Pop Quiz!

(No Googling or peeking at other people's answers before responding, please - this is a closed-book quiz. Also: this will not go on your permanent record.)

With reference to the long multiplication problem above, please answer the following questions.

Q1) What does the "4" that is circled mean?

Q2) What does that zero in the second row mean (the one that is circled)?

We'll come back to this later.

Common Core Math Standards: What's the Point?

Our topic today is the Common Core Math Standards.  They seem to have some people in a tizzy.

The picture at left, for example, has been making the rounds on the internet. Evidently someone snapped this picture of their child's math homework because they were enraged by it, and lots of people are hopping on the bandwagon.

This is curious to me.

The intent here should be pretty clear - the point is to develop the child's conceptual understanding of subtraction. This happens in parallel to developing their fluency in the standard subtraction algorithm that we all learned as children (not shown on this page, but also covered in the Common Core standards).  The standard subtraction algorithm is efficient for calculating, but doesn't support understanding.

Do We Really Need to Change the Way We Teach Math?

Why would we want to teach a child to understand the concepts behind the algorithms?  Isn't that just a waste of time?  We all did fine just learning the algorithms by rote, didn't we?

Well, no.  It turns out adults in the U.S. aren't very good at math.

Speaking of which - let's take a moment to reflect on the quiz above.  How did you do on it? How quickly could you answer? Did you step away to search the internet for some clues or look at the poll results before submitting your own answers? (Naughty monkey!) How confident were you in your responses?

The simple fact is we are failing our kids in math education, and we have been for generations.  Here are some more fun facts to illustrate the point:
It's pretty clear we're doing something wrong. Solving big problems like this requires big changes.

How Can We Do Better?

If we look at some of the top performing countries, like Singapore, and ask what they are doing that we aren't, the most obvious difference is that they focus on conceptual understanding a whole lot more than we do, and they spend a lot less time teaching things by rote.  They teach the kids several models and methods for thinking about numbers and operations on numbers, for example - methods like the "Counting Up" method represented in the picture above. It's not a secret. We have just stubbornly refused to do it (until recently, anyway, with the arrival of the Common Core math standards).

But Aren't We Just Complicating Things?

Teaching concepts may take more time than memorizing a few recipes for calculating without understanding (at least initially), and some people seem to object to spending the extra time.

This is extremely short-sighted.  There may be more time spent initially developing understanding, but that investment will pay dividends many times over across the student's time in school, from kindergarten through high school (and beyond).  Understanding the concepts makes later learning far more efficient - and effective. By spending more time developing foundational understanding, we could actually get better outcomes while spending less time on the math curriculum overall. Not only that, but students who understand what's going on have a much better experience, are more engaged, and are more confident. All good.

Why Can't We Just Do What We've Always Done?

What does it look like when we fail to teach children conceptual understanding?  The video below shows one example of a second grader working on some grade-level math problems.

Notice that she knows the numbers and she knows how to count - these are typically learned by rote. Her conceptual understanding is very weak, however, and as a result she has to go through a laborious process of counting up from zero to answer simple questions like "How much do I have if I add 10 to 35?"  And then she gets the wrong answer. Repeatedly.

Time invested in developing authentic understanding is not a waste of time.  Quite the contrary. The real waste is time spent teaching without developing understanding - which produces the kind of disjointed, brittle, and tentative knowledge shown in the video - which is ultimately quite useless and will likely fade away rapidly.

This is all too common.  This could well be any of our children.  The maddening thing is that there's every reason to believe this child - and virtually every child - is completely capable of understanding the concepts she would need to reason fluently about the questions being asked of her in this video. She's not failing at math. Our education system is failing her. The same way it has failed generations before her.  In large part by teaching math by rote, without conceptual understanding. (Don't scapegoat the teachers, by the way - the root problems here are systemic.)

We could try to press ahead as some people are advocating and just teach this child the algorithms for long addition, multiplication, and division as we have always done.  But with such a weak foundation of understanding, what would we really expect to achieve that way? Her performance on those would quickly come to look like her performance here - labored, uncertain, and error-prone. Eventually she could well stop using the algorithms for lack of confidence, or even forget them altogether.

This is the sort of large scale, systemic problem the Common Core Math Standards are meant to rectify.

Could the explanation in the textbook shown in the picture above be edited for clarity?  Sure it could.

Is that evidence that the Common Core is a failure and should be trashed?  Far from it. We've been doing math education wrong for a very long time. The Common Core Math Standards represent a big step in the right direction - in the direction of what Singapore and other top-performing countries do, in fact.

Some people seem to think the textbook image above is crazy. What's really crazy is recognizing that the status quo is not acceptable while repeating the same educational processes generation after generation and expecting a better result.

Come to think of it, wasn't that literally Einstein's definition of insanity?

1. If your readers watched all the way through the video, they will have noticed not only task avoidance, but her heartbreaking affect. It makes me both sad and angry because I have seen this in far too many students. No child should reach first grade and hate or avoid math.

The push back for Common Core, I think, is a push back against what adults don't understand. Like many, I was taught math by memorizing procedures. I had no conceptual understanding of computation. I followed a pattern by way of imitation. Babies do this when they are acquiring and practicing spoken language. In fact, unless we are mute, we can speak without a conceptual understanding of how it is that we put sounds together to produce words, words to produce sentences, and sentences to paragraphs of thoughts/articulated ideas. (Do many of us sit and contemplate how we produce language?). But articulating an idea, an argument, a stance, requires reasoning. Reasoning is not parroting back memorized replies to specific questions. When I lived in another country and was learning this language, I knew how to conjugate verbs, I had a growing vocabulary, I could produce questions and replies in simple conversation; but I was not yet “thinking” in that language.

Thinking mathematically, to me, means problem solving by reasoning. I have a saying in my class, “Solve for the unknown: in math and in life.” In order to do that, you must recognize there IS an unknown, you must recognize what you DO know, and then you must experiment and reason through to a solution. When we give our students multiple experiences in reasoning through their solutions, that is a life skill that transcends math. There is not ONE right method for solving a math problem. When students understand why a solution is correct, and have multiple ways of reaching the solution, they are learning to problem solve. I wish life was simple enough that every problem we encounter had one easy solution. Giving students an algorithm to memorize is like learning how to conjugate verbs, throwing in some vocabulary, but never reaching the fluency to converse in a meaningful way.

I will agree that in most of our day to day math experiences, using a calculator, relying on an old learned procedure, will help us; but not acquiring the deep reasoning skills required when wrestling with understanding a math (or science, or social studies, or…) problem leads to a very shallow thinking society which is destined to make poor choices for the good of all.

2. I’m not sure what is most troubling, the mindless allegiance to outdated thinking or the adamant refusal to acknowledge the evidence of concern.
First, I am not defending Common Core, simply the evolution of our mathematics education. Second, the math components would not receive as much criticism if it weren’t packaged with Common Core. It carries with it all the political biases that inhibit rational, productive conversations.
That being said, I’m tempted to just say that our society would rather continue its stagnant performance in education because it’s familiar and predictable. Anytime an adult relies on the phrase, “That’s not how I did it when…”, they concede that they have nothing better to say.
In each PISA performance since 2000, the US’s rankings have dropped. In 2012, for the first time, we fell into the lower 50%. Yes, more than half of the 65 countries outperformed us. Sadly, our performance can be traced back to international testing pre-PISA, and that would show an even farther fall. The leading nations have significantly different designs from each other but offer commonalities. This knowledge is not new, but change is expensive.
Our system is designed to pass as many students as possible across a finish line of “minimum basic standard”. For the majority of their career, students are capable of performing successfully enough on math tests by following recipes to get an answer or recognizing some facts; understanding not required. The proof is that we live and die by multiple choice tests.
These new expectations causing the tizzy contradict math routines from our society’s schooling experience. Needing to do something different implies that what was being done was wrong, which is scary and threatening. These demands are not just about solving problems different ways or showing more work that a calculator can do. The push is to build flexibility in thinking and deep investigation into relationships. These strengths transcend math tests or math classrooms and more adequately prepare young learners to be young adult learners which we need to become the next generation of adult thinkers. Adult thinkers do not suddenly appear after surviving all their developmental years in “monkey see, monkey do” teaching environments.

1. Hi, Jeremy.

Thanks for joining the conversation.

> Needing to do something different implies that what was being done was wrong, which is scary and threatening.

I agree this is part of it. Also, change itself can be threatening. There seems to be something particularly challenging about making change in education, though. I think a large part of it is that historically we have not had a scientific framework that could help people understand the relationship between teaching and learning. Without that explanatory framework, people do not have confidence that proposed changes in educational practices would make things better instead of worse. Part of what is puzzling to me about education is what happens next, though. Instead of recognizing or admitting they don't understand and that that creates angst for them, many people seem to get adamant that their point of view is correct and opposing views are obviously wrong. They also seem to become more susceptible to the fallacy of "appeal to an illegitimate authority" in an attempt to back up their claims. The whole system seizes up. It's very counter-productive.

3. Old article and I doubt this will ever be seen, but I read your position and I was compelled to respond.

Your position is asinine. I direct you to Whitehead and Russell's 300 page proof of 1+1=2. A level of intuition and understanding you will never find in an elementary classroom. Your example of 325-38=287 may not seem so simple... even for a mathematician.

Intuition and understanding require experience. Experience requires time. Unfortunately, you don't have any extra time in U.S. schools. The time in a math class is the same as it has always been. You're just introducing a new convoluted algorithm (here the counting up subtraction method) or many algorithms for solving subtraction that replaces the traditional algorithm. Or rather, you're replacing the universal manual with a stack of manuals with pretty pictures and cumbersome math manipulations hoping that magically the child "gains intuition."

Procedure first... Intuition later.

This concept doesn't change at any level in life. Rote memorization happens first, whether it’s a math model, a foreign language alphabet, a sound, or a pronoun. Intuition comes later. For example, I've returned back to school in my early 30s to earn a second degree, this one in Electrical Engineering. I'm currently learning techniques for solving first and second order ordinary differential equations. I'm not entirely sure I understand it fully, but I can follow the procedures to solve math problems. I expect to acquire intuition slowly after taking more classes that utilize these techniques. This has been the trend for every complex topic I've learned for 30 years.

More time... Less crap

The problem isn't how math has been historically taught. The problem is a long summer and short school hours. The reason why other countries do better in math is because they spend more time on it. Further, South Korea, for example, requires kids to have basic reading skills “before” attending school, illustrating a level of parental dedication to education you won’t find in a lot of U.S. homes.

Math frustration and embarrassment...

Kids need to be taught "grit", not some magical math panacea. You see that in the video with the second grader. The lesson isn't "we need to fix math teaching", the lesson is "math is hard, let's learn to tackle hard problems and not give up." The most important lesson you will ever learn is to pick yourself up and try again.

Your argument has major problems - three examples:
1) You commit the anecdotal fallacy when you treat your own personal experience as if it's representative of the human species - do you have any evidence that it is?
2) You commit the naturalistic fallacy when you use your post hoc description of how you think you learned math into a prescription for how we should therefore teach all children math
3) You seem to assume that your interpretation of your own learning experience - how the learning happens, that is - must be true ("rote memorization first..."). But there's plenty of evidence that people's folk psychological notions of how they learn are just plain wrong.

>I direct you to Whitehead and Russell's 300 page proof of 1+1=2. A level of intuition and understanding you will never find in an elementary classroom.

This may be true but it's irrelevant here. I'm talking about elementary understanding of the addition procedure, you're talking about the deep philosophy of mathematics. No one needs to understand Russell and Whitehead's 300-page proof to understand how to add 1+1. In fact, the proof doesn't help with that one bit - no more than understanding the grammatical theory and entire etymology of every word in the sentence "Please pass the potatoes" makes me better able to get what I want for dinner.

>Unfortunately, you don't have any extra time in U.S. schools. The time in a math class is the same as it has always been.

This is wrong on two counts. First, in well-documented cases learning time has been compressed by ten or a hundred times using what we know about how people learn. It's very much like software design - sometimes simply moving or tweaking one line of code can speed up a program by a factor of ten or more. On the other hand, a math curriculum that I designed for kindergarten was used by some students for 100-200+ hours during the regular school year. The limitation is teacher time, not learning time. Adaptive technologies are a multiplier of effective learning time.

>Procedure first...Intuition later.

Not according to the evidence. To take one example, understanding that a number is a quantity (and not some arcane symbol with arbitrary meaning) is a central intuition that is not a stretch even for young children, but many children don't develop this understanding as early as they should and that lack of understanding has been shown to lead to problems later. Furthermore, the evidence suggests that math instruction is most effective when skill/procedure and concept/intuition are taught together and used to develop each other. See this excellent review of the research, for example:
http://www.nap.edu/openbook.php?record_id=9822

>>> The reason why other countries do better in math is because they spend more time on it.

Evidence? How much more time? Where are they getting that time? How do you know this? We have clear evidence that changing the curriculum can take the kids from the bottom of the class and put them at the top - in the same amount of time. (See, for example, Chapter 7 in this book: https://books.google.com/books?id=EkCq8P0_ZGAC&pg=PA143&lpg=PA143&dq=gradients+in+math+learning+griffin&source=bl&ots=TAMQd-3yGY&sig=-CPUIlKxVoNpYOIevGFhQIuV5UA&hl=en&sa=X&ei=NMKhVaXHC8nu-AGFxbHYAQ&ved=0CEUQ6AEwCA#v=onepage&q=gradients%20in%20math%20learning%20griffin&f=false)

>Kids need to be taught "grit", not some magical math panacea.

Kids do need to be taught grit. That’s true but irrelevant here since it's not an either-or. They also need to be taught math in an effective way.