (7 = 4 + 3). This is an important idea that is often missed. Can you see how recording this way helps to discourage the interpretation of = as a command to give an answer?
   
   
6. Allow sufficient time for groups to come together to share their discoveries and their different strategies for recording/thinking. All strategies should be valued. You may also find it interesting to hear students’ responses to this question: “How many different ways do you think are possible?”

 

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But don’t stop there!

For real understanding to blossom, students of all ages must appreciate each symbol as a way to represent a specific idea, and a combination of symbols as a way to record a ‘thinking path’. We as teachers have the ability to influence that by teaching ideas (instead of digit-dances), then using the symbols as a way to record those ideas. For many of us this represents a ‘seed’ change; a change that comes not from a mandate, but from a deep personal belief that it will make us better teachers and that our students will profit by it.

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

So whether it’s 3 or 300 or 3³ remember… the mark is not the math!

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