My oldest son, Rory, is about to start Grade 3! Good-bye primary! Hello elementary! I can’t believe it! Where did the time go???
What does this mean in terms of his math? It means he is now ready to learn the algorithm for addition and subtraction; and I am so grateful to his teachers for delaying it so far. To find out some of the benefits of delaying, read my earlier post here: Double Digit Addition: Delaying the Algorithm.
How do I know Rory is now ready to learn the algorithm? Before I answer that, it would be easier if you could see him solve a few problems.
What stands out to me most, is Rory’s flexibility in mathematical thinking. Rory solves each problem in a different way. Had his teachers forced him to use an algorithm, Rory wouldn’t be able to think as flexibly. These skills will be a huge benefit to him as a problem-solver and as a mathematician! In fact, this is one of the weaknesses I often see in kids who are doing math outside of school, where they practice how to do math quickly, but not necessarily flexibly. Whereas if you allow kids to create their own methods, the understanding is deeper. (Van de Walle).
How will you know if your students are ready to learn the algorithm? They will have the following necessary precursors:
1) An understanding of place value.
Rory demonstrated this in Problem # 1 when he said, ‘pretend you only see the 2 and the 5, so 20 + 50 is 70’. In Problem # 2, he did the adding separately (4+6=10) and then answered with ‘so that means 40+60 is 100’. He will have no difficulty going back and forth when adding different place values in short-hand (i.e. 4+6) but recognizing that the answer (10) depends on the digit’s place in the number (in this case 10 tens or 100).
2) A comfort with solving problems that involve regrouping.
Rory demonstrated this when he tried to leave the ones place blank with the first problem; but when he added his ones (8+7), he realized he couldn’t put 15 in the ones place because it had two digits. In other words, he recognized that he had to regroup; he doesn’t know this is called regrouping yet, but he understands that is what has to happen. He showed an even stronger understanding when he tackled problem # 2. He was going to start left to right, but then realized the problem didn’t require regrouping so he could go from right to left instead.
3) An ability to think flexibly.
Rory used multiple strategies to solve these problems which means he will have no trouble with the multi-step nature of the algorithm. Problem # 1 he solves by going from left to right. Problem # 2 he solves by going from right to left (like the algorithm) and how does he tackle Problem # 3 (an extension using 3 digit numbers)? His solution is to decompose the second number and add each place value separately! This shows that he can transfer his skills to new problems (i.e. he applied what he has learned for adding 2-digits to add a 3-digit number). The surprising thing is that none of this was coached! He chose a strategy that he thought suited the problem at hand.
4) An ability to bridge 10 accurately.
Rory did this in Problem # 3 when he added 50 to 564, although he struggled to get the correct answer right away. This tells me he would benefit from some more practice with this; however, he easily did it with 2-digit numbers so I’m not too worried at this point.
Ready for the algorithm!
Now, can you picture his excitement when he finds out there is an algorithm that works on every problem; and it makes addition and subtraction more efficient and sometimes even faster?! It might look something like this:
Imagine if his primary teacher had forced him to learn the algorithm before he was ready. No enthusiasm. No freedom to problem-solve. No chance to build understanding. Not to mention the many lessons required to learn it. This is because students who aren’t ready require a lot of re-teaching. For example, if you watched Problem # 3, you will see exactly what I’m talking about. I had taught Rory about inverse operation this summer before he was ready. As a result, he doesn’t have the understanding to go with it, so he got confused and needed reteaching when he tried to apply it. This is exactly what would happen if I taught him the algorithm before he was ready.
So am I going to teach him the algorithm now that he is ready? Why deprive his Grade 3 teacher of all that fun?! But more importantly, why rush it?! Although it might be worth it to see that ridiculous face again!