Archive for June, 2014

Here we go again…another false statement about the “dumbing down” of American kids…about doing math in ways that seem foolish…about the “idiocy” of the Common Core Standards.

If you want to see the original post, go here: toprightnews.com/?p=3958

Most of what is discussed here has already been addressed in Parts I – IV. But I will use this post to highlight some of the issues and why I claim that folks like this (in this case, Jason DeWitt of Top Right News) are misrepresenting the Common Core Standards and that the root of the issue is their misunderstanding of the Common Core Standards.

Point #1 – The standards do NOT tell teachers/students how to do arithmetic

In the article, we see the following:

One student got tired of being dumbed-down by Common Core’s convoluted “standards” to do basic arithmetic.

“Standards” like how to add two numbers, which the student was told to do like THIS:

I challenge any reader to show me where in the Common Core Standards that students are told to “do basic arithmetic” in any specific way. Certainly, you will see standards that encourage students to make sense of mathematics…to understanding computational algorithms…to explain their thinking. Also, you will see standards that encourage computational fluency. What this is really an example of is how young learners might begin to make sense of the traditional algorithm for adding two numbers. Mathematics educators would NOT expect students to remain at this stage…rather, we expect students to develop computational fluency by thinking about the mathematics involved. Here is the example shown in this article:

CCSSM #5

I must be crystal clear…this is not an example of how students HAVE TO do this computation…this is an example of how a student MIGHT THINK about the computation.

Can we all agree that it is easier to do mental math that involve numbers like 20, 30, or 40? So a student might think…”26 is four less than 30…so let’s think about using the number 30.” Since  we are adding 4 to 26, we would have to subtract 4 from 17 in order to get an accurate result for the final sum. Instead of thinking about 26 + 17, think about 30 + 13…which is 43.

I challenge you to ask 10 adults how they might think about 26 + 17 mentally…some will do this method! Some will imagine the algorithm in their minds eye (6 + 7 = 13…carry the one…2 + 1 + 1 + 4…so 43). Some might take 17 up to 20 by adding 3…which means we would have to subtract 3 from 26 to get 23…therefore, 26+17 is the same as 23+20 = 43. Can we all agree that 23 + 20…or 30 + 13… are easier to do mentally than 26 + 17?

Point #2 – There is no such thing as “the common core way”

In the article, another example is given and the following statement is made:

So when he was given his next basic arithmetic assignment, to find the difference between 180 and 158 (180-158), this 5th grade student just did it his own way — the right way – 

Exactly…the student “did it his own way…” That’s what we want! We want students to make sense of the mathematics involved in these computations and to develop a fluent and efficient way to compute. While the article makes this statement as a way to diss the common core…I argue that this is exactly what the common core would encourage! As long as “his own way” was a way that makes sense to him and to others.

Ok…ok..I get it…the point of the article is that the student “stuck it to the common core man” by explaining how he found the answer…using one word…MATH! How cute…how clever…

Please…common core bashers…explain to me…what is so wrong with asking a student to explain his/her thinking? My hunch is this…you can’t do it so you don’t want your kids to do it. That is, you are uncomfortable explain your thinking so why should your kids have to explain their thinking? And if your kids are asked to explain their thinking and you don’t know how to help them do it, then there is something wrong with the assignment/teacher/standards/textbook/test/whatever.

Hmmm…mathematical thinking…not mathematical doing…mathematical thinking…

Kaye Stacey writes about mathematical thinking this way

The ability to think mathematically and to use mathematical thinking to solve problems is an important goal of schooling. In this respect, mathematical thinking will support science, technology, economic life and development in an economy. Increasingly, governments are recognising that economic well-being in a country is underpinned by strong levels of what has come to be called ‘mathematical literacy’ (PISA, 2006) in the population. 

Do we want students who can “use mathematical thinking to solve problems”? Do we want to encourage our kids to engage in mathematical thinking to support “science, technology, economic life…”? Then we must stop showing kids how to do mindless procedures and instead help them to develop the mathematical thinking that leads to procedural fluency. Procedural fluency does not just happen magically. It is developed…and can be developed in such a way as to also develop and support mathematical thinking.

Think about the last time you had to divide two fractions…like 7/8 divided by 3/4.

First…this probably last occurred on your 7th grade final exam! Just kidding…but think about it…would you just “invert and multiply”? Is it important to understand the mathematics entailed in this computation? I say yes….

Scott

Raising Arizona Kids

Posted: June 16, 2014 in Uncategorized

This article appeared in the online magazine in June 2013:

Math teachers prepare for Common Core standards

By Daniel Friedman | June 3, 2013

Your kids need to be smarter and better educated to get decent jobs when they graduate from college. Many states, including Arizona, have adopted Common Core State Standards to better prepare children for the job market.
Implementing new standards in education isn’t like flipping a switch. At the most basic level, teachers need to know if the textbooks and teaching materials they already have will work for the new standards. On a more complex level, they need to modify their the lessons, sequence and teaching approach to meet the new standards.

The Arizona Mathematics Partnership, led by Scottsdale Community College, was awarded an $8.7 million, five-year grant by the National Science Foundation to help 300 middle school math teachers understand and teach the new mathematics Common Core Standards.
Math gets more difficult during the middle school years. Algebra rears it’s ugly head and equations move beyond basic operations. Parents often bow out of helping their kids in middle school because it gets more complex and they haven’t factored polynomials recently.
I spoke to Scott Adamson, a math teacher at Chandler-Gilbert Community College. He is working with teachers this summer at a five-day Summer Institute at Scottsdale Community College, but that is just a small part of the program. He also visits teachers at their schools, observing lessons and giving feedback on how to implement the new math standards.
The new standards include the usual content standards. like this one for sixth grade:
6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
This is a standard for third grade:
3.OA.7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8 ) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
Adamson says the current math textbooks don’t really line up with the new standards but having teachers create their own entirely new lessons to match the new standards isn’t practical. It takes hours to design a lesson, create the materials and the assessments to go with it.
He works with teachers to help them implement the new standards as efficiently as possible. If they try to create all new lessons “they’ll be burned out by October,” he says.
In addition to the content standards, the Common Core standards include Mathematical Practices:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

Adamson says getting the correct answer is not enough “The heart of it is, what do we want out of our students? It is not to mindlessly perform steps and arrive at answers without understanding why and how they got their answers,” he says. One example he gave is students should understand why multiplying a negative number times a negative number yields a positive number. (You remember that, right? -3 x -7 = 21)
Really, the new standards with the mathematical practices are building an intellectual framework within which the content standards are learned and understood, says Adamson. A calculator won’t help “construct viable arguments and critique the reasoning of others.”
The new standards will demand more of your children as well as their teachers.
Teachers in the Arizona Mathematics Partnership program will get 200 hours of professional development, and a $2,500 stipend for their efforts. In addition to SCC and Chandler-Gilbert, other partners in the project are Glendale Community College, Chandler Unified School District, Deer Valley Unified School District, Florence Unified School District, Fountain Hills Unified School District, J.O. Combs Unified School District, Salt River Pima-Maricopa Indian Community Schools and Scottsdale Unified School District.
Parents will need to support their kids and send them to school willing and able to work to meet the new standards in math and English language arts which will be assessed for the first time in the spring of 2015.
Read the Arizona Common Core Math standards.
Find out everything you can about the standards.
Tags: Arizona Mathematics Partnership, Arizona schools, Common Core Standards, math, Scottsdale Community College, teachers

Daniel Friedman
Daniel Friedman is a staff writer and photographer for RAISING ARIZONA KIDS magazine.

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