Posts Tagged ‘common core standards’

Here we go again! I just saw the following on Facebook:

CC X

What was wrong with the old way? Before I answer…IT”S NOT THE OLD WAY! It’s just A WAY! If you want to read much about misunderstanding the common core, consider the other posts with a title “Misunderstanding the Common Core Parts I-IX”. I will reiterate, recap, and re-explain for anyone new to this blog.

When we subtract as shown at the bottom of the image, 568 – 293 = 275, there is a lot going on in terms of understanding the algorithm. Understanding the algorithm is what allows us to be efficient and accurate in computing by hand. Which brings us to an interesting issue: most people would use a calculator to do this!

Suppose you were going to attempt to compute 568 – 293 in your head. Or, suppose you were estimating. You might start with 568 – 200 = 368 just to get the thinking started. You might estimate/round 293 and say it is nearly 300 so 568 – 300 is about 268. Here’s my point…the most important digits are the digits in the 100’s place not the 1’s place! It is much more natural to start with the 100’s place than with the 1’s place as is traditionally done.

Let’s get back to it…once we think 568 – 200 = 368, the next thing to think about is the 10’s place. Let’s subtract 90 more. 368 – 90….hmmmmm…well…368 – 60 would be 308….hmmm…10 more…308 – 10 = 298….hmmm…20 more…298 – 20 = 278. It’s hard to do in writing but what I am trying to convey is that when people think about computations mentally, they might first focus on easier computations…subtracting 60, then 10, then 20 is much easier to do in my head that subtraction all 90 all at once.

So, we have subtracted 568 – 290 so far and have arrived at 278. Now subtract 3 more…278 – 3 = 275. There  you go! 568 – 293 = 275!

It would happen much faster in my head in reality than it seems when you read through everything that is happening in my head. Here it is in one, smaller space:

568 – 293….well, 568-200 = 368….now 368 – 90 is 368 – 60 = 308 and 308 – 10 = 298 and 298-20 = 278. Finally, 278 – 3 = 275…done…and fast!

What’s wrong with the old way? Nothing! What’s wrong with thinking, focusing on place value, mental math, understanding algorithms, and number sense? Nothing!

Here is the bottom line to me when it comes to this particular image. When we try to explain our thinking and show all the details of doing a simple subtraction problem, it might look convoluted and confusing. When we really think about how things work and try to understand the structure of the algorithm, it can make perfect sense!

One more thought…at the bottom of the image, we see this…

common core x part 2

People look and say…wow, look how easy that was compared to that new, crazy math that my kids are coming home with! Yea, right! There is so much missing that many 3rd graders would not understand!

Likely, people would start on the right and find that 8 – 3 = 5. Fine.

Next, people would look at the 10’s place…6 – 9…hmmm…you can’t take 9 from 6! Sometimes kids see this and their brains tell them to think about 9 – 6 and they put 3 in the 10’s place! But we know, you have to “borrow” from the 100’s place and actually subtract 16 – 9 = 7 (note…I am ignoring place value here since that is the traditional thing to do).

Finally, move to the 100’s place. Remember we borrowed 1 (ignoring place value and meaning) from the 5 to make it 4. So, 4 – 2 = 2 so put a 2 in the 100’s place.

There is so much going on here that a 3rd grader has to figure out. The creator of this image chooses to omit it in order to present a stark contrast from “the old way” to the “common core way.”

Making sense of mathematics (in this case 3rd grade subtraction) as I have tried to do in this blog post is not crazy…it is not new….it is just MATH!*

 

*Credit to Dr. Ted Coe!

 

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.

“Five percent of the people think;
ten percent of the people think they think;
and the other eighty-five percent would rather die than think.”

-Thomas Edison

I don’t know exactly what Edison meant by this or how he determined the percentages, but the message, to me, is this: thinking is not what people do well!

On a recent Sunday news show, Texas governor Greg Abbott claims that the Common Core State Standards are bad because “it takes a teacher more than a minute to teach a student how to learn that 9+6=15”. He is referencing this video. Check it out.

Let’s think about this. How is this an argument against Common Core?

Is teaching a student how to think about 9+6 = 15 a bad thing? There is a difference between just doing 9+6 and thinking about 9+6. The Common Core Standards clearly have an emphasis on procedural fluency. Students should be able to determine that 9+6=15 quickly. But, how will such fluency happen? It will happen when students have a way of thinking about 9+6. Because it takes a minute to explain this thinking does not mean that it will take a student a minute to compute! Furthermore, the expectation is to push mental math…a good thing! For a student to flexibly be able to see 9+6 as equivalent to 10+5 is very good!

Governor Abbott does not have the opportunity to explain how he believes students should be taught that 9+6=15, but I will make a conjecture that he will argue that students should just memorize it. And in saying this, what would he mean by “just memorize it”? Perhaps something like this (and I embellish):

Teacher: “OK Johnny, this shape right here (9) is called the number 9 and this shape right here (6) which kind of looks like an upside-down 9 is called the number 6. And when you put this symbol in between this (9) and this (6) the answer is two shapes…a 1 which sits right next to a 5. So 9 + 6 = 15”
Johnny: “But why?”
Teacher: “Don’t ask why…just memorize it. Let me get the flashcards.”

We might as well do something like this:

Teacher: “OK Johnny, let me tell you something that I want you to memorize. If you take purple and add dog the answer is banana.
Johnny: “But why?”
Teacher: “Don’t ask why…just memorize it. Let me get the flashcards.”

The teacher in the video is just unpacking the mathematical thinking necessary for a student to make sense of what 9+6=15 means and how this fact might be memorized in a meaningful way. Certainly, we expect students to be able to respond to 9+6 in just a few seconds. This memorized fact will be retained when the mathematical reasoning and thinking, done by the student, allows the student to make meaning and make sense of the computation.

And by the way…this kind of thinking, to allow students to memorize basic math facts, has been around long before the Common Core Standards! This is not a Common Core thing…this is just good teaching.

Diane Briars, president of the National Council of Teachers of Mathematics, says this about the issue: “Teachers have used techniques like splitting a number into parts of 10 for addition–rather than straight memorization–since the 1950s at least, and the research showing its benefits goes back to the 1920s. It has long been best practice for early childhood math.”

I encourage everyone…please read the Common Core Standards and understand their purpose. Then, if you want to debate the issue of the federal government offering federal money tied to the implementation of these standards, great! If you want to debate standardized testing, its cost, its taking instructional time, its impact on the mental health of students, teachers, and parents, great! But please…do not conflate these issues. Standards are standards…teaching is teaching…curriculum is curriculum…testing is testing. My experience is, when people actually consider the Standards, the issue is not the Standards! When they see that students are expected to become effective problem solvers, mathematical thinkers, and fluent in computations, most agree that the Standards are not a problem.

When people try to show that the Standards (not the role of the feds, not standardized testing, etc.) are bad, it is clear that they are not really thinking…

 

 

There seems to be a common theme among the complaints about the Common Core Standards. Parents say, “that’s not the way I learned it…why not just do it the “old fashioned” way?”

Part of the reason is this: Because we are trying to prepare students for a new, modern world! Years ago, it was very important for people to become very accurate with paper and pencil computations. That’s just not the case anymore. Now, it is important that we take a balanced approach…a balance between computational fluency AND conceptual understanding AND the ability to apply mathematical ideas to solve real-world problems. This is exactly what the common core standards have as a major focus…a balanced approach!

Here is a question that we all should think about…how does a young learner develop computational fluency? Does it just happen magically? The following videos seek to answer the question in the context of addition algorithms, subtraction algorithms (videos made by Dr. Scott Adamson) and multiplication (video by Dr. Raj Shah).

ADDITION ALGORITHM

http://www.screenchomp.com/t/1CXpvMoLDq

SUBTRACTION ALGORITHM

http://www.screencast.com/t/NULauMxJ6

MULTIPLICATION ALGORITHM

http://vimeo.com/110807219

Scott

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!

common-core-math

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…

AE

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.

CaptureWith 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

Our good friend, Dr. Ted Coe, had the following article published on azcentral.com:

 

Why Arizona needs Common Core standards

Math professor: It’s all about how we think and learn

sr-online0706  06270230
By Ted CoeMy TurnTue Feb 25, 2014 12:54 PM

In a recent surveyconducted by Expect More Arizona, 43 percent of the state’s likely voters said they favor Arizona’s College and Career Ready (Common Core) Standards, which define the knowledge and skills necessary for K-12 students to succeed in college and careers.

After being given a short description of the standards, support rose to 71 percent.

Yet just last week, the state Senate Education Committee passed bills to abolish or allow schools to opt out of these standards and literally revert back to the level of standards from the last century. This is not the direction Arizona should be moving.

To understand why, you must first understand three key areas of mathematics teaching and learning: ways of doing, ways of thinking and habits of thinking.

Ways of doing: I have been a math educator for 20 years. I have seen how our culture views mathematics simply as ways of doing. Consider the typical math classroom experience: A teacher explains a particular mathematical process and students complete the process 30, 40 and sometimes up to 100 times.

The next day, the teacher introduces another mathematical process, perhaps unrelated to the previous one. Students collect these pieces for up to five months and the system rewards those who are able to memorize the greatest number of processes.

Think about it: When you finished the last of your mathematics classes, did you see math as a collection of random pieces or as a completed puzzle?

Ways of thinking: For too many, math ends up being merely a collection of random pieces, when in reality the pieces should always fit together. After all, mathematics is the one academic subject that makes logical sense.

Unfortunately, many lose the notion that what we do in math should follow from how we think about math. Our ways of doing need to develop from ways of thinking.

Arizona’s College and Career Ready Standards set a new expectation in this regard because they are built on progressions of thinking. That is, the standards emphasize not only the mathematics processes but also conceptual understandings that are consistent across the grades.

Math instruction should make sense both day-to-day and year-to-year. Teachers who use clever shortcuts to supplant thinking do their students a disservice in the long run.

Habits of thinking: It is important to encourage good mathematical habits of thinking. Arizona’s College and Career Ready Standards will develop mathematical practices such as problem-solving, persevering, reasoning abstractly, constructing arguments and attending to precision. The standards will require our students to think about and apply their math skills to real-world contexts.

Educating our state’s future leaders has never been more important. Arizona’s College and Career Ready Standards provide a framework that will ensure our youth get the quality education needed to prepare for college and the workplace.

In the realm of mathematics, that means doing more than helping students ace an exam. Rather, it’s about encouraging and supporting ways of doing, grounded in ways of thinking, while developing essential mathematical habits.

Dr. Ted Coe is an assistant dean at Grand Canyon University, a member of the Partnership for Assessment of Readiness for College and Careers Math Operational Working Group and a co-director of the Arizona Mathematics Partnership, an $8.7 million National Science Foundation grant to promote excellence in middle school math and improve student achievement.

The Common Core State Standards Initiative is a state-led effort coordinated by the National Governors Association Center for Best Practices and the Council of Chief State School Officers. To date, forty-five states have adopted the Common Core Standards and are working to develop new assessments to measure student success based on these new standards.

Content standards have been a part of educational reform at the local, state, or national level for many years. However, the Common Core State Standards in Mathematics offer a new and exciting focus that has not been seen before – Standards for Mathematical Practices. These mathematical practices focus on the ways of thinking and habits of thinking necessary for students to understand mathematical ideas that can be used as they prepare for college and career.

The philosophy of the getrealmath team is in sync with the Common Core Standards for Mathematical Practices. Let’s look at the eight mathematical practices and see how our  approach can support the development of these practices.

Common Core Standards for Mathematical Practices

  1. Make sense of problems and persevere in solving them.
    Solving problems is at the heart of our  approach. Students can better make sense of problems that are based on real-world situations. As students work to solve these problems, they learn to persevere. Perseverance is best learned by, well, persevering! Initially, perseverance may mean that a student sticks to it for just a minute or two. As students find success solving interesting problems that are connected to motivational contexts, they are apt to stick to a problem for longer periods of time before giving up.
  2. Reason abstractly and quantitatively.
    Rather than to only mimic procedures, students using activities based on real-world contexts must engage in reasoning. In fact, the activities often ask students to explain their thinking, to make connections, and to justify their arguments. These kinds of activities will help students to reason abstractly and quantitatively. Also, the activities make appropriate use of mathematical symbolism to help students move from the concrete to abstract.
  3. Construct viable arguments and critique the reasoning of others.
    When working to solve real problems, students often need to construct an argument defending their thinking, their problem solving approach, or their conclusions. This is a skill that will be beneficial for preparing students for college and career. All of the activities can be easily incorporated into group work and/or classroom discussions and instructors are encouraged to do so. When working together with a partner or a team, students will be afforded the opportunity to listen to the reasoning of others and to provide feedback. Learning to provide constructive criticism of the reasoning of another is a valuable skill that can be developed while solving real-world problems.
  4. Model with mathematics.
    The Common Core Standards for Mathematical Practices states, “mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” Teaching with a focus on real-world activities provide students the opportunity to develop this problem solving ability.
  5. Use appropriate tools strategically.
    These tools include paper and pencil, calculators, Internet resources, measuring devices, and even mental images. Students need to learn not only how to use these tools but also when to use these tools appropriately. Some computations can and should be performed mentally (e.g. 6 x 8 = 48) while others can be done efficiently, in the context of a real-world problem situation, using a calculator. In some problem situations, students may need to conduct research to learn more about the situation as they make sense of the problem. Collecting data, making measurements, using concrete tools to model a situation are real-world problem solving strategies.
  6. Attend to precision.
    Certainly, computing accurately is part of attending to precision. Another important part of attending to precision is helping students to develop precise explanations and descriptions of their thinking, their conclusions, and their strategies. As students construct viable arguments, write down their conclusions, or orally present their thinking, we can help them to do so precisely. Describing the mathematics as it connects to a real-world situation can be helpful in developing this precision.
  7. Look for and make use of structure.
    One of the goals of our approach is to help students to develop a conceptual understanding of the mathematical ideas that they are learning. For example, the idea of a linear function is strongly connected to the idea of a constant rate of change. As students work to make sense of linear functions in a real-world context, they can better see the structure of the mathematics and make important conceptual connections.
  8. Look for and express regularity in repeated reasoning.
    Mathematics is often described as the science of patterns. Students who learn to identify patterns can develop a profound understanding of the mathematical ideas. While working in real-world contexts, students are often encouraged to practice reasoning about the solution methods so that the ideas can be developed in a profound way.

The Common Core Standards initiative is committed to assessing students’ development of these mathematical practices. That is, student success on standardized tests at the state level will soon require that students work to develop them. Using the Make it Real Learning Activities, students will be led through a development of the above-mentioned mathematical practices while working in real-world contexts.

Scott