Knowing How and Knowing When

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

This post could be titled Arithmetic Meets Mathematics.  What’s the difference?  Arithmetic is commonly thought of as the nuts and bolts of mathematics.  It’s the computation piece. Mathematics is the broader picture.  It’s the relationships, patterns, etc.  In other words, arithmetic is usually considered a subset of mathematics.  I read somewhere on the Internet that arithmetic is to mathematics what spelling is to writing.  I thought that was a great analogy.

The problem is that often arithmetic, the computation algorithms, are taught in isolation.  Let me think about how I want to say this…NO!  We have to give equal weight to both computation and context.  It does no good for a child to know the algorithm if they can’t recognize which operation to use.  Likewise, if a child recognizes a problem as subtraction but can’t subtract, they will still miss the problem.

So how do we make sure our kiddos know how and know when?  First, we start at an early age developing strong number sense and fluency with numbers, and we get them modeling (concrete) and drawing (representational) problems as soon as them can hold a pencil.

Consider the problem below.  It’s going to be tricky for a lot of kids because it’s not take away subtraction.  It’s not even comparison subtraction.  It’s a missing part problem.  So the point is here is that we have to have variety in our problems.  Now, if kids have been having part-part-whole experiences from a very early age, this problem is no big deal.  Once they recognize that they are looking for a missing part (be careful of saying it’s subtraction…), they still need to find a solution.  If a child chooses to subtract using the standard algorithm, they need to regroup. Yikes!! I guess that rules out 1st graders doing a problem like this, right? What if a child could look at this problem and say, Well, if I add 2 to 8, that gives me 10. And from 10 to 25 is 15, so 15 + 2 is 17. Seventeen balloons were not red. That’s some mean number sense, and a 1st grader with strong number sense could do it.

Computation simply cannot  and should not be taught in isolation.  And we can’t keep skipping the concrete and representational stages to get to the algorithm (abstract).  Let kids see the connections and the beauty in math!

Check out these scripted mini-lesson units for teaching computational skills using CRA and in the context of real-world problems:
2-Digit Subtraction with Regrouping
Teaching 2-Digit Multiplication (Concrete, Representational, Abstract)

7 Comments

  1. luckeyfrog

    I *just* posted about how important it is for us to challenge kids with different types of problems, and not just the ‘how’ but also ‘when’ to use it! (I love the way you worded it, though!)

    What resources do you use with model drawing? My former school used Math in Focus (commonly called Singapore Math) and liked some of the elements, and I borrow from it now that I’m in a different school… wondering if you use the same process or another! 🙂

    Jenny
    Luckeyfrog’s Lilypad

    Reply
    • Donna Boucher

      Hey Jenny! I have always just drawn problems–kind of free form, but recently I’ve incorporated more of the Singapore process to make the models more consistent and focus more on parts and wholes. I think you have to adapt any process to what makes sense to you!

      Reply
  2. Tammy

    My kids just did a problem like this. (There are 26 letters in the alphabet. 5 are vowels. How many are not vowels?) I didn’t see any sophisticated strategies from them, but they figured it out without any troubles.
    ❀ Tammy
    Forever in First

    Reply
  3. Tammy

    My kids just did a problem like this. (There are 26 letters in the alphabet. 5 are vowels. How many are not vowels?) I didn’t see any sophisticated strategies from them, but they figured it out without any troubles.
    ❀ Tammy
    Forever in First

    Reply
  4. Steve

    Seems like it needs to say “a total of eight” for the problem to be answered. Otherwise, more than 8 COULD be red. Maybe 12 are red and the problem only describes the first eight. Precision in stating the problem is nice, too.

    Reply
    • Donna Boucher

      If a first grader thought that way and could explain his thinking, I would be thrilled! I think it would open up a great discussion.

      Reply
  5. Terri Fontenot

    We are doing our addiction subtraction unit now (3rd grade) and we are all attempting a guided math framework in our class. We were just talking about the “naked numbers” and the problem solving piece! I moved down from 4th and it has taken me a minute to realize where they are supposed to be and I am/was struggling with the group set up, but the last two days have gone so well! I am able to see 2-3 groups in my (approximately 60 mins) and we I can see where they are and adjust groups quickly! We do a word problem going through UPS (understand plan and solve) every day at the beginning of class! It was amazing to hear them talk through it yesterday! At first I was hearing, “it is subtraction because it says altogether or how many more,” but when I said “if I change the wording, is it still subtraction?” They said, “yes! Because you are finding the difference between or you are taking away from!” We are also teaching them interchangeably! Not in isolation- as the inverse operations! It had been a learning curve for 3 weeks getting expectations set up, but I think it will help in the long run!

    Reply

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