Misunderstanding the Common Core Part VII

Here is the latest (for me) bashing of the common core standards:

http://toprightnews.com/?p=6326

Again, the subject of the bashing is subtraction…I find it interesting that subtraction seems to be taking the biggest hit when it comes to common core bashing!

The blogger says, “Here is the insane “method” they came up with to utterly confuse 4th graders about subtraction:”

Insane? Utterly confuse? I think not!

Consider this: Albert Einstein was born in 1879 and died in 1955. How old was he when he died? Think about it…then pay attention to how you thought about it…

Many adults think about this situation like this:

• Starting with 1879, it takes 21 years to get to 1900.
• Then, another 55 years takes us to 1955.
• So, Einstein was 21 + 55 = 76 years old (or would have turned 76) in 1955.

This is the “counting up method” described in the textbook! This method seems SANE and UTTERLY sensible to me!

As I have explained in other posts (Part I, Part II, Part III, Part IV, Part V, Part VI), the main issue is how one thinks about subtraction. If one only thinks about subtraction as “take away” then their understanding is incomplete. We can also think about subtraction as a comparison…which is really what is happening here. We are comparing 1955 to 1879 to see how many years have elapsed. Furthermore, think about the method (yes, it is a method!) described early…first add 21 years…then add 55 years…and compare that method to the standard subtraction algorithm from a “take away” perspective.

With the double borrowing, mistakes can be made. In addition, I am a fan of mental math…it can absolutely be a helpful and healthy thing to keep track of quantities mentally!

The goal…and who can really argue this…is for students to become fluent, flexible, and efficient with computations.

• Fluent in a way similar to becoming fluent in a foreign language…comfort, ease, speed, and confidence are all hallmarks of fluency.
• Flexible in the sense that a student will use an algorithm if necessary, use a counting up strategy if possible, or other strategies learned in school.
• Efficient in the sense that a student will choose a strategy that is appropriate for the task at hand. For example, in the Einstein example, I argue that the counting up strategy is more efficient than the traditional algorithm…at least for those who have been trained to think that way!

Scott

Misunderstanding the Common Core Part IV

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.”

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…

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

Misunderstanding the Common Core – Part II

I say Part II fully expecting that there may be Parts III, IV…LVI…coming just like the Super Bowl! There is much misunderstanding and bashing that is not justified when considering the actual content and practice standards.

Consider an image that is going around Facebook:

Misunderstanding #1: This is NOT an example of common core somehow promoting some new, “ridiculous” way to perform subtraction. In fact, a big part of the common core in the elementary grades is computational fluency. That is, students absolutely should be expected to subtract the problem presented here in “under 5 seconds”!  To say otherwise shows a complete ignorance of the common core standards. Go read the standards and their supporting documents and then feel free to engage in a dialogue about how they can be improved.

Standards – http://www.corestandards.org/Math/

Progressions – http://ime.math.arizona.edu/progressions/

Misunderstanding #2:  Many of the Facebook comments to the image are of the form, “yea…why not just show the kids how to do the problem in under 5 seconds??!!!?” As stated earlier, computational fluency is the goal…but how do kids get there? They get there when they are able to think about the problem, make sense of the situation, engage in methods that make sense and are grounded in foundational mathematical thinking such as place value and the meaning of subtraction. Developmentally, 2^nd graders are not ready to just jump into the formal, traditional algorithm. To deeply understand how to subtract, students are encouraged to use appropriate tools strategically so that they can work toward a deep understanding of the formal algorithm.  Young learners can benefit greatly from the use of manipulatives, drawing, and the use of number lines to help them to understand subtraction. For example, young learners might start with base-10 blocks and represent the problem as follows. They will see that they need to “take away” 316 from the starting value of 427. Using the blocks, they can model 427 and then take away the needed amount (316) to see the result of 111. Later, as shown, students might draw out the blocks as they transition from the concrete manipulatives to paper and pencil to mental math. To arrive at the goal of fluent mental math computations, students need to have had developed mental images of what they are doing mentally! Otherwise, there is nothing mentally to connect to and the goal of fluency cannot be achieved.

1. Base-10 blocks: The “1” block is just a single tile. The “10” block contains ten “1” blocks stacked vertically. The “100” block consists of ten “10” blocks stacked horizontally.

2. Model the quantity 427.

 3. Take away 316 from 427 (represented by the red X’s).

4. Write the value of the remaining blocks.

 

With time and repeated reasoning (not repeated mimicking or doing without thinking), students will transition from using the blocks to sketching the drawings to just doing the computation in “under 5 seconds”.

Note: This example did not require “borrowing” (aka regrouping). The base-10 blocks and/or drawings can help students to develop a deep understanding of this as well. Maybe this can be part of the next post!

Misconception #3: Not all problems require thinking about subtraction as “take away”. Subtraction can also be thought about as an additive comparison. The common core helps students to develop strong problem solving strategies and this is a great example. Suppose the problem was something like:

Samantha had $427 and spent $316. How much money does she have left? This problem is clearly a situation where $316 are being “taken away” from $427.

What if the problem was something like:

Samantha has $427 and her sister Gloria has $316. How much more money does Samantha have than Gloria? This is not a “take away” situation. This is a comparison situation. We might think about this situation by counting up from $316 until reaching $427. Or, we might count down from $427 until reaching $316. With young learners, a number line can help them to visualize this strategy.  As students become more and more sophisticated with their thinking, they might count down by 100’s, 10’s, and/or 1’s as needed. If a student were to verbalize their thinking, they might say, “start with 427 and count down by 100 to 327…count down another 100 to 227…count down another 100 to 127. Then, count down 10 more to 117. Finally, count down 6 more…117, 116, 115, 114, 113, 112, 111.” Samantha has $111 more than Gloria.

I re-emphasize that we will not leave the learning at this place. Good teachers who are implementing the common core will continue to deepen student understanding by making connections to subtraction. Note that as we compare $427 and $316, we can argue that both quantities have at least $316. The difference (and that is why we call the result of subtraction the difference!) between the two quantities is the $111 additional that Samantha has!

Misconception #4: Some people might think that the status quo is fine with regards to teaching math. We should just “teach” students mathematics in ways that it has always been done. As a life long mathematics educator, I just can’t support this view. I currently teach at a community college where many students enter unprepared for college level mathematics. Internationally, the US scores poorly in mathematics in comparison to other nations. Students make decisions about college majors and careers based on programs of study that require the least amount of mathematics. Math anxiety is a real problem and is often caused when people try to mimic procedures that don’t make sense. When things don’t make sense, students struggle to succeed and have negative feelings toward such things. People complain that the cashiers at McDonald’s can’t make change.  This has been the case without common core! If people have a complaint about how the common core standards are organized, it would be wonderful to hear a better plan!

In summary, the common core standards absolutely do promote fluency (fast and efficient computations). This does not happen on its own but has to be developed! The common core, through the Standards for Mathematical Practices, provides students the opportunity to develop a deep, well-connected understanding of fundamental mathematics so that they can use mathematical tools and skills to solve problems.

Scott