When most people are asked how to define fluency, they think of speed and accuracy. I think everyone would agree that students who have automatized their math facts are in a much better position to learn more difficult concepts because they have more brain energy to concentrate on understanding the concept they are learning rather than worrying about the math facts embedded in the problem. There is a third part to this definition, though, that is the key to laying the foundation for future math learning – flexibility! When children are taught strategies that allow them think flexibly about numbers, we are helping them develop their number sense and unlocking students’ access to more difficult concepts later. So not only are we helping them learn the math facts, but those very same strategies can be applied to larger numbers, and even fractions and decimals down the road.
In our district, we have been following a progression of strategies for each operation developed by Dr. Nicki Newton in her upcoming book Running Records due out in the spring (2016). I took a webinar this past summer to learn about this instructional assessment tool and it has simply transformed everything! Combined with implementing Number Talks, I believe we have the chance to change the trajectory of math education for our students.
Essentially, the running record is a math interview where we get a measure on the students’ accuracy, flexibility using strategies, and speed (although this part is not nearly as important to me as accuracy and flexibility since I believe, like Jo Boaler’s, that you don’t need to be fast at math to be good at math). Once we figure out where the student falls on the progression, I have been developing a collection of games and activities for each strategy so we can begin working with the student on the strategy they need and help them move forward on the progression. It has been awesome to watch the progress our students are making!
To teach the strategies, we use math tools such as 10 frames, Cuisenaire rods, rekenreks, counting objects and dry erase board 10 frames. Students need to see concretely how and why this strategies work. I have created a set of videos (click here for the webpage) that shows exactly how we are using these tools to teach these strategies. Here is the progression of the strategies we are using:
Plus 1 – We want student to understand that adding one to a number results in the next counting number.
Plus 0 – We tend to combine the Plus 0 examples with the Plus 1 examples since adding 0 is such a difficult concept for our young kiddos.
Count on 2 or 3 – For this strategy, we focus on starting with the larger number especially when the larger number comes second…so 2 + 6 I would want my student to start at the 6 and count on 2. The last thing we want to do is encourage children to count, but if we are just adding 2 or 3 and we have this strategy in place, it will take about the 3 seconds we are looking for to get an accurate answer.
Adding within 5 – We want to be sure that students have a good handle on adding numbers whose sums are up to 5 before we move on the sums within 10 so this is a great checking point.
Adding within 10 – We want to be sure students are comfortable working with sums within 10 before moving on to the next strategy.
Adding to make 10 – The most crucial skill! You can’t practice it often enough even when children pass this strategy. This is a strategy in which we want automaticity. Our students play lots of games like “Go Ten” which is played just like Go Fish but you are asking your partner for the pair that makes 10 with yours or we place 12 cards face up on a table and students are choosing two that combine to make 10. We assess this strategy by giving the students a number and they tell us the number that adds with it to make 10. Before we move a student on from this strategy, we want to be sure that if we say “2” they say “8” and so on without hesitating. This forms such an important foundation for future math concepts.
(disclaimer here – at this point with our K-2 kiddos, we then start them on the subtraction strategies within 10…once they finish those, they reenter this progression here)
Plus 10 – we want our student to develop automaticity with 10 plus any single-digit number since that, too, is a foundational skill embedded in more difficult concepts later on
Doubles – the last of the strategies that we look for automaticity since it is also a foundational skill for future strategies
Doubles plus 1 – We want students to recognize that if they are adding two numbers that are 1 apart, they can double one of the numbers and adjust accordingly. For example, 6 + 7 can be thought of as 6 + 6 + 1 or some children prefer to think of it as 7 + 7 – 1.
Doubles plus 2 – We want students to recognize that if they are adding two numbers that are two apart, they can do Doubles Plus 2, Doubles Minus 2 or even Double the middle. Let’s say we have 6 + 8….Student 1 can think 6 + 6 + 2 = 14, Student 2 could think 8 + 8 – 2 = 14, and Student 3 could think of taking one away from the 8 and giving it to the 6 to change the expression to 7 + 7 = 14. There is no one right strategy..the idea is that children are thinking flexibility about numbers and are using a strategy easiest for them.
Decompose to make 10 when adding a 9 – These last two strategies are the most important strategies because they translate perfectly to larger numbers, adding fractions and decimals, and within concepts of adding units of measurement. Since one of the addends is a 9, we can take away 1 from the other addend and rename the expression with a 10 and whatever is left over. For example, 9 + 6 would be changed into 10 + 5 to make it easier for our brains. Focusing on just the 9 first helps the students develop the ability to think about taking an amount from the other number to make a 10.
Decompose to make 10 when adding a 7 or 8 – This is the last strategy on the progression and forms the foundation for our work with larger numbers down the line. We need to take either 3 from the other number to make a 10 with the 7 or 2 from the other number to make a 10 with the 8.
The decomposing strategy with 7, 8 or 9 is truly the most versatile, so if a child is using this strategy when they are working with any of the ones that fit into doubles plus 1 or doubles plus 2 I would not make students work on the doubles strategies. But, I have found students to stick with the doubles strategy for many expressions such as 5 + 8. Students will say they know 5 + 5 + 3 makes 13. For them, I do want to work with them on decomposing to make a 10 since it is so versatile to future math concepts.