How a second grader got me to stop teaching and start listening

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 Last year, I had a second grader who came in having no concept of subtraction.  In my beginning of the year assessment, he could not answer questions such as 7-6 = or 10-5.  After a great deal of work on conceptual development and hands on practice, he was doing a great job with his subtraction facts.  Except for some of the most challenging "facts" that our program teaches as the up to 10 facts.  (basically a teen number take away a 6, 7, 8 or 9.  Despite repeated attempts to "teach" this student "how to do it" he just wasn't making any progress.  After backing off on "teaching him how to do it" for a few weeks and doing more exploring of lots of different solution strategies with other students, he suddenly could answer all of these types of questions and the best part was that he was doing it fluently.  Could he explain the up to 10 strategy to me?  NOPE!  He invented a strategy of his very own, and I think he had been doing it all along.  I think my teaching him to do it my way was all symbolic to him and gave no meaning to the problem.  Let's take a look at his strategy and see how he has used it as he has moved into grade 3 and more complex subtraction problems.

At the Fact Level

This is where this student first invented this strategy.  See the picture below for this example (17-8) He sees a number such as 17 as a ten and seven ones.  GREAT!  He knows he has to take away 8 ones.  He says to me, there isn't enough ones to take away 8, so I am going to take the 8 ones away from the ten.  That leaves me with 2 plus the 7 ones I ignored.  2 + 7 = 9.  He does all this mentally and fluently!!!  He is able to explain his thinking to me in a clear and concise manner.

Another example: 15-7.  You can't take 7 ones away from five ones so take7 ones away from the 10. That leaves 3 ones left from the ten plus the 5 I ignored.

I can't tell you how much of a breakthrough this was for him (and me!)  I really wonder how long he had been solving problems this way and I was thinking he wasn't doing them right!  Did he try to explain this to me before-hand and I told him it was the wrong way to do it?  I don't know.  I do know that a current second grader has been solving problems like these the same way!  How many years have I had kids doing this without even realizing, or acknowledging their strategy?

Taking it up to the next level

By the end of grade 2, I was working with this student on a lot of 2 digit minus one digit type problems like the ones shown below.  To my surprise (and delight) this student extended his fact strategy and it worked very well for these types of problems.

Look at the example of 23-7.  "I can't take 7 ones away from 3 ones so I am going to ignore the 3 for a minute and take 7 ones away from 20 which is 13.  Now I put the 3 I ignored back with the 13 and that is 16."

Did you follow that?

Another example.  42-8. "I can't take 8 ones from 2 ones so I am going to take 8 ones from 40, which is 32.  Then put the 2 back I ignored and it is 34.

 Of course, this student does these problems mentally and writes down just his answer.  The horizontal number sentences you see off to the side are just my notes about his strategy.  This kid is a tier 3 kid in math and needs a lot of intervention.  It is pretty amazing that he invented this strategy and that it works so well. 

Moving onto the very end of grade 2 and a lot of grade 3 is more multi-digit subtraction.  When we began working on 2 digit subtraction, he continued to use his strategy.

In this example, he says "I can't take 46 away from 2 ones but I can take it away from 80.  80-40=40 then minus 6 more is 34.  When I put back the 2 I ignored, it is36."

Three digit computation

Here is a look at how he uses his strategy with 3 digit numbers.  See picture below.  The red example is 138-79.  He takes 79 from 100, leaving 21 and then puts the 38 back and gets 21 + 38 = 59.  Still doing these completely in his head.

Another example.  Take 176 away from the 200 which leaves 24.  Put the 40 back with the 24 and that is 64

The Pinnacle of Third Grade Subtraction

As we head towards the end of his third grade year, we have been spending time with four digit subtraction. I was thinking he would abandon his strategy on this but it actually works quite well for him even with very large numbers.  Take a look at the picture below. 5350-1995.  Wow.  This is a problem I see much older students getting wrong all the time.  This student did it in under 30 seconds without using a pencil or writing anything down.  "5000-1995 is like 5000-2000 but then you have to put 5 back so it is 3005.  Then I add back the 350 I ignored and it is 3355." Wow!  Did you follow that?  Isn't that a great way to do this problem mentally?

This is traditionally seen as a triple borrowing problem and causes kids a lot of headaches.  The traditional algorithm can get messy and kids really need to understand what they are doing when they do all that borrowing.  This kid's strategy uses place value and combinations of 10 and 100 and can all be done mentally!

The big take away from working with this kid for me has been stop trying to teach strategies and let kids invent them!  This gives them ownership of their learning and helps them develop really strong conceptual understandings that will take them a lot farther than learning a trick.  See how his idea works for all subtraction problems?

Have you ever had a hard time understanding a student's strategy?  What did you do about it?  Please share in the comments section below!