## September 21, 2006

### Conceptualize This

The Trib is running another NCTM backpedals story today. Here's the set-up:
For nearly two decades, a battle has raged over the best way to teach elementary and high school math.

On one side sit fundamentalists, who prefer old-fashioned drilling and a focus on the basics. On the other side are the so-called "new math" proponents, who care more about understanding the concepts than performing the calculations.
Then the Trib gives us an idea what the new-math loonies really mean when they tout that they "are more about understanding the concepts." Behold constructivist math in all its glory (click to enlarge):

This monstrosity is called lattice multiplication. Supposedly, if you teach expose kids to multiple ways to multiply, such as lattice multiplication and a few other loopy methods now consigned to the ash heap of history, kids will be able to conceptualize math better.

Nonsense. I see no conceptual advantage in learning the lattice method either in lieu of or in addition to the standard algorithm. The only "advantage" to lattice multiplication I can see is that it separates the one digit multiplication steps from the carry steps which our standard algorithm combines.

This means that students have to keep less facts in their head to solve the problem. This benefits kids who do not have quick recall of the multiplication facts and prevents them from over-taxing their working memory. The new math focal points supposedly rectify this "quick recall" shortcoming in the previous NCTM standards.

So does the lattice method help students conceptualize multiplication better or is it just a crutch for students that haven't mastered their basic multiplication facts? You be the judge.

I'd say it's a ham-fisted "solution" to a problem they created for themselves by downplaying practice in favor of calculator use.

MikeZ said...

Somehow, I don't find that any easier. More than that, what's wrong with Russian Peasant Multiplication?

KDeRosa said...

Don't get me started on russian peasant multiplication. Isn't there an egyptian multiplication algorithm too?

KDeRosa said...

Indeed there is.

Now someone tell me how learning all these methods helps a student conseptualize multipication.

Anonymous said...

Most Russian peasants that I met used an abacus for just about everything. In fact, they would calculate everything on an abacus even though they had a calculator handy. I doubt any Russians use the Russian Peasant method. From what I saw, they all abandoned it in favor of the abacus.

Anonymous said...

"This means that students have to keep less facts in their head to solve the problem."

This is the big driving force in selecting algorithms - not conceptual understanding. Actually, the partial products method provides more insight than the lattice method. Then again, partial products doesn't force kids to keep track of carries in their heads. This "less brain-power" approach can also be seen in the popular "Forgiving Division" approach. Kids don't have to figure out how many times 23 goes into 96 in their heads. And they always carry on about developing "number sense".

"So does the lattice method help students conceptualize multiplication better or is it just a crutch for students that haven't mastered their basic multiplication facts?"

The answer is obvious. Last year, my son got both the Lattice Method and the Partial Products methods in EM. If they were truly interested in understanding, they would have the kids figure out why both techniques work and come up with the same answer. They would also add in the traditional method to figure out. Of course, kids only have a small clue as to why any of these algorithms work. They could do it 25 different ways and they would still not know. Why not wait until algebra when the teacher can spend one day on the task?

Conceptual understanding is a false position and the only straw they can grasp at. It gives them surface credibility and allows them to avoid the real issue. The true problem is the lack of practice and mastery of ANY algorithm. They see no linkage between mastery and understanding. Mastery is too much like "drill-and-kill" and that is considered to be a filter for math. More kids can get further in math if you reduce or eliminate all of the specific, tangible requirements. With no real understanding and no mastery of the basics (or anything), what do you have left? Fuzzy Math.

KDeRosa said...

More kids can get further in math if you reduce or eliminate all of the specific, tangible requirements.

That's why there is such an emphasis on data analysis. It wastes a lot of time and is not nearly as cumulative skill-wise as math is.

Mastering fractions and decimals means mastering lots of skills that build on each other. Mastering data-analysis means mastering counting a first grade skill.

Anonymous said...

Isn't there a need and value to data analysis and probability? Aren't you selling these skills a little short?

After all, statistics, economics, trade, business. . .all of these fields rely heavily on people that can handle data. Data is increasingly important given the sheer volume that is generated by computers and the internet. Perhaps these state tests are simply acknowledging a need for a foundation in data handling.

Don't shoot me . . .

Anonymous said...

"Isn't there a need and value to data analysis and probability? Aren't you selling these skills a little short?

After all, statistics, economics, trade, business. . .all of these fields rely heavily on people that can handle data."

In business school, yes, but trust me on this one, because I know from lots of experience, they can't do it in business school unless they have those math skills.

Nobody should be doing simulations in high school. No high school student has the knowledge to understand what a simulation is, much less set one up or interpret the results.

Anonymous said...

"Don't shoot me . . ."

Ken's getting a reputation here.:)

I'll try to answer some of what you're saying. Yes, I agree that data analysis is becoming a more important skill and that maybe some introduction to it isn't a bad idea in the early grades. The problem comes when it starts pushing out fundamentals that are critical for success in algebra, and that's what I think is going on. Add that to the education system's disdain for doing anything that's been done before (traditional=bad) and you really can have a mess.

I've had two kids go through this so I've spent a lot of time thinking about it. One of the flaws in the execution of it(pushing middle school skills down to grade school) is how quickly multi-step problems with statistics and graphs enter the picture. Multi-steps with algorithms (too rote) or regular word problems have not yet been mastered or even touched on in some cases, but somehow the ability to analyze graphs on several different levels, is developmentally do-able.

Word problems don't have more than one or two steps in the early grades, but I've seen word problems with graphs and charts containing several steps in them that come home with no advice or lesson on how to even approach such a problem.

And it isn't so much that it's a bad idea altogether, but that it's often sequentially out-of-sync with other skills that need to be taught and mastered at that time.

The teachers are definitely between a rock and a hard place in many cases because the state tests are loaded up with baby stats and graph analysis. The state tests also want some children to literally go backwards in their math development by answering some math problems with big pictures or essays when they already can do it in the abstract, a skill that is necessary for algebra and beyond.

When the Singapore curriculum starts to move to multi-step problems, it makes a big deal out of it. Whenever it moves to even more steps, it again makes a big deal out of it. The curriculum seems much more cognitively aware of how children learn than most of ours.

SusanS

KDeRosa said...

This is great.

The commentors are scared of me.

In anotherpost, two commentors did the research for me that I was too lazy to do.

And here, rightwingprof and Susan are answering questions for me.

All that's left for me to do is sit back and collect the paycheck. Oh wait ...

MikeZ said...

Speaking of Egyptian multiplication - but it looks amazingly like Russian.

This can only mean that the Russians came from Egypt.

As an aside, many Aztlan-oriented schools (public charter schools) are teaching "Aztec Math", which is really just the base-20 number system. This is supposed to get them in touch with their "cultural heritage".

KDeRosa said...

yeah that's a skill employers are clamoring for.

Anonymous said...

How can we talk about learning statistics and probability in high school? I could not imagine learning probability topics and solving probability problems without calculus. The topic is much more than counting coin tosses.

TurbineGuy said...

I aint scare of you. I have decided to convert to the dark side and embrace the obvious superiority of constructivism. I started with my 7-8 year old soccer team. Instead of traumatizing the children by teaching them meaningless skills like kicking and dribbling, I now just give them the ball and tell them to discover the game of soccer on their own. I don't even bother explaining the rules because as we all know soccer rules are based on an elitist white European culture. Instead, I let my kids create their own rules. To protect their self-esteem, each player earns a goal by breathing. One breath = 1 point. To address individual learning differences I have scheduled 17 different practices, because every kid is different. Since games (the sports equivalent of standardized testing) are culturally biased and ran by obviously prejudiced sexist referees, I developed my own standards of learning. I rated my kids against kids who learn by a more regimented style, my kids came out on top. For each kid who picked up the ball with his hands thereby demonstrating creative thinking, I awarded that team a point. Cold hard scientific facts proved that my style of coaching had a positive effect of 1 million standard deviations. My next step is to write a book...

Anonymous said...

"I could not imagine learning probability topics and solving probability problems without calculus."

I think this is the difference between continuous probability distributions and discrete probability distributions. It makes sense to start with discrete probability distributions, which map nicely to tossing coins and rolling dice.

Discrete probability is perfectly reasonable to teach high schoolers.

-Mark Roulo