Is there a happy and fun process to success in math? Is there a natural process to mastery? No. Students have to realize that learning may require hard work. It may require flash cards. It will require nightly homework sets in math. If kids are unhappy because learning is not fun all of the time, they are getting the wrong message. They need to know that hard work can lead to so much more fun. What’s worse is when students spend class time in fun group learning and then find themselves completely lost when it comes to applying math skills. It’s one thing to not like math, but be able to do it, but another to think you like math, but can’t do a thing.

So what does rote mean in math when you get into fractions? Can you do all sorts of problems without understanding? I’m not talking about pie chart conceptual understandings. Let’s look at the supposedly rote rule of “invert and multiply” for dividing fractions. What level of understanding is required here? Does this understanding come just from talk of concepts or by mastery of skills? How formal do you want to get? Reform math sticks with conceptual understandings that leave kids high and dry when it comes to solving all of the variations one might encounter. Neither rote learning nor general concepts will get the job done. However, is rote learning all that happens with a traditional (skills first) approach to math? Is there no linkage between skills and understanding? Does success in solving problems show no understanding?

What if you have something like:

5 / (2/3)

or

(5/3) / (-3/8)

or

(2 1/2) / 4

or

3.25 / (8/3)

or

(3x/2y) / 5x

Can all of these problems be done with just rote understanding or just concepts? What process does reform math use to go from general pie chart concepts to the understanding needed to do these problems? How does reform math ensure mastery of skills? It doesn’t. It sees little linkage between understanding and mastery of skills.

What happens when students get to word problems such as DRT, work, and mixture? Maybe you can get away with a general Polya-type process, but what happens when you add geometry and angles? How do you develop the understandings required to properly apply specific mathematical skills? You can’t use Polya to solve fluid flow problems in pipes without understanding and mastery of Bernoulli’s equation. It takes a lot of practice and understanding to develop the flexible skills to apply mathematical definitions, identities, and algebraic rules to any situation. This process is glossed over in K-8 reform math education.

Motivation and engagement don’t get the job done. It’s like getting fired up by diet commercials. The next day, you are back to eating mathematical Twinkies, and educators will start looking for the next big mathematical diet. Project Lead the Way in high school will never motivate away the gaps students have in math skills. Colleges care about math, not Project Lead the Way. Engagement is a cover for putting the onus of mastery on the students.

Balance is a false god. Balance doesn’t ensure mastery of skills. Mastery of skills is the goal. Understandings can be improved, but gaps in skills are very difficult to diagnose and fix. They are also a sign of missing understanding. A top-down approach will never get the job done because educators think of skills as rote and as something that is really not quite that important. They think that understandings can drive problem solving. They cannot. You need the understandings that ONLY come from mastery of skills. Educators need to ask the parents of their best students what they do at home. We ensure mastery at every step of the process even if our kids are “math brains”.

As Barry explained:

“… there is no chicken-and-egg problem: more often than not, skill must come first…”

Bottom up, not top down. Flexible skills are never rote. Skills can be made more flexible with additional understanding and practice, but conceptual understanding without mastery of skills is nowhere and leads to a dead end.

But more importantly is what is meant by “teaching for understanding”.

There are natural progressions of learning throughout the elementary school curriculum. In some cases, understanding comes first. In others it comes much later. Artificially imposing the sequence as in “understanding MUST come first” will sometimes work. But it is unhealthy and unbalanced.

Consider how a child learns to count. There are two aspects: (i) the mechanical act of pointing at objects and reciting the number sequence: one, two, three, … and (ii) the abstract conception of cardinality and cardinal equivalence as established by one-to-one correspondence, and of course eventually (iii) the marriage of (i) and (ii) into a coherent and fundamental concept of “number”.

Now, (i) is simply a skill. It is learned largely by rote. (ii) is, arguably, the “understanding” element. It is learned by reflection and analytic interaction with ideas. Which comes first? Perhaps there are exceptions, but I have NEVER seen a child who fully grasps the number-concept “understanding” embodied in (ii) prior to learning the fundamental counting skill (i). This is a clear instance in which skill precedes understanding.

More to the point, it is the skill itself upon which understanding is built, in this case. The child develops his understanding (ii) by repeatedly practicing the pure skill (i) over and over until it is realized conceptually through familiarity and tactile experience that forges pathways and connections in the brain.

Daniel Ansari, a leading scholar of cognitive developmental psychology who studies brain activity during the learning of mathematics said, at a recent conference, that neither skill nor understanding should be underemphasized — they provide mutual scaffolding and both are essential. But in terms of sequential priority, there is no chicken-and-egg problem: more often than not, skill must come first, because it is difficult to develop understanding in a vacuum, whereas skill can be developed in this fashion, and provides the appropriate experiential context within which understanding can be developed.

I teach an elementary math methods course for pre-service teachers and this debate is exactly what I am trying to help my students dig into carefully. We are going to have a debate in our next class specifically looking at the benefits of teaching for understanding vs. the benefits of practicing/focusing on basic skills.

I don’t pretend to have a perfect answer to this, but I wanted to share a few of my thoughts.

I think that one aspect of this issue that the author of the article hasn’t addressed is the motivational aspects of the different epistomologies. One of the main features of a traditional math program that focuses on basic facts is the idea that if you do enough practice, the students will eventually get those basic facts down. Assigning a large amount of practice for basic skills is essentially built on the cognitive model that when something is done correctly often enough then it becomes encoded in our brains. This has worked for many students and I don’t think should be abandoned entirely.
The problem with this focus to me is that in my own experiences in math, my teaching of 4th graders for 6 years and most recently my 2nd grade daughter is that too many students end up dreading math. Motivation to continue with math in any way becomes pretty rare except for the most successful students.

Teaching for understanding and critical thinking also assumes that information is being encoded and that practice is needed, but the focus is simply on encoding (thinking about) the deeper understandings of why the algorithim works. If done poorly in class, this can be pretty useless, as many have argued, but when it is taught well, I have found that more of my students are motivated to work with math problems. My biggest concern with this method is the time that it takes to really let students think about and discuss math at a deep level. (One way to help with this is to have students explain their thinking in pairs or small groups while the teacher circulates to assess understanding)

Overall, my view as a former elementary teacher and current professor is that creating a balance between these issues really would be the best math learning environment. The benefit of balance would be most likely when we reduce (but not eliminate) the drilling of basic facts and also provide opportunities in class and in homework for students to explain their thinking thus building fluency and understanding. Not easy or simple to do but I think it is possible.

Thanks for your comment. I find it interesting that you are being told you are not allowed to teach a traditional algorithm in your school. Are you told this despite it being included in the textbook you are currently using?

You may be interested in an article I just wrote for online Atlantic on math teaching. It is located at:
http://www.theatlantic.com/national/archive/2012/10/its-not-just-writing-math-needs-a-revolution-too/263545/