Misunderstanding the Common Core Part IV

Posted: June 8, 2014 in Teaching and Learning Philosophy
Tags: , , ,

I was recently made aware of the following video bashing common core math…

There are many reasons why they video represents a misunderstanding of common core math. Let’s start with some possible mathematical understandings represented in the video. The host, Caleb Bonham, shows people how common core purportedly wants students to calculate 134 – 52.

Right away, Mr. Bonham says “four minus 2…2, 4 minus 5…carry the one…” What? Carry the one? Imagine a student learning (and by learning I mean mimicking) this algorithm for the very first time.  If Mr. Bonham was the teacher, he would say, “alright kids…4 minus 5…carry the one…” What is the mathematics that supports this? What does it mean to “carry the one”? Why is he carrying the one? Where is this one coming from? Is it one because somehow 4 minus 5 involves the number one?

When showing the so called “common core way” of computing the difference, he says, “you can’t just do it like this (the traditional algorithm)” and he proceeds to show what he believes is the common core way in a very mocking way. What he fails to realize is that this is not “the common core way.” What he is showing is just one way to make sense of the traditional algorithm! If students are thinking (note…this is what we want students to do…not just “carry the one” without any understanding or reasoning) about subtraction as taking away one quantity from another and if students are learning about the idea of regrouping (aka borrowing) for the first time, then the representation he shows makes a lot of sense! The dots provide a visual representation of the computations in the traditional algorithm…very useful for a young learner who is trying to make sense of the ideas.

The student starts by subtracting (taking away) 2 of the dots from the ones place. Great! The student moves to the tens place and needs to take away 5 tens…but only has 4 tens…what should they do? In the traditional algorithm shown by Mr. Bonham, he proposes that the student just crosses off the one in the hundreds place and “carries” it to the tens place. He says, “thirteen minus 5 is eight.” While this produces the right answer in the end, the articulation of the mathematics is wrong (I will explain below). The dot method he shows explains the mathematics.

When the student comes across the dilemma of taking away 5 tens from 4 tens (in reality, taking 50 away from 40), the solution is to look to the hundreds place and exchange one-hundred for 10 tens. The ten dots (each representing a quantity of 10) is placed in the tens column. Think about it…13 tens is one-hundred and thirty which is equivalent to one 100 and 3 tens. But now, we can take away 5 tens (50) from 13 tens (13) leaving 8 tens (80). So, when Mr. Bonham “carries the one”, he is actually exchanging one 100 for ten 10’s so that the subtraction (as take away) is possible. Nice!

Mr. Bonham makes a mathematical error when he asks the gentleman, “how many 10’s in 134?” and they respond “3” and make “three little dots.”

3 tens

Let’s think about this…something that the common core wants students to learn to do…to make sense, reason, etc. How many tens are in ONE HUNDRED AND THIRTY FOUR? Only 3 (I think the man on the right says 30!)? What? I would say that three tens make 30! That’s not even close to ONE HUNDRED AND THIRTY FOUR! Let’s think about that again…how many tens are there in ONE HUNDRED AND THIRTY FOUR? If we had 10 tens, that would be 100…13 tens is 130…so there are 13 tens in ONE HUNDRED AND THIRTY FOUR…13 tens and 4 ones. Suppose a student was thinking (not just trying to mimic an algorithm) about both numbers in this way. That is, we have 13 tens and 4 ones…take away 5 tens and 2 ones…13 tens take away 5 tens is 8 tens (80) and 4 ones take away 2 ones is 2…so the answer is 82! No algorithm needed! Now that is efficient! If efficiency is what we are after, then let’s do it my way!

Mr. Bonham likes to make the point that doing math is all about being efficient. Let’s compare his method with mine…

efficient

On the left is the traditional method shown by Mr. Bonham:

  1. Think…what’s 4 minus 2? Write down 2
  2. Think…what’s 3 minus 5? Can’t do it so carry the one (whatever that means)
  3. Think…what’s 13 minus 5? Write down 8 (note…it is really 13 tens (130) minus 5 tens (50)…not 13 minus 5.

On the right is the way I describe:

  1. Think…what’s 13 tens minus 5 tens? Write down 8 in the tens place realizing that 130 minus 50 is 80.
  2. Think…what’s 4 minus 2? Write down 2.

One less step…and much more understanding!

Now…here is the real bottom line in my opinion…Mr. Bonham makes a claim that there is the “efficient way” (according to him) and the “common core way”. That is just not true. The common core pushes for a balance between conceptual understanding (perhaps thinking about place value and understanding the mathematics behind carrying the one) and procedural fluency (computing answers quickly and accurately). It is absolutely false to believe that the common core is proposing some long, inefficient method to compute. But…how does one develop computational fluency? Young learners would need to do things like use manipulatives, draw dots, represent their thinking in a visual way, explain their thinking. Then…and only then…can a young learner possible move to using Mr. Bonham’s algorithm or perhaps even better (it IS more efficient) using the left-t0-right algorithm that I proposed above! Computational fluency…the “efficient way” does not just magically happen!

A challenge: download the common core math standards at http://www.corestandards.org/Math/. Search for the word “fluency” and see how often students are expected to demonstrate computational fluency! Or, continue to stick your head in the sand and listen to people like Mr. Bonham who probably have never even read the common core math standards!

Scott

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Comments
  1. Chuck Schwan says:

    Alright, I’ll bite:

    The common core method makes it very hard for a parent to assist their child in the learning process. Even with the parent/family help sheets sent home by the teachers I am often left scratching my head on how my 2nd Grade (now 3rd) son comes up with the answer and worse, I can’t help him because it confuses him.

    Keep in mind I am a very mathematical individual. I breezed through basic math, algebra, geometry, trig and calculus in school. Typically my brain is faster than the calculator.. to the point that my wife makes fun of me for it (in a joking way of course).

    If I was given the option to never have my son taught common core I’d sign up for it in an instant.

    • getrealmath says:

      Thanks for your reply!

      I am curious…do you see any mathematical benefit in making sense of the subtraction algorithm as shown in this video? For example, when they “carry the one” a young learner may think, “oh…I get it…I cross out the one and put it above the 3.” But what if the number was 234…would you cross off the two and put it above the 3? No. What is the mathematics of “carrying the one” as demonstrated in the video?

      Also, did you take the challenge of reading the common core standards? The word “fluency” is throughout the entire standards! But…how does one become fluent? I argue that fluency comes (and includes) a mathematical understanding of what the procedure is doing.

      One more point…there is no such thing as “The common core method”. The common core are standards…methods will be provided by students, teachers, curriculum, classroom tasks, etc.

      You might be interested in my blog post about how parents can help students with their homework.

      Scott

  2. Antaeus Feldspar says:

    One thing I don’t understand about your method: how would it work if the problem was not “134 – 52” but “134 – 56”? If I started by subtracting 8 from 13 to get a 5 in the tens place, I’d be left with “4 – 6” which works out to “-2”. I can’t see how working with negative numbers results in an algorithm more suitable for students who are still learning subtraction than “carrying the one”.

    • getrealmath says:

      Hi!

      Before I explain, I want to be clear that I am not promoting one method over another. In my thinking, it is not about somehow choosing “the best” method that teachers should show their kids, the focus is on making sense of mathematics…which ever method is being used. Then, students can develop computational fluency (efficiency).

      In your example – 134 – 56 – students could see that 134 contains 13 tens and 56 contains 5 tens. 13 tens minus 5 tens is 8 tens. So far, we have 80. As you say, 4 ones minus 6 ones is negative 2. So now we have 80 from the tens place and -2 from the ones place…a total of 78. So, 134 – 56 = 78.

      I think it is interesting to compare what happens when we have a negative number in the process to when we don’t. In the first example, 134 – 52, we can subtract 5 tens from 13 tens and get 8 tens (80 so far). We subtract 2 ones from 4 ones and get 2 ones. A total of 82.

      Likewise, in the case of 134 – 56, we total the results from subtracting in the tens place (80) and the ones place (-2) and get 78. Same idea.

      To reiterate my point, I am not saying that the traditional algorithm of borrowing is bad and that teachers should teach this “work from left to right” method in its place. But…it does work!

      Scott

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