# A defense of the Common Core

I read the following defense of the pedagogical strategies behind the Common Core: http://www.vox.com/2014/4/20/5625086/the-common-core-makes-simple-math-more-complicated-heres-why

I really have no issue with the article itself. Sadly, the article does not address the two great deficiencies in the implementation of the Common Core: (1) homework problems and other assessments to gauge the depth of a student’s conceptual understanding of mathematics in ways that are age-appropriate, and (2) the direct tying of high-stakes tests based on the Common Core standards to the assessment of teachers.

I don’t feel like replicating my previous posts on this topic, so I’ll refer to my past posts here: https://meangreenmath.wordpress.com/2014/12/18/common-core-subtraction-and-the-open-number-line-index/

# Why Do Americans Stink At Math?

This is a long op-ed from the New York Times, but it’s pretty good: http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html?_r=1

# The Smarter Balanced Common Core Mathematics Tests Are Fatally Flawed and Should Not Be Used: An In-­Depth Critique of the Smarter Balanced Tests for Mathematics

My biggest critique of the Common Core is not the standard themselves — it’s the ham-handed way that publishers attempt to assess students’ knowledge. This recent article by Steven Rasmussen echoes these thoughts and is an utterly disturbing look into the way high-staking testing in mathematics is being implemented: https://dl.dropboxusercontent.com/u/76111404/Common%20Core%20Tests%20Fatally%20Flawed%2015_03_07.pdf

Here’s the introduction:

This spring, tests developed by the Smarter Balanced Assessment Consortium will be administered to well over 10 million students in 17 states to determine their proficiency on the Common Core Standards for Mathematics (CCSSM). This analysis of mathematics test questions posted online by Smarter Balanced reveals that, question after question, the tests:
• Violate the standards they are supposed to assess;
• Cannot be adequately answered by students with the technology they are required to use;
• Use confusing and hard-to-use interfaces; or
• Are to be graded in such a way that incorrect answers are identified as correct and correct answers as incorrect.
No tests that are so flawed should be given to anyone. Certainly, with stakes so high for students and their teachers, these Smarter Balanced tests should not be administered. The boycotts of these tests by parents and some school districts are justified. Responsible government bodies should withdraw the tests from use before they do damage.

The full report is 34 pages long, giving example after example of horribly written test questions. This example was my personal favorite:

Question 2: A circle has its center at $(6,7)$ and goes through the point $(1,4)$. A second circle is tangent to the first circle at the point $(1,4)$ and has the same area. What are the possible coordinates for the center of the second circle? Show your work or explain how you found your answer.

In Question 2, the test makers ask students to solve a geometric problem and show their work. In general, asking students to show their work is a good way to understand their thinking. In this case, would anyone begin the problem by not sketching a picture of the circles? I doubt it. I certainly started by drawing a picture. A simple sketch is the most appropriate way to show one’s work. However, there’s just one major issue: There is no way to draw or submit a drawing using the problem’s “technology-enhanced” interface! So a student working on this problem is left with a problem more vexing than the mathematical task at hand—“How do I show my picture by typing words on a keyboard?”

I highly recommend reading the report in its entirety.

# Math Coach’s Corner

I found the following blogpost very intriguing. This sounds like an engaging way of teaching children how to multiply, whether or not the Common Core was the educational fashion of the day: http://mathcoachscorner.blogspot.com/2014/08/a-peek-inside-theres-nothing-alien.html.

# Common Core, Subtraction, and the Open Number Line: Index

While the implementation of the Common Core has left much to be desired (understatement of the day), I do endorse — whether it’s done through Common Core or not — the fostering of deeper conceptual understanding when teaching mathematics to elementary school students. I have plenty of opinions on teaching for conceptual understanding, Common Core mathematics, and (where the Common Core has utterly failed) assessing for conceptual understanding:

Division 1: A discussion about the usefulness of unorthodox ways of teaching long division.

Division 2: A continuation of the above discussion.

Subtraction 1: Introducing a viral picture about the Common Core, and its easy solution.

Subtraction 2: The pedagogical rationale for using an open number line (even though I personally do not endorse this technique as superior to other ways of teaching subtraction).

Subtraction 3: The abject failure of current developmentally inappropriate ways of assessing the depth of a student’s mathematical knowledge.

Subtraction 4: The importance of engaging parents when unorthodox methods are used to teach mathematics to children.

# Jonathan Katz on Some Problems of Common Core Mathematics

Courtesy of Diane Ravitch:

Jonathan Katz taught mathematics in grades 6-12 for 24 years and has coached math teachers for the past nine years.

He prepared this essay for the New York Performance Standards Consortium, a group of high schools that evaluates students by exhibitions, portfolios, and other examples of student work. The Consortium takes a full array of students and has demonstrated superior results as compared to schools judged solely by test scores.

What is of special concern is his description of the mismatch between the Common Core’s expectations for ninth-grade Algebra and students’ readiness for those expectations.

Here is a key excerpt…

[The Common Core standards seem] to honor the idea of problem solving and the many ways a student might engage with a problem. It seems to value the process of problem solving, the ins and outs one goes through as one tries to solve a problem and that different students will engage in different processes.

To implement such a standard, a teacher would need to present students with problems that allow for and encourage different approaches and different ways to think about a solution—what we call “open-ended problems.” Yet, when you look at the sample questions from the Fall 2013 NY State document you would be hard pressed to find an example of a real open-ended problem. Here is one example in which a situation is presented and three questions are then posed.

Max purchased a box of green tea mints. The nutrition label on the box stated that a serving of three mints contains a total of 10 Calories.

a) On the axes below, graph the function, C, where C (x) represents the number of Calories in x mints.

b) Write an equation that represents C (x).

c) A full box of mints contains 180 Calories. Use the equation to determine the total number of mints in the box.

A situation is presented to the students but then they are told how to solve it and via a method that in reality few people would even employ (who would create a graph then a function to find out the number of full mints in the box?). If you are told what to do, how can we call this solving a problem? (This would have been a very easy problem for most students if they were able to solve it any way they chose which is what we do in real life.) In fact, all eight problems in the same of Regents questions follow the same pattern. Students are told they have to create the equation (or inequality or system of inequalities or graph) to answer the question. Thus there is no real problem solving going on—merely the following of a particular procedure or the answering of a bunch of questions. Why don’t we use problems where there is a real need for an algebraic approach? Why would we ask students to look at a simple situation then force them to use an algebraic approach, which complicates the situation? We should be helping students to see that the power of algebra is that is gives us the means of solving problems that we would have great difficulty solving arithmetically.

If we were truly trying to find out if our students are developing the ability to problem solve, we would never create questions of this nature. They would be more open-ended so students had the chance to show how they think and approach a problematic situation. But that can’t happen on a test where everyone is instructed to do the same thing so we can “measure” each student’s understanding of a particular standard. This is not real mathematics and a contradiction of the Common Core Standards of Mathematical Practice!

Why does this matter? The consequences are huge, and not just for students. Consider the message we are sending to teachers. Since students will be assessed on following given procedures rather than how they strategize and reason through a problem, then teachers’ lessons will become all about following procedures to prepare their students for an exam they must pass in order to graduate. This will simply perpetuate the same failing math teaching practices we had in the past, will compound the dislike that students already have for math class, and will not in any way help our students to develop mathematical thinking.

# Common Core, subtraction, and the open number line: Part 4

The following picture has been making the rounds lately.

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

1. The solution of this particular question.
2. The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
3. The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
4. The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

This is Part 4 of this series of posts: the importance of engaging parents and caregivers when unorthodox techniques of mathematics are taught.

I’ve only witnessed the implementation of the Common Core from afar, but there’s absolutely no doubt that the professional development of teachers who have been asked to teach math in a new way has left a lot to be desired. Ditto for explaining these new approaches to parents and caregivers who want to help their children.

True story: I personally did not know about the open number line until the “Meet the Teachers” night that was held for parents near the start of the school year. The teachers explained that they would be doing math a little differently and did a couple of examples using the open number line. I could feel many eyes in the room looking back at me (people know I’m a math professor) with facial expressions saying “Is this stuff really going to work for our kids?”

As this was my first exposure to the open number line, I was skeptical (I would have preferred using base-10 kits) but I held my tongue and listened carefully to the presentation. After the presentation, I was convinced that this was a completely legitimate way of teaching the subject and that the teachers had the requisite depth of understanding to teach arithmetic using this technique. After the presentation, I told anyone who’d listen that this technique was mathematically sound and pedagogically sound, even if it was different than the way that “it’s always been taught.”

Parents generally bought into the technique that evening. I’m not sure that they would have bought into it if its rationale had not been carefully explained to them.

And, as a reminder, Texas is not a Common Core state.

The failure to explain to parents and caregivers unorthodox but correct ways of teaching mathematics has been perhaps the greatest failure of the roll-out of the Common Core. It’s unacceptable that children are crying over their math homework and parents feel powerless to help (a common theme that I’ve heard over and over again from my friends).

Teachers and parents ought to be natural allies in wanting children to have a greater depth of understanding of arithmetic that will prepare them for algebra later. However, because the strategies of teaching the “why”s of mathematics have generally not been carefully explained to parents, they naturally feel somewhat helpless when trying to help their children with their homework.

My own field of research is not mathematics education. So, at professional conferences, I’ve asked friends and colleagues the same question over the years:

Letting children use their own natural curiosity to get at they “why”s of mathematics is good. Letting children use their own natural curiosity and also having the support of parents at home is better. So what research has been done on strategies on successfully engaging parents with how mathematics is currently taught versus how it was taught a generation ago (or, more accurately, what parents remember of their own experiences from elementary school)?

To my surprise, people that I greatly respect did not have an immediate answer to my question. So I’m guessing that while there’s been a lot of research into successful strategies for teaching mathematics in the classroom, there hasn’t been a lot of research into how these strategies can be supported when children are away from the classroom and asking their parents for help on their homework.

I’ll repeat the close of yesterday’s post: I won’t defend the indefensible way that the Common Core has been rolled out. Voters will be more than justified in voting out anyone who supports the Common Core if its implementation isn’t fixed in the very near future.

# Common Core, subtraction, and the open number line: Part 3

The following picture has been making the rounds lately.

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

1. The solution of this particular question.
2. The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
3. The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
4. The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

This is Part 3 of this series of posts: assessing the depth of a student’s knowledge of mathematics in ways that are developmentally appropriate. To be blunt, this has been one of the great weaknesses of the roll-out of the Common Core in the early years of its implementation.

Where I agree with most critics of the Common Core is that, even though I want students to understand why (for instance) subtraction works, 2nd and 3rd graders are still learning how to write complete sentences. So of course they will get easily frustrated with explaining their reasoning in paragraph form. I think there are better ways of assessing whether young children really understand subtraction that is more developmentally appropriate. (Here in Texas, students in this age group are instead asked to explain their reasoning by drawing pictures.)

I’m happy to defend the standards of the Common Core themselves, but it’s patently obvious that the implementation of the Common Core standards were driven by textbook publishers and not educators. I don’t doubt that much of what’s assigned for homework under the guise of the Common Core is developmentally inappropriate.

I’m perfectly happen to defend the logic behind a specific pedagogical technique like the open number line. But I can still be genuinely annoyed that the first years of the Common Core has been so badly botched. And I’m especially annoyed that most public defenders of the Common Core seem deaf to the legitimate and thoughtful complaints of its critics (as opposed to the braying expressed in the original picture).

Here’s another example that made the rounds in recent months, for which I have a considerable amount of sympathy.

The problem in question that forms the basis of her argument:

Mr. Yumata’s class has 18 students. If the class counts around by a number and ends with 90, what number did they count by?

The “correct” answer requires students to divide 90 tick marks into equal groups of 18, counting the number of tick marks in each group. Again, the textbook publisher missed the mark. The phrasing of the problem asks for just the answer (5). If the problem wants to know a rationale for the answer, then it should have been phrased in a different way.

There absolutely is a place in the curriculum for the method described in the video — dividing $X$ objects into piles of $Y$ to see how many objects are in each pile. This works well when children are first exposed to division and $X$ and $Y$ are reasonably small integers. However, by the time students get to $X =90$ and $Y =18$, another method should be used. By this point, dividing objects into piles is logically correct but pedagogically questionable, as it requires precisely 108 steps (as noted in the video). Any little careless mistake in counting will lead to an incorrect answer.

Flatly, I won’t defend the indefensible way that the Common Core has been rolled out. The textbook publishers have clearly missed the mark on how to assess the depth of a student’s knowledge. Voters will be more than justified in voting out anyone who supports the Common Core if its implementation isn’t fixed in the very near future.

# Common Core, subtraction, and the open number line: Part 2

The following picture has been making the rounds lately.

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

1. The solution of this particular question.
2. The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
3. The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
4. The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

This is Part 2 of this series of posts: the pedagogical reasons for using an open number line.

I firmly reject the premise, implied by the picture, that the only goal of teaching mathematics is just getting the right answer. If that’s the goal, then we can eliminate much of the K-5 curriculum by handing out 19-cent solar calculators to every student. That will pretty much guarantee that the students will solve elementary arithmetic problems like the one shown. It will also fail to develop their higher-order reasoning skills and leave them hopelessly unprepared for algebra.

A great example of a teacher using children’s natural curiosity to foster mathematical thinking can be seen in this video: https://www.teachingchannel.org/videos/common-core-teaching-division.

I absolutely agree with Common Core critics that students in elementary school should, at the end of the day, be able to quickly compute 427-316 using the standard algorithm in 5 seconds or less. I would also argue that it’s also important for students to learn why is algorithm works and not just how to use the algorithm. This deeper level of conceptual understanding begins to lay the foundation for the more abstract thinking required in later years.

The open number line predates the Common Core by about 25 years; for more information, see http://www.k-5mathteachingresources.com/empty-number-line.html. When used properly (a big “if”), it will lead students to the standard algorithms for addition and subtraction. I’ve personally seen the open number line successfully used in the elementary schools where I live in the hands of skilled teachers. Bottom line: the students had no problem with the technique. (By the way, Texas is not a Common Core state.)

So I take great issue with the quote “The process used is ridiculous and would result in termination if used.” Of course the process is ridiculous for people who already know how to subtract. For students who are first learning how to subtract multi-digit numbers, however, it is a completely appropriate way to introduce students to the topic which will lead them to the standard algorithm for subtraction. Once learned, then there will be no need to revert back to the previous method.

I have heard some critics say that teaching the “why”s of mathematics, using the open number line and other techniques, merely favors the stronger math students and the expense of the weaker students. Perhaps this happens in some classrooms. However, this has not been my personal experience. I have supervisory authority over aspiring teachers who have taught math to literally thousands of third, fourth, and fifth graders — all in a non-Common Core state. With careful supervision, they have used the children’s natural curiosity and inductive reasoning (but not deductive logic — that’s an important distinction) to get at the “why”s of mathematics as well as the “how”s. And I’ve not personally witnessed anything like the frustrations that I’ve heard as the Common Core has been implemented. Hence my “don’t throw out the baby with the bathwater” approach when I hear and appreciate legitimate concerns and frustrations with how the Common Core has been rolled out.

One more thing: a great motivation of the Common Core has been to improve student achievement in algebra by fostering mathematical habits of thinking in the lower grade levels. It’s certainly debatable about whether or not this ideal has been achieved. But the motivation for the Common Core and improving conceptual understanding at the lower grade levels is to increase access to higher-level mathematics, not to favor the high achievers.

Though I’ve seen it used effectively, I’m personally not an advocate for the open number line. Given the choice, I would use a base-10 kit (shown above) for students who are first learning subtraction. I would start by collecting 4 plates of 100, 2 rods of 10, and 7 cubes of 1. I would then remove 3 plates of 100, 1 rod of 10, and 6 cubes of 1. The remaining pieces (1 plate of 100, 1 rod of 10, 1 cube of 1) represents the answer of 111.(This particular problem didn’t require trading a plate of 100 for 10 rods of 10 or else a rod of 10 for 10 cubes of 1; this would be handled later in the curriculum.)

I do recognize that teaching children multiplication and especially division with a base-10 kit is very cumbersome and is significantly easier using an open number line. So I see the rationale for using an open number line for addition and subtraction if it will be used in later years to teach multiplication and division.

I don’t support requiring children to learn subtraction or any other concept in mathematics by any one specific technique. The big issues are (1) getting the right answer and, of equal importance, (2) understanding why the answer works. I take issue with the original picture because it favors (1) but not also (2), and a surface level of understanding will not serve students well when they hit algebra.

Bottom line: I have absolutely no problem with schools using a technique like the open number line, as it is a mathematically correct way of explaining to students why the answer works.

# Common Core, subtraction, and the open number line: Part 1

The following picture has been making the rounds lately.

My bedrock position is simply stated: I’m for teaching any technique in elementary school that’s (1) logically correct, whether or not it’s the way it’s (mythically) “always been taught,” (2) encourages students to think mathematically, as opposed to mindlessly following a procedure with no real conceptual understanding, and (3) prepares students for algebra in a few years’ time.

That said, I have a lot of opinions about this picture, which does not necessarily align with our society’s impatient obsession with 10-second sound bites and 140-character tweets. So be it. I will divide my opinions into several categories of increasing scope.

1. The solution of this particular question.
2. The pedagogical reasons for using this technique (called an open number line). In other words, do we only want Jack to get the right answer, or do we want Jack to understand something about the logic behind the answer?
3. The difficulty of assessing the depth of a student’s knowledge in a way that is developmentally appropriate.
4. The importance of engaging parents with unorthodox ways of teaching mathematics.

Some of my opinions will line up nicely with supporters of the Common Core. Other opinions will align with the Common Core’s thoughtful critics.

This is Part 1 of this series of posts: the solution of this particular question. Here it is:

Jack correctly started at 427 on the number line. He then correctly understood that 316 consists of 3 groups of 100, 1 group of 10, and 6 groups of 1. He then correctly subtracted 3 groups of 100 (for an interim answer of 127) and then correctly subtracted 6. However, he forgot to subtract 10. That’s why he got a wrong answer (121) that was 10 more than the correct answer (111).

Just to make sure I wasn’t completely missing the mark on this, I rewrote the problem (without the handwritten commentary) and showed it individually to a few elementary school students. They all saw Jack’s mistake within 15 seconds. They may not have been able to explain what Jack did right and what Jack did wrong in the form of a letter (more on that in a later post), but they certainly identified the core problem quickly.

I understand a parent’s frustration with knowing how to subtraction but seeing a child learning subtraction in a different way. (More on that in a later post). I also understand that some may argue with this technique of teaching children how to subtract. (More on that in a later post.) But there’s no way to sugarcoat this: an engineer who took differential equations and read this problem but couldn’t figure out that Jack forgot to subtract by 10 has little conceptual understanding of mathematics.

So let offer some free advice to critics of the Common Core who want to share this picture to vent their complaints. I am totally sympathetic with frustrations expressed in this picture. Sharing this picture with your fellow critics may feel good, perhaps with the self-justification “If an engineer can’t figure this stuff out, then how can I?!?!” However, sharing this picture is not going to persuade anyone who disagrees with you to your cause. Remember: some children can solve this problem in 15 seconds or less. If anything, sharing this picture only communicates to those who disagree with you that the critics of the Common Core are the people who have little conceptual understanding of elementary school mathematics. Once again, I am sympathetic to the emotions expressed in this picture, but there are better ways of criticizing the Common Core and persuading its unabashed supporters to your cause.

In the posts that follow, I will provide plenty of criticism of how the Common Core has been implemented in its initial years.