Misunderstanding the Common Core Part XI

I stumbled across another misinformed, incorrect video purporting to show how “frightening” the common core standards are.

The mathematical issues brought up in this video are the same as the issues I address in many of the other posts about Misunderstanding the Common Core. Please read those. But, here are some new thoughts that this video created in me.

Thought #1 – The anti-common core movement really likes to pick on the idea of subtraction! I wonder why? There must be hundreds of other standards and yet, apparently, only subtraction brings up these emotionally charged challenges to the common core standards. Are we to think that if subtraction is “frightening” then all other standards are “frightening as well?” Is that good thinking?

However, as I have tried to explain in the other posts, the method of subtraction shown in this video is not bad! The lady in the video shows how 43 – 13 is computed. First, 3 – 3 is zero…write a zero below the 3’s. Then, 4 – 1 is 3…write a 3 below the 4 and 1. The answer is 30. Done.

The so-called common core method to subtract is a method related to counting back change. It is a method where one counts back, starting at 13, to 43.  Thirteen plus 2 is fifteen. Fifteen plus 5 is 20. Twenty plus 10 is 30. Thirty plus 10 is 40. Forty plus 3 is 43. Note, we counted back 30 units. Therefore, 43 – 13 = 30. No big deal.

Thought #2 – Even though I have no problem with the “counting back change” method as ONE POSSIBLE method to subtract 43 -13 = 30, this is not the “common core way” as reported in this video. I challenge any reader to show where, in the actual common core standards document, it says that subtraction must be done in this way or any particular way! In fact, I find the following standard:

To say that the common core standards require one method of subtraction over another is just not true.

Thought #3 – Just because a person doesn’t understand something does not mean this is a bad thing! I think that the lack of mathematical content knowledge of the general population in the US is a big reason for the so called “frightening” aspects of the standards. As I have tried to explain in all of these posts, there is nothing frightening about the standards! Rather than to react to something that one doesn’t know much about, I challenge everyone to become more aware of the standards. Read them. Ask questions. Study. Then if you have a legitimate, well-reasoned concern, then certainly engage in a respectful discussion about the issue!

This sounds like something we could all learn to do in other areas of our lives as well…

Scott

 

 

 

 

Common Core Standards ARE accepted by Conservatives too!

I am one of the most conservative political persons you can imagine and as such I have been quite frustrated with those that are aligned with me politically and the closed-minded stance that they have taken to the Common Core standards. From Conservative political candidates for president to politically active Conservative individuals, many have exhibited their ignorance (defined as: one with a lack of knowledge or information) in many instances. Therefore, it was refreshing to learn that the Lutheran Church Missouri Synod, of which I am an active member, formally published a document recently that demonstrated a positive and accurate reaction to the CCSS from a Conservative Christian perspective. Here are some of the important points made in “Should Lutheran Schools Consider Adopting the Common Core State Standards?”

1. Lutheran schools have a rich heritage and long standing commitment to academic excellence that is rooted in the Christ-centered mission that flows from the Lutheran Confessions. The Common Core State Standards may serve as a tool that enhances curriculum development. They could provide additional guidelines upon which the rigorous programs of study for our students could be designed. The rigor is not the product of the CCSS standards alone; rather, is connected with the instructional process provided by Lutheran educators.
2. The Common Core State Standards suggest a set of high-quality academic expectations that all students should master by the end of each grade level. They propose consistent grade-level learning goals for all students and inform parents about learning outcomes, thereby making it easier for parents to collaborate with teachers in helping their children achieve success.
3. The Common Core State Standards are not a curriculum. (emphasis added) A curriculum includes what, when and how subjects are taught and what materials to use. These matters are not dictated by the Common Core State Standards. For Lutheran schools, these elements will continue to be determined by individual schools, working to meet the needs of their students.
4. The Common Core State Standards represent a fundamental shift in the teaching and learning process. They establish clear, measurable goals for students that assist teachers in making instructional decisions. They place emphasis on creativity, critical and analytical thinking and application to curriculum content. They may serve to assist our schools in guiding the way that instruction takes place in each classroom while allowing the school to develop its own unique curriculum content.
5. The Association of Christian Schools International (ASCI), Christian Schools International (CSI) and The National Catholic Educational Association (NCEA) support the move toward the Common Core State Standards. The reason? – Its focus on rigor, relevance, best practices, and college and career readiness.
6. Ultimately, Lutheran schools will determine independently which standards to follow. The process of that discovery must remain true to The Lutheran Church—Missouri Synod’s (LCMS) Christ-centered mission: to provide a solid, Christian education developed with standards that drive instruction so that every child can be reached with an effective education – one that centers on the LCMS faith. The Common Core State Standards may be considered a tool that would allow LCMS schools to more fully prepare its students for service and witness to Christ and the world.

What I find the most reassuring about this document is that if one will take an honest look at the standards themselves with an open mind to assessing their quality removed from the political process and scheming that many see how powerful and helpful they are!

Trey Cox, Ph.D.

OpEd Piece by Jo Boaler

The following article was written by Dr. Jo Boaler:

Jo Boaler is a professor of mathematics education at Stanford University, co-founder of http://www.youcubed.org, and author of What’s Math Got To Do With It: How Teachers and Parents can Transform Mathematics Learning and Inspire Success. (Penguin, 2015)

Memorizers are the lowest achievers and other Common Core math surprises

Opinion

Memorizers are the lowest achievers and other Common Core math surprises

by JO BOALER

May 7, 2015

In this Feb. 12, 2015 photo, Yamarko Brown, age 12, works on math problems as part of a trial run of a new state assessment test at Annapolis Middle School in Annapolis, Md. The new test, which is scheduled to go into use March 2, 2015, is linked to the Common Core standards, which Maryland adopted in 2010 under the federal No Child Left Behind law, and serves as criteria for students in math and reading. AP Photo/Patrick Semansky

It’s time to debunk the myths about who is good in math, and Common Core state standards move us toward this worthy goal. Mathematics and technology leaders support the standards because they are rooted in the new brain and learning sciences.

All children are different in their thinking, strength and interests. Mathematics classes of the past decade have valued one type of math learner, one who can memorize well and calculate fast.

Yet data from the 13 million students who took PISA tests showed that the lowest achieving students worldwide were those who used a memorization strategy – those who thought of math as a set of methods to remember and who approached math by trying to memorize steps. The highest achieving students were those who thought of math as a set of connected, big ideas.

The U.S. has more memorizers than most other countries in the world. Perhaps not surprisingly as math teachers, driven by narrow state standards and tests, have valued those students over all others, communicating to many other students along the way – often girls – that they do not belong in math class.

The fact that we have valued one type of learner and given others the idea they cannot do math is part of the reason for the widespread math failure and dislike in the U.S.

Brain science tells us that the students who are better memorizers do not have more math “ability” or potential but we continue to value the faster memorizers over those who think slowly, deeply and creatively – the students we need for our scientific and technological future. The past decade has produced a generation of students who are procedurally competent but cannot think their way out of a box. This is a problem.

We don’t need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms. All of these ways of working are encouraged by the Common Core.

Mathematics is a broad and multidimensional subject. Real mathematics is about inquiry, communication, connections, and visual ideas. We don’t need students to calculate quickly in math. We need students who can ask good questions, map out pathways, reason about complex solutions, set up models and communicate in different forms. All of these ways of working are encouraged by the Common Core.

Technology leaders are publically arguing that calculation is not math, and that math is a much broader subject. Conrad Wolfram, one of the leaders of one of the world’s most important mathematics companies, Wolfram-Alpha, urges schools to stop emphasizing calculating and focus instead on problem solving, modeling, thinking, and reasoning as these are the mathematical abilities that students need in the workplace and their high tech lives. This broad, multidimensional mathematics is the math that engages many more learners and puts them on a pathway to life long success.

Part of the problem in the U.S. is the desperation of many parents to advance their children in math, pushing them to higher levels of math faster and sooner, somehow believing that a resume packed with advanced math courses will guarantee their future. Bill Jacob, a mathematics professor at the University of California, Santa Barbara, speaks openly about the dangers of students being pushed to higher levels of mathematics too soon. “I know it is hard to persuade parents that their students shouldn’t race to get calculus, but I really wish they wouldn’t. Too much content and depth is left out when they do.” said Jacob, who is not alone in saying that he would rather have students in his university mathematics courses that have breadth in their mathematical experiences than any additional Advanced Placement courses. Experts in England are giving the same advice to parents of high achieving students. Geoff Smith, chairman of the British and International Math Olympiads warns that accelerating children through the system is a “disaster” and a “mistake”. He, like me, recommends that high achieving students explore the mathematics they are learning in depth, instead of rushing forward.

Mathematics is not a subject that requires fast thinking. Award winning mathematicians talk about their slow, deep thinking in math. Fields Medal winning mathematician Laurent Schwartz wrote in his autobiography that he felt stupid in school because he was one of the slowest thinkers in math. Eventually he realized that speed was not important – “What is important is to deeply understand things and their relations to each other.  This is where intelligence lies. The fact of being quick or slow isn’t really relevant.”

Some school districts, such as San Francisco Unified, are trying to slow down the math experience, requiring that advanced students go deeper rather than faster. Students still reach calculus but the pathway to calculus consists of deep understanding rather than procedures and memorization. This is an important move. There is no harm in students being introduced to higher-level mathematics earlier, as long as the mathematics is enjoyable and ideas can be explored deeply. Third graders can be fascinated by the notion of infinity, or the fourth dimension, but they do not need a race through procedural presentations of mathematics.

New brain science tells us that no one is born with a math gift or a math brain and that all students can achieve in math with the right teaching and messages. The classrooms that produce high achieving students are those in which students work on deep, rich mathematics through tasks that they can take to any level they want. No one is told what level they can reach and no one is held back by narrow questions that limit students’ mathematical development and creativity.

Many people across the U.S. have gross misconceptions about mathematics learning, thinking that mathematics is a narrow subject of memorization and speed and that people who do not calculate fast or memorize well are not ‘math people’. We need to change the conversations about mathematics, communicating to all children that they can learn. We also need to change the way math is taught, valuing the different ways of thinking that are so important to the subject. Mathematics, itself, needs this and although change is hard, it should be embraced.

We need to broaden mathematics and open the doors of mathematics to all students. When we do this we will see many more creative, energized young people equipped to think quantitatively about our ever-changing world. We all need this.

Jo Boaler is a professor of mathematics education at Stanford University, co-founder of www.youcubed.org, and author of What’s Math Got To Do With It: How Teachers and Parents can Transform Mathematics Learning and Inspire Success. (Penguin, 2015)

This story was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Read more about Common Core.

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

The following picture has been going around Facebook as an example of how crazy the new Common Core Standards must be:

The argument is that in the “old fashioned” way, we can get the answer in just a very few steps. In the “new” way, it take many more steps and is more confusing. This superficial view and example of the common core standards misses the point entirely. Please consider the context by which either method might arise in the thinking of a student.  For example, suppose the situation was that Ken had $32 and spent $12. Ken took away $12 from $32 and one might be thinking in the so called “old fashion” way. This is the “take away” model for subtraction. Suppose the situation was that Ken has $32 and Scott has $12. How much more money does Ken have than Scott? We are making a comparison (comparison model of subtraction). To make the comparison…to find the difference between how much Ken has and how much Scott has, one might start with Scott’s amount ($12) and add up to your amount: 8 more than $12 gets me to $20…another $12 gets me to $32…a total of $20 more. There are misconceptions that we are teaching new ways over old ways…not true! We are recognizing how one might think about the solution to a problem and that, depending on the situation, we might think in different ways.

People sometimes look at this example and make comments like, “no wonder our kids are confused when it comes to math!”

Actually, the issue is this: if the problem says (as in my second example) “how much more money does Ken have than Scott?” students see the word “more” and want to add 32 + 12! They are taught to look for “key words” like “more” which means to add. What I am saying is that we want students to think and make sense of the situation and solve the problem accordingly. My thinking in the “how much more” question might be to start with my $12 and add up to Ken’s $32. In the “Ken spent $12” question, I am thinking about subtraction because I am thinking about taking away $12 from the original $32. We are not showing new ways to subtract…we are solving problems in ways that might make sense.

By the way, in the given example, a young learner might count up to $32 from $12 in more or less sophisticated ways. In this case, it appears that that the student first counted up to $15 so that she could then count by 5’s (which may be more comfortable than counting by 8’s or 12’s). Once they counted up to $30, she only needed to count 2 more to get to $32 for a total of $20 more. This is great reasoning that respects that particular child’s thinking process!

For those that think that the common core is all about teaching “new” and confusing ways to “do math”, don’t worry…we are still teaching “old fashioned” subtraction! But, we are also teaching students to think, reason, make sense, solve problems, etc. You might consider the Common Core Standards for Mathematical Practices. This is the “core” of the Common Core:

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.

Scott