Our world is struggling with education of all sorts. We know that STEM fields are in trouble. Not enough people are excited about taking the science and math classes needed to jump into them.
But there’s another piece of the education puzzle that’s missing — writing.
I talked to a primary school teacher at a social function today. She was telling me how students these days were often taught the algorithms of writing. One of which is the classic intro-supporting paragraphs-conclusion that I remember.
I said, “Oh, so they don’t get taught to write in their own voice, and stuff like that?”
“That’s just it,” she said. “They naturally write in their voice. But with the systematic methods we push on them it destroys it!”
That sounds familiar!
It pulled on my heartstrings — there was another basic subject that was competing with the almighty and all-powerful queen of subjects — reading.
Seems writing has the same challenges as math — people think there’s a formula to it when really it’s about voice, personal preference, beauty and art.
And writing, like math, is something people always do when they have to but rarely do just for fun.
What do we learn — really?
It made me really think about learning and teaching. All learning can be boiled down into one of three categories:
Learning information (historical facts, spelling, times tables)
Learning how to get information (reading, researching, googling)
Learning how to give information (writing, speaking)
Reading clearly has its place in #2, as does writing in #3.
But where is math?
When you teach math, are you only teaching facts (like in #1)? Or are you teaching children how to get or discover information (#2)? Or are you teaching them how to give or share information (#3)?
I’ll leave my answers for next time. Until then — what do you teach?
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, Practice Phase 2 and 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
\(3x + 3y = 3(x + y)\)
Or with letters:
\(xy + xz = x(y + z)\)
Practice Phase 2 – Big Picture
Here the skill is integrated with factoring numbers:
At some point a student might notice that combining like terms is equivalent to factoring the common term then combining the numbers, like this:
\(3x + 5x = (3 + 5)x = (8)x = 8x\)
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.
