Do Common Core Methods Teach Understanding Better than Traditional Methods?

With the current wave of Common Core this and Common Core that, there's a tremendous amount of confusion, especially among parents.  The purpose of this blog post is to not only clarify things, but to also suggest improvements.

Common Core is neither new nor traditional

The Common Core standards ask for a deep understanding, but they don't specify how that understanding must be acquired.  The Common Core standards also specify some very traditional requirements, such as memorization of the addition math facts in 2nd grade and the memorization of multiplication math facts in 3rd grade.  The Common Core also specifies the standard algorithm for things like long multiplication.  

In short, the Common Core specifies all of the qualities of an A+ student at each grade level, meaning that if a student were to master all of the specified standards, that student would absolutely be not only computationally correct, but also deeply understand the operations.

The strength of the Common Core standards is also its deepest flaw.  They fail to specify what a B-level student should know.  That interpretation has been left up to the greater teaching population, so what's actually being taught in schools is actually like cherry picking.  Teachers are teaching to the standards they are attracted to, but they aren't teaching to 100% of the standards, largely because there's no guidance to distinguish the relative importance of each standard.  Granted, teachers believe they are teaching to 100% of the standards, and they probably really do touch all of them, but frankly, some skills are more important than others, and we're often failing in extremely vital areas, such as the memorization of math facts, which has been proven through research to be a prerequisite to acquiring a deep understanding.  In 2006, the National Council of Teachers of Mathematics (NTCM) published the Curriculum Focal Points, which said exactly this.  If you know the history of the NCTM, that sparked controversy because the NCTM is known for having a Constructivist emphasis on mathematics. 

Constructivism - what is it?

For starters, here's a good definition:

http://www.thirteen.org/edonline/concept2class/constructivism/

Today's teachers are being trained to teach using Constructivist techniques.  As applied to math, they are trying to teach a deep mathematical understanding by approaching number theory from a variety of angles.  It sounds amazing.  

If you are really good at math, and you've recently been exposed to Constructivist techniques, you may have discovered that the steps being taught in school closely mirror the process you would use to solve math problems in your head.  When you look at it from that standpoint, then yes, this looks wonderful.

However, for a brief history lesson, please read this page:

https://en.wikipedia.org/wiki/New_Math

One of the goals of history is to not repeat our mistakes, but we appear to be doing exactly that.  I'm not saying Constructivist approaches are all bad (I actually agree and use some of them), but I do think the balance between deep understanding and traditional methods has been thrown off.

What's a good common example of Constructivist math?

A good example of Constructivist math is to focus on base 10 place value understanding.  A common problem might be 23 + 9.  The Constructivist does it this way:

• 9 = 10 -1
• 23 + 9 = 23 + (10 -1) = (23 + 10) - 1
• 23 + 10 = 33
• 33 - 1 = 32

You wouldn't necessarily write out every single step I just wrote there, but essentially, that's what's actually going on.  

The Traditionalist does it this way:

• 9 + 3 = 12
• 20 + 12 = 32

What I like and don't like about this example

It's very easy to construct an argument that the traditional method is better because it is faster in this case, but the counterargument is that the Constructivist approach leads to a deeper understanding.  My standpoint, which you may find surprising, is that neither argument is correct!

The problem that is happening in schools is that the students often are being forced to write out nearly every step.  It's actually insane to write out every Constructivist step for a problem as trivial as 23 + 9.  This causes parents to be especially angry (rightfully, too!).  Furthermore, is it really teaching a deep understanding?  No, it's not.  Instead, standard approaches have been simply replaced by newer approaches.  Students are being taught a step-by-step approach that if they see the number 9, they should replace that with "10 - 1".  So instead of that understanding becoming intuitive (which would imply a deep understanding), students are learning a new set of steps.  The new method is not better than the old way in terms of actual results achieved.

How to teach the same understanding better

I have two approaches to improve this topic so that it actually teaches a deep understanding.  Remember that the real goal is to make the place value concept actually intuitive and not a series of operations.

The first method

I have been using this method when I teach my own children.  The prerequisite to this approach for this example is that the student has not yet memorized the addition math facts.   

The first question I will ask is 23 + 10.  That's super easy, so the student will say 33 without much thought.  

The second question I will ask is 23 + 9.  If the answer doesn't come out immediately, I then say, "You know the answer to 23 + 10", so how does that help you get the answer to 23 + 9?

Inevitably, the student discovers the answer, not by counting on fingers, but by converting 9 into 10 - 1.  When I ask the student "how did you know the answer?", it's not always easy for the student to first explain the reason.  That's actually a great sign.  When you intuitively understand something, it's not always easy to explain what you know.  That's how I know the deep understanding has actually been acquired.  

Why this works

For starters, I make the student answer the problem without using pencil and paper and without counting on fingers.  Isn't the entire point of this exercise to break down the problem so that you can easily solve it in your head?  I have guided my own children to achieve a deeper base ten understanding through self discovery.  I never explicitly taught my children to take 9 and convert it into 10 - 1 even though that was my entire goal.  I consider my approach to be a purist Constructivist approach because the solution was self-discovered.  I sincerely believe that leading students to self discovery can raise IQ because you challenge your students to apply original thought, rather than recall of some "9 = 10 - 1" algorithm.  So the heart of my criticism of today's Constructivist approaches as seen in school is the very possibility that students are memorizing a new rule like "to add 9, first add 10, then subtract 1" instead of intuitively grasping it.

The second method

The next step to make this concept gold is to now ask questions that are too difficult for most young children to compute in their head using standard approaches.  I am likely to ask questions like these, in progression:

• 63 + 19
• 156 + 9
• 345 + 100
• 345 + 101
• 345 + 99
• 345 + 98
• 567 + 98

This is the same concept, only with bigger numbers.  If your child can add 345 + 100 easily, then there's no reason why the same child can't solve 345 + 99 if the concept is understood.  If your 5-year-old can add 23 + 10 and is aware of the hundreds, then my viewpoint is, 345 + 99 is fair game if the deep understanding is there. 

The bottom line is that working with small numbers is a good way to first teach a concept, but if you don't apply that concept in a more challenging format, the truly deep understanding might never occur.

The flaw I see in schools

Even if schools force their students to write out all of the steps on paper, the flaw I see is the failure to advance things to larger numbers as seen in my second method above.  If you want the actual understanding to sink in, you need to put your students in a position where they have to really apply it.  Take this problem, for example, and apply the same Constructivist steps:

685 + 298 = ?

• 298 = 300 - 2
• 685 + 298 = 685 + (300 - 2) = (685 + 300) - 2
• 685 + 300 = 985
• 985 - 2 = 983

Now, writing out all the steps doesn't seem so insane now, does it?  Truthfully, I really want you to solve that problem in your head, but I'm still pleased if you can write this out.  I think you'll find far less objections from Traditionalists regarding the solution above compared to the same approach when applied to 23 + 9.  

My Suggestions to School Teachers

I understand that there are few chances to work one-on-one with students, which is why you require things to be worked out on paper, so my first method for teaching an understanding through self discovery of the algorithms (as opposed to directly teaching the steps) might not be easy to achieve.  The second method, however, is what I really want to push due to its feasibility.  A lot of these concepts frankly don't make practical sense unless you apply them to larger numbers.  Here are some examples of what I'm saying:

• I don't want to estimate the answer to "9 x 8", but I do want to estimate the answer to "89 x 78". 
•  I don't want to estimate the answer to 98 - 19, but I do want to estimate the answer to 59738 - 19873.
• Please don't take 7 + 7 and convert that into a number bond question where the second seven becomes 3 and 4 and the 3 is added to the first 7 to get 10, allowing you to arrive at 10 + 4, which results in 14!  But, if you take 317 + 87 and show how that converts to 320 + 84, I'm quite OK with that.
• Please don't take 145 - 62 and turn it into a visual 2-minute problem with charts. I know the point of the problem is to solve the sub problem of 140 - 60, where you can't take 60 away from 40, so you have to get ten 10s from 100, but to actually construct a chart to show this is ridiculous.  Would you solve that problem that way in the real world?  Heck no.   Do you want to test real understanding?  Make that an oral question and make your student solve that on the spot without pencil and paper.  Otherwise, let the student solve the problem on paper using any method known to that student, and provide full credit for getting the right answer.

The bottom line is, for homework and tests, please do not require an approach that looks ridiculous on paper in the eyes of a Traditionalist.  If the problem and its solution looks ridiculous, would the solution look useful if the numbers used were bigger?  If so, use bigger numbers.  One of our flaws with K-3 mathematics is the reluctance to work with bigger numbers.  My point is that the deep understanding doesn't exist unless the student can apply the same concepts to larger numbers, and frankly, you don't really know if a student has the understanding unless the student can solve it without paper and pencil.

Let's Marry Traditional Approaches with Constructivist Approaches

I really want to marry the best of both worlds.  I hope by now I've convinced you that I am a big fan of self-discovery because I apply that approach to my own children.  At the same time, I cling to some Traditionalist beliefs, as should be evident in the next paragraph.

If I could mandate one thing to change US math education, it would be mastery of the addition math facts by the end of second grade at a rate of recalling each math fact within 3 seconds, and similar mastery of the multiplication math facts by the end of 3rd grade.   These thresholds would ideally be tested at the federal level.  I'm being specific on purpose and I'm citing a low threshold.  To be really good, recall of each math fact ought to be nearly instantaneous, at a rate no slower than 1 problem per second.  Imagine trying to solve the following algebra problem without knowing your math facts:

Here's the truth: you can't solve this efficiently with a normal calculator, and you are downright doomed if you don't know your math facts.  Furthermore, this isn't even calculus!  Imagine trying to do calculus without knowing your math facts.  It's not possible.  There is simply no way around the fact that you need to know your math facts, so you might as well learn them early.

Here's the answer, by the way, as generated by MathScore.com:

Outside of the math facts, I see Constructivist approaches as potential improvements when applied to situations where using traditional methods would be inconvenient or merely impractical.  Why fight a traditional method when it is clearly more efficient?  If you know a superior algorithm exists, why teach second best? Deep understanding is supposed to be intuitive.  Artificially writing out a bunch of steps to solve a simple math problem is not intuitive, and that type of "new new math" is always going to anger parents.  My observation is that today's teachers are trying too hard to teach a deep understanding while using numbers that are too small for the real lesson to sink in.  If you want your Constructivist teaching methods to not only be effective, but also not anger parents, please work with bigger numbers!

At the same time, when it comes to traditional math problems, there comes a point where computing bigger numbers becomes silly.  It's important to be able to add and subtract at least 3-digit numbers, and probably useful to add and subtract 4-digit numbers, but there's no point in drilling 5-digit long subtraction!  There's no point in drilling 3-digit multiplication!  When the numbers get this big, any normal adult would whip out a calculator.  But that's exactly the point where we can focus on a deeper understanding.  Take something like 385 x 212 and ask for an estimate of the answer, and you can get the student to say "400 x 200 = 80000, so 385 x 212 is something close to 80000".  Now that skill is practical, and applicable to real world situations.

In conclusion, here's my advice as concisely as I can say it:

• Stick with traditional methods for small numbers, and focus on Constructivist techniques for bigger numbers
• Work with bigger numbers at a younger age so that the deeper understanding can really be achieved.