Math is not a spectator sport.

Who said it first? Maybe Jerry Mortensen. Maybe George Phillips.

But hundreds, maybe thousands, of math teachers have said it and put it on their syllabus.

*Because it’s true.*

But does this mean that students have to “practice, practice, practice”?

**And what exactly does that mean?**

Say to me “practice, practice, practice,” and I’ll growl at you.

You’d might as well say to me, “I don’t know squat about your learning style and aren’t interested in finding out. But I know that you’d better do all 50 problems or you’ll lose points on your homework grade.”

So I prefer to think of this triad mantra in three phases: ** Practice Phase 1**,

**and**

*Practice Phase 2***…**

*Practice Phase 3*### Practice Phase 1 is isolated practice.

When a student learns a new math concept, he or she should apply it a few times, by itself. This is what’s offered in the problem sets of textbooks: isolated practice of the new math skill.

This practice is also an assessment phase. At many points here, the student can ask, “Can I do this? Do I understand this?”

Regardless of the answer, the student can proceed to Practice 2.

### Why “regardless of the answer”?

Practice phase 1 is what’s commonly referred to when grownups say “practice, practice, practice.” But over practicing can be detrimental.

Some kids need to nail the isolated practice before moving on.

*Some kids.*

And some kids will become annoyed with it and need to move on to see the value of the skill.

Refrain from giving 50 problems just because “practice makes perfect.” It doesn’t. **Practice sometimes makes frustration.**

Let your child move on to practice phase 2. You can even **let them skip assigned problems**. She or he may return to the isolated practice in phase 1 later or mentally isolate the skill within the “bigger picture” in phase 2 or 3.

### Practice Phase 2 is “big picture” practice.

This practice phase helps solidify the math skill. It should teach the integration of the skill into the full toolbox of math skills.

The math skills ancillary to the main skill being practiced, are themselves being practiced (hopefully in phase 2 or 3). How fluent a student is in these ancillary skills has an impact on his or her perception of achievement of the main skill.

Textbooks offer a problem set for this phase, but it’s often disguised. It just looks like harder problems.

It’s helpful to **point out the ancillary skills** being practiced. This helps affirm how much a child is learning and aids in building confidence.

### Practice Phase 3 is ongoing.

Real skill is realized when it’s integrated without thought. For me this was always at least six months after first learning a skill. Sometimes it happened years later.

And there are some things I didn’t “get” until decades later.

Which means that **practicing a skill must be continuous** – even if a student doesn’t fully understand the logic underlying the process.

Of course this phase is subtle if not completely invisible. But instructors can point out when the “old skills” come into play long after they are learned.

### Example: Factoring

**Skill:** Use the distributive property to remove a common factor from an expression.

**Practice Phase 1 – Isolated**

**Practice Phase 2 – Big Picture**

Here the skill is integrated with factoring numbers:

Here it’s integrated with rules of exponents:

This one’s integrated with combining like terms:

Here there are fractional exponents:

**Practice Phase 3 – Ongoing**

At some point a student might notice that combining like terms is equivalent to factoring the common term then combining the numbers, like this:

### So, really – how much practice and when?

A student should practice in phase 1 until he or she is comfortable. Remember, over practice can be annoying and seem punitive.

And punitive math is never effective.

A student should practice in phase 2 based on his or her comfort level. This is a great time for parents/instructors to notice weaknesses in other skills, too.

Practice phase 3 is, well, ongoing.

It’s really helpful to note, out loud, where old math skills come into play. Instructors should always be on the lookout for ways to do this.

Even things like long division show up in the more advanced skill of polynomial division.

### Pay attention to your student.

Some kids may respond to the “practice, practice, practice” mantra. Some may get annoyed at it (like me). If your child needs and wants more top-down learning, allow it.

Don’t force phase 1 practice – offer it.

And if you see a deficiency in an ancillary skill in phase 2, bring them back to phase 1 on that one.

Math really isn’t a spectator sport. But that doesn’t mean you have to drill like crazy on it. Adjust the teaching and practicing according to the child.

*Thanks to my math cousin, Vijay, for inspiring this article. *

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I was a student who resented the 40-60 nightly homework problems. (Hello? Did they think that MATH was the only subject we took??) I generally refused to do the homework if I was comfortable with the material (yes, even in elementary school…. I started my rebellion in 4th grade). I would generally get 97% or better on my tests, but I’d earn steady C’s in the class because my homework grade would be next to zero. I frustrated my math teachers. I wish I could say I feel bad about that, but they frustrated me with busy work.

Many of my classmates made C’s and D’s in math.. and they actually DID the homework. What good are all of those math problems if the student doesn’t “get” the concept in the first place? It was STILL busy work… it was just busy work they didn’t quite understand.

I’m sure that some students really did benefit from the “practice” that the homework provided, although I’d be willing to bet that for most of them it was purely a rote, procedural benefit and not one of understanding. Similarly, there were the top students who aced the tests AND completed the unspeakable homework assignments. I still envy their fortitude. They still make me feel lazy.

Thanks for this!!!

Oh – RockerMom! You are inspiring me to admit something! I’ll write it up in a post and link back soon!

(teaser: I resented the huge homework sets too, and I was BAD in how I dealt with it.)

I like how you laid out the three stages of practice, but something you said struck me slightly. As you were talking about practice being continuous, you say, “Which means that practicing a skill must be continuous – even if a student doesn’t fully understand the logic underlying the process.”

Do you mean that students may not understand the logic behind the practice itself, or do you mean that they may not yet understand the logic behind what they are practicing?

To the latter, shouldn’t our goal be to get students to have a strong conceptual understanding behind the mechanics that they are practicing?

Knowing how to factor an array of polynomials would mean little if the student did not understand why the operations they are performing work and make sense.

If you meant the former, then I can understand how students may not fully grasp why we are taking them through the three stages. I don’t think that we should keep up a smoke screen to hide our methods, but I don’t think you were suggesting that.

Thanks for asking, Chris.

Indeed I didn’t think about the variations of that. I meant that a student doesn’t need to know how the math works or why it works. I have many examples of times in my own life that I hadn’t a clue why something was the way it was. The first was learning to take derivatives of polynomials when I was 7 years old – and loving it! (Purely accidentally – I was watching a foster kid do her homework.)

You can see the pattern or “game” of doing something and be able to apply it elsewhere without understanding what’s going on with it. In fact, as we grow in mathematical maturity, we discover amazing things over and over. Often these things could be considered “understanding” skills or tasks we would have been doing for years!

Math is so incredibly complex that “understanding” all the bits under the skills is just flat impossible for anyone. So why force it? Why not allow kids (and grownups) to accept that perhaps they might not understand it now (whatever that really means). But eventually they can – in their own way.

Know how to do the steps, watch where similar things come up and one day you’ll see how it all works together. And enjoy the journey.

(I’ll likely expand on this in an article soon. Thanks so much for the thought provoking question, Chris!)

I am wondering what your take is on today’s on-line tools for math practice. Khan Academy, for example, let’s me move on if I get five in a row correct. I am not sure that is the right formula, but something along those lines would avoid a lot of this debate over practice.

In the past when students worked on paper at night and brought it in the next day for assessment there was not much room for abbreviating practice at the point of mastery.

Do you think automated tools can help on this issue?