All things are not created equal!

All things are not created equal…including the understanding of the equals sign! Did you know that Americans have a very poor understanding of the equals sign, especially when compared to students in other countries?

“Ninety-eight percent of the Chinese sample correctly answered 4 items indicating conceptions of equality and provided conceptually accurate explanations. In contrast, only 28% of the U.S. sample performed at this level.” (Li, Ding, Capraro, & Capraro 2008)

Students on our continent tend to see the equals sign as meaning ‘the answer to’. In other words, the answer to 2+3 is 5.

What’s wrong with that you ask? The true meaning of the equals sign is as a symbol that indicates equivalence; it does not mean ‘an answer’. It means that one side is equivalent to the other. The equals sign acts as a balance in that the two sides must balance; it represents sameness as in the same amount, but not necessarily using the same things.

Still don’t see the difference? You’re not alone! Most of our students don’t understand the difference either. When students think that the equals sign means ‘the answer’, this is what happens:

Want another example? How many of you have written a problem like this on the black-board: 10 = 6+4 and had your kids tell you “that’s wrong!” because you wrote it backwards? If so…you have some work to do!

But don’t fret! You can quickly transform your students’ thinking by incorporating a few new habits into your routine.

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Experiencing equivalence using a balance scale!

Rory quickly and easily conceptualized the true meaning of the equals sign using this home-made balance scale. I was surprised at how accurately it worked too! It was also a great model to demonstrate equivalence such as: 4+4 is the same as 5+3; and Rory was very able to prove their equivalence to me, by rearranging the towers of 4 +4 to look like 5+3, with unifix cubes. In later lessons, I would use the balance scale to ensure concrete understanding of the other relationship symbols, ‘less than’ and ‘greater than’, as well.

Here’s what you can do:

  • Most importantly: Use your words carefully! When reading equations, reinforce the idea of equivalence by reading the equal sign as ‘is the same as’ or ‘is equivalent to’. Don’t ask for the answer to a number sentence, ask for what it is equivalent to.
  • Write number sentences backwards and forwards. In other words, alternate which side has the operation to be performed. For example, write 9 = 5+4; don’t always show just 5+4=9.
  • Give questions that ask children to find equivalent expressions, not just questions with one number answers. For example,  5 + 4 = ? + 1 instead of 5+4 = ?.
  • Reinforce the commutative property: 3 + 2 = 2 + 3.
  • Cuisinairre rods! These are great for finding and showing equivalent representations of expressions.
  • Use a balance scale! The more concrete and real you can make the understanding, the better. And not just for early grades, upper levels appreciate the visual as well.
  • Give experiences with true AND false number sentences.
  • Watch for textbooks and worksheets that don’t promote this way of thinking. The study mentioned earlier, blames a lot of the misinterpretation of the equals sign on North American textbooks.

Building the understanding of the equals sign, as a relationship symbol, starts in kindergarten and therefore the earlier you can promote the proper understanding of it, the better. The equal sign is the primary symbol used to understand relationships in our number system. Understanding its meaning promotes algebraic reasoning and gives students access to powerful relationships for working with numbers. With a thorough comprehension of the equal sign, other representations, such as the symbols for  ‘less than’ and ‘greater than’, make more sense. And that’s our goal! To make more sense!

Stay balanced!

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What the heck is a rekenrek?

Never heard of a rekenrek? You’re not alone, but I’m here to tell you, it is one of the best manipulatives for developing number sense in early primary, and you can make it yourself! Originally developed in Holland, it looks like a mini-abacus, but functions quite differently. Although you can have many rows in a rekenrek, it usually comes with just 2 as seen below.

 

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There’s an app for that!

 

Why do I think it is the best manipulative out there? Well, let’s start with pre-school. It is a great instrument for working on one-to-one correspondence, counting and cardinality. Watch how Oliver (3 years old) easily subitizes a group of 5 on his first try with a rekenrek!

Not convinced yet? This math model offers teaching opportunities ranging from pre-school all the way to grade 2! Here’s a list of the other math milestones you can use it for:

  • Counting strategies: count all, count on, see groups.
  • Visualize numbers 1-20.
  • Develop benchmarks of 5 and 10.
  • Commutative property.
  • Number conservation.
  • Decompose and recompose numbers.
  • Subitizing.
  • Addition – subtraction relationships.
  • Math facts to 20.
  • Visualize doubles and doubles plus one, one or two more and one or two less .

Here is a video of Rory and I exploring some of these things on our home-made rekenrek.

Are you still thinking to yourself: “I can do all of this using 10 frames or unifix cubes” so what’s the big deal?” I’m definitely not saying: ditch the others; children should be exposed to many different manipulatives, but here’s a few reasons why rekenreks should be one of your first choices:

Rekenreks are concrete. There is often this push to move students to think abstractly too soon despite Piaget’s theory which states: children may remain in the pre-operational stage, and thus not able to mentally manipulate information, until the age of 7! Rekenreks are a concrete model that children can use to do their work, to communicate about their work, and to assess their work. Worksheets should be a method to RECORD their work. If you move too quickly to worksheets without giving the developmentally-appropriate materials to complete them, you may be hindering the development of true understanding. Rekenreks help develop the true understanding at a developmentally appropriate level.

Rekenreks have a 5 structure instead of a 10 structure; as a result this is a useful first step to learning place value. Young children find unitizing a group of 10 (seeing a group of 10 as one ten) difficult and this is what place value is. A rekenrek helps children unitize a group of 5 first, and then 10. In other words, it establishes those major benchmarks of 5 and 10, which will aid in subitizing skills (knowing the number without counting) and later in unitizing skills (seeing a group of 10 as one 10).

Rekenreks allow students to count in groups. When using unifix cubes or counters, children are manipulating one block or counter at a time in order to put them together. With the rekenrek, children can move a group of objects (i.e. 4) at one time. This means moving from a manipulative that requires counting all (counters), to one that encourages counting groups.

Rekenreks help in the development of pattern recognition. Pattern recognition is the basis of number sense and its development begins in the early years. Identifying patterns in pre-school leads to pattern recognition with numbers, which ultimately leads to greater number sense all around. By seeing what 5 is on a rekenrek, and then 5 and 1 more, and then 5 and 2 more, children begin to ‘see’ these patterns in their mind and thus master math facts. This leads to the next point…

Rekenreks help develop automaticity with basic math facts. This is the ability to produce answers in a few seconds by relying on thinking of the relationships among the operations rather than recalling answers. Because rekenreks support subitizing skills, they help kids achieve mastery of addition facts. Students begin to ‘see’ the answer, and don’t have to calculate the answer.

Rekenreks reinforce  the different relationships between addition and subtraction. There are three categories of problem structures for these relationships: change problems (join and separate), part-part-whole problems and compare problems (how many more or less). The rekenrek is an excellent manipulative to use to model all of those structures using rich real-world problems to support them.

Rekenreks allow students to think flexibly about numbers and construct their own strategies. They are an ideal tool for number talks. Starting a class off by saying: I made the number 11, can you guess how I did it, opens up the floor to hear and discuss alternate ways of making 11. For example, I made doubles of 5 and 5 and then added one more or I used 10 on the top and added one more on the bottom. “Not all children invent their own strategies. So strategies invented by class members are shared, explored and tried out by others.” Van de Walle.

Still a non-believer? Well believe this: evidence abounds!

Well…actually it doesn’t… but what little research I could find showed that there definitely was a difference between children taught with a rekenrek and those taught without.

“Results indicated that students in Group 1 (who were taught with the rekenrek) scored significantly higher on an addition and subtraction test with numbers from zero to 20 than did students in either Group 2 (taught without the rekenrek) or 3 (not taught).” Tournaki, N., Bae, Y., & Kerekes, J. (2008).

So what are you waiting for?! Start using rekenreks in your home or classroom today!

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