The Facts about Facts!

Why bother?!

In this age of technology, do we still need to learn number facts? Absolutely! Thinking flexibly about numbers opens up thinking for more complex mathematical problems. Mental math is also a survival mechanism when you are out in the real-world! Think of how often you use it: How many km am I going over the speed limit?! Do I have enough money to pay for the groceries? How many points is Hillary Clinton up by?!

How do you teach mental math?

Traditionally learning math facts has always been focussed on drill, drill and more drill. In recent years, however; teachers are straying from this model and realizing that drill is not as effective as other methods for ensuring that deep understanding and number sense.

“Posing a story problem…followed by a brief discussion of the strategies that children used, can improve children’s accuracy and efficiency with basic facts.” Rathmell, Leutzinger, & Gabriele, 2000

In fact (no pun intended!), it has been found that these children actually master more facts than children in a drill program! In addition to story problems, number talks that expose children to multiple strategies goes much further than drill alone. A number talk is when you pose a question (ideally in a story problem) and engage the students in a dialogue about how they solved it.

Playing games has also been shown to increase students’ recall and accelerate their understanding of facts. This is only true, however; if the use of reasoning strategies is explicitly built into the games and reinforced through interactions with others. So, if you are a fan of worksheets, drills, timed-tests or Kumon – hopefully this post will change your mind!

Addition strategies

It may interest you to know that there is a huge difference between the way we present facts in North America, compared to China. In China they teach fact tables (for example, the 6 fact tables are all the facts with 6 as an addend: 6+1, 6+2, 6+3 etc.). In America, we teach fact families (for example all the ways to make the sum of 12: 1+11, 2+10, 3+9 etc.). The result? Using fact tables, there are 81 facts to learn (although if you know the commutative property that can be reduced to only 45!!!); using fact families, there are a whooping 153!! (W. Sun and J. Zhang, 2001) So I encourage you to rethink the way you present facts in the early years. Your child may thank you 153 minus 45 times!

Although students should be exposed to multiple strategies, it is important that they are allowed to invent and use their own strategy and not be pigeon-holed into a teacher-chosen one. That being said, there is usually one strategy that is more preferable to use for the given scenario. Discussions about efficiency should take up a lot of your time. You’ll find children are quick to realize that one strategy is better than another and your job is to make the opportunities for those discussions to happen.

The main mental math strategies are:

  • Counting all
  • Counting on
  • One more or one less
  • Two more or two less
  • Making 10
  • Doubles and near doubles

Does order matter? Kind of….the first few methods are counting skills so children should find those mental math skills easier to master. The next few are reasoning methods which require a lot more higher thinking. In order to master those, students must be comfortable counting on and back as well as familiar with composing and decomposing numbers. At school, we just started a unit with making 10’s, but only because they had just finished place value and it seemed like a natural progression. But even within that strategy, we can make it easier or more difficult for students that need it. To make it easier – start with the 9 fact table (easy to make 10 by counting one more). To make it harder – start with the 6 fact table (how many more to bridge 10?).

The important things to remember? Ensure repeat exposure with each strategy and have patience! “It can take between 2-4 lessons before most students really internalize the reasoning strategies discussed in class.” Steinberg 1985

Isn’t using manipulatives with mental math cheating?

I like to call it reinforcing! Child development naturally progresses from concrete to pictorial to abstract, so do that! Help build the mental picture in their minds by using concrete manipulatives. Here are some good ones to use for facts to 18:

  • Five and Ten frames
  • Rekenreks!
  • Abacus
  • Base 10 Blocks
  • Number lines
  • 100 chart
  • And one of the favourites! Fingers!  Read an interesting article HERE on the importance of fingers and mapping within the brain!

Here’s a video of me introducing addition facts to Rory using 5 frames. My first check is to see whether he knows what addition is; then I follow to see whether he has the ability to count on. If he doesn’t, then the only mental math strategy accessible to him at the moment will be counting all. I was super surprised and excited when Rory automatically made groups of 5 with his 5-frames and even more impressed when he made the group with the larger number (i.e. 4 instead of 3), especially since he has never seen this manipulative before. This just shows the power of a manipulative in learning addition strategies.

Keep in mind...

Kids need continual practice throughout the year. Look for ways to incorporate mental math into your day naturally. Even exceptional students benefit from conversations about the efficiency of each strategy. For example, when should we use bridging 10 instead of making doubles?

Can you expect your 4-year-old to master the facts? Not usually and not with full understanding. The ability to reason and achieve full mastery for addition and subtraction facts up to 18, usually occurs in Grade 2, but it is different for each child.

What can I do with Rory at the kindergarten level? I can expose him to problems of joining and separating and teach him the meaning of the words add (join) and subtract (separate). I cannot teach him reasoning strategies until he has the ability to count on and is developmentally ready, but I can see where he’s at. In Grade 1, I can use 5 frames and rekenreks to help him develop his facts up to 10 and help him achieve mastery of those before entering grade 2. And then in grade 2, expose him to different strategies so that he can think flexibly and easily, mentally.

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The good, the bad, and the ugly about number bonds

What are number bonds? Number bonds are just another way to visualize addition facts and are a huge component of Singapore math. What is Singapore math? It is a collection of computation strategies that arose out of Singapore, where students are supposedly ranked among the best in the world in math achievement. So are number bonds all they are cracked up to be? I think they might be! But I’ll let you decide. First, here is a video of Rory discovering number bonds for the first time so you’ll know what I’m talking about.

THE GOOD…

“Emphasizing number relationships is key to helping children fully develop number sense.” Van de Walle

And the most important relationship to develop? Part-part-whole. Number bonds are a great visual to see the part-part-whole relationship.  Focusing on a quantity in terms of its parts is a major milestone for young children and number bonds can help them get there. Just make sure you keep the big idea in mind by consistently using the “part-part-whole” terminology.

The building block of number sense is to think of numbers flexibly. Number bonds help develop number sense by showing different ways to decompose and recompose numbers. By showing different number bonds for one whole number, children see multiple ways of making (or unmaking!) a number. 5 can be made from 2 and 3, or by 4 and 1. Number bonds show this connection well.

However, what is my favourite attribute of number bonds? Number bonds are a great way to teach addition and subtraction at the same time. Rory and I continued our lesson with questions such as: “With 5 whole cars, if you had 4, how many did Philian get?” (one). Number bonds allow children to link the inverse operations easily so that they develop the two skills together. This creates fluency in both operations at the same time, and not a weakness in subtraction, which we often see when the two operations are taught separately.

THE BAD….

“Primary teachers have the tendency to rely too heavily on textbooks, workbooks and photocopied support materials.” Charlesworth

Number bonds lend too easily to this. It is too easy to photocopy a bunch of circles and have the students fill it in with little attention to problem solving. As a result, students are not given the chance to discover the meaning of the relationships on their own. Instead, why not have the students choose their number and discover the number bonds that connect it? Or why not use number bonds as a method to record their work for a problem, but not as the problem itself?

Number bonds shouldn’t be presented independently; use manipulatives to make it real. Adding is putting together groups of objects to find out how many, and students need practice actually doing this with concrete objects. Notice with Rory, I don’t start with the number bond visual – I start with the concrete manipulative to mimic a real life situation. I use that to build the number bond and whenever Rory got stuck, where did he go? He referred back to his concrete manipulatives: the cars. Number bonds help you to see the two parts, but manipulatives make the two parts real.

“Research has demonstrated that when kindergarten and first-grade children are regularly asked to solve word problems, not only do they develop a collection of number relationships, but they also learn addition and subtraction facts based on these relationships.” Van de Walle

Number bonds should not be presented first and then problem-solving second. Instead, allow students to discover the facts for themselves and then use number bonds to make sense of their work. Rory was able to solve the problem for himself and will have a deeper understanding of ways to make 5. Using real problems makes the learning more engaging for the child, especially if the problem involves them. And as the research shows, it also helps them achieve mastery as well!

THE UGLY…

You might be tempted to use number bonds as flashcards for a drill or a timed test or prolonged practice. Number bonds used as drill creates anxiety and stress and doesn’t encourage an understanding of the part-part-whole relationship. Children should learn facts through discovering patterns and relationships. By focusing on families of facts and their relationships, in a problem-solving environment, you are encouraging mastery of facts through exposure. Rory quickly noticed the patterns that make 5 – as one part gets smaller, the other gets bigger. He giggled when he saw that 3+2 is the same as 2+3, but he was discovering these relationships under the guise of a real problem. There was no anxiety or pressure for him to memorize number bonds.

Some kids aren’t going to be ready for number bonds because they are too abstract. To aid with this, start with 5 or 10 frames and concrete manipulatives to help them see the facts more easily. Keep the manipulatives going and don’t switch to the abstract (pencil and paper) until much later. Allow the child to determine the pace of the learning….as long as you are providing opportunities for them to engage in problems that reinforce the facts, you’re good!, You will know mastery is achieved when they don’t have to count how many are in each group, they just know and this typically doesn’t happen until 3rd grade!

THE BOTTOM LINE…

So are number bonds good, bad or ugly? Well, you always need to keep in mind your big idea…you are trying to build number fluency which involves efficiency, accuracy and flexibility. Number bonds are another tool in your toolkit to help students visualize part-part-whole relationships. As long as you remember to focus on those relationships and surround the students learning in problem-solving and not drill, then number bonds are a great resource. Just beware of what could make them bad or ugly as well.

What does Rory think? He loved creating them so much, he carried on and made his own number bonds independently afterwards…some more abstract than others!

Have you had any experience with number bonds: good, bad or ugly?! I’d love to hear about it! Tell me in the comments section below!

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