Question

Dear Mitch,

We just found an SAT tutor for our daughter, but my husband is already a little concerned. My husband is an engineer and approaches math with paper and pencil (and calculators and computer programs, of course), but he is very suspicious of any of the new methods a lot of people use now. Our daughter only had one lesson, and this young man used what he calls "hands-on" methods that require the student to use blocks and plastic shapes and things like that to understand the questions.

As I understand it, the theory behind it is that if you involve real objects to come to the answer then it's more likely to make sense. But my husband thinks it's a lot of nonsense, and he's been mumbling that he can't believe we're paying for her to spend half the session playing with toys. This fellow comes highly recommended but is definitely known for using his own approach rather than the type you find in the review books.

Is my husband right? All we want is for our daughter to gain confidence for the test, because she's always done well in her schoolwork but is very nervous about the test and thinks she might even go blank.

Sincerely,

Pat D.

Bardstown, KY

Answer

Dear Pat D.,

Thank you for hitting upon a subject that is very dear to my heart!

I hate to say that anyone is "wrong", especially if we're not together so that both parties have a chance to defend his position and maybe even persuade the other to see something he might not have seen, and your husband surely knows your daughter better than I do!

However... without meeting any of the parties involved, I will say that in this case my bet would be that the tutor IS CORRECT.

Just because your husband is an engineer (and possibly a brilliant mathematician), he is NOT your daughter's math teacher, and he is NOT your daughter's math tutor.

In fact, I would also guess that math probably came naturally to your husband from the beginning, and so for him it might seem an annoying waste of time for HIS teachers to have brought in 'toys' to bring the lessons to life.

But you mention that your daughter's confidence is not high for the test despite fine grades. This is a hint that although math may have presented no particular issue to her in her math classes, she senses that there are little gaps of understanding, the type that can hide and stay dormant on a deeper level than the type that shows itself in schoolwork, but that tends to reduce one's confidence for all new problem solving situations.

First of all, 'hands-on' manipulatives are not that 'new'. In fact, in some ways, mathematicians have used them in private for centuries, cutting up pieces of wood, etc. to derive new formulas or to check to see if they were on the right track. I think many people know the story of Archimedes discovering the way to determine an odd shape's volume or capacity when he noticed the water rise as he entered his bathtub to relax from a day of frustrating, failed-attempts to arrive at his answer!

When presented with enthusiasm by a teacher or tutor who deeply appreciates the value of hands-on learning, teaching math with manipulatives is an EXTREMELY effective way of helping students of all ages and all levels learn math. I will give one specific example of a way that I teach SAT students using manipulatives, and I think it might give your husband a clearer idea of the connection between toys and multiple choice questions that require number two pencils to complete.

First, it is worth noting that the Japanese method of teaching math, used in virtually all Japanese math classes (and yes, that includes high school) is VERY heavily weighted by a hands-on approach, and students sit around tables solving problems with pitchers of water they pour into different sized cups to grasp volume, division, re-combinations, and other concepts. They use scales to check the balance of equations, using manipulatives to make abstract numbers into real items, etc. And in England, where there is less of an emphasis on the use of hands-on learning for mathematics after the very early stages, produces students who tend not to do as well on international competitions as the Japanese problem-solvers, the American problem-solvers, as well as the problem-solving students from many other countries. The English do, however, produce students who can rattle off the multiplication tables more quickly than the same aged students of many other countries, but not as quickly as a standard four-dollar calculator. Unfortunately, calculators -- and students who mimic them – tend not to be as sought-after by most employers as are students who have developed a set of skills to solve new problems!

Here is one example of a hands-on approach that I use to help students grasp a common question on standardized tests. Those tests, of course, include the infamous SAT Test.

I present a bunch of metal bottle tops (specifically, discarded lids from baby food jars). These lids can be balanced on top of each other to form a stack; eventually, of course, they inevitably come tumbling down -- but by then the idea has (hopefully) been communicated and understood!) I have the student estimate the height of one cap. Estimates range from one-quarter inch to half-an-inch. The truth is between those two, but either estimate works perfectly for the lesson. So, going with the quarter-inch estimate, I have them close their eyes (after all, the SAT people do not permit students to bring such 'toys' into the exam), and I have the student see if he/she could figure out how tall a stack of eight lids would be. (Remember, first we've taken a minute to 'play' with the caps so that they are as familiar to students who have no young siblings as they are to their peers.)

"Eight?" the student usually repeats, wisely making sure the question is not misunderstood at step one, "Well, eight times one-quarter... well, four times one-quarter would be one inch, so eight times one-quarter would be two inches... TWO INCHES!"

The eyes open and the student proudly smiles. I then hand over eight caps and ask the student to stack them. Somewhere between the second cap and the fifth there is a pause, then a gasp, then this:

"Oh!"... pause... "No, I'm an idiot!"

I assure the student that he/she is the opposite of an "idiot" or he would never have shouted that out so quickly. Then I ask, "What's wrong?"

The student points out that the shape of the cap allows part of each to go inside the cap placed on top of it (a subtler version of what happens when you stack paper cups from a sleeve of paper cups that has fallen down and come apart).

So?

Well, this, believe it or not, is a mathematical concept called "nesting", because each object is partially nested in the one above it. So, while one cap may indeed be one-fourth inch tall, two caps is less than 2 x 1/4^{th}. And so on.

And on the SAT exam?

Well, I can't recall ever seeing bottle caps, but look out for other objects with the same property. A mathematical favorite is shopping carts. If the shopping carts in grocery stores did not "nest", there would be no way to fit them in front of the store.

To accommodate fifty of the three-foot-long carts you would need one-hundred-and-fifty feet! And that would make quite a barrier for consumers to get through, don't you think?

But on an exam like the SAT, how would this show up?

Like this:

John Smithstone is designing a grocery store.

He is being offered a good price on shopping carts for the store. The shopping carts described in the offer are the perfect width for the design he has in mind (they are wide enough to accommodate any particular product he expects to sell, yet they are narrow enough to allow people to pass each other easily between aisles.) However, he is wondering about the length of the carts.

Each cart measures three-feet in length, and he wants to figure out how much room will be necessary to line up the 3-foot-long carts in front of the store. If he calls the ideal number of carts "n", which of the following formulas should he use to derive his answer?

A) 3(n)

B) 1/3 (n)

C) 3(n) + 1/3 ( n)

D) (n) + 1/3 (n-1)

Even if the student is not sure that 2/3 of each cart nests inside the previous one, leaving 1/3 of the cart beyond that point, there is only formula above that accounts for the fact that there is nesting involved, but that nesting does not begin until the second cart.

NOTE: Since we do not know the man's ideal number of carts, it could be one cart. (And that is possible, as some small city stores indeed contain only one or two carts, and the rest of the consumers are expected to use hand-held baskets).

The correct answer is D.

Just try it for 0 carts, 1 cart, 2 carts, and any other number you like. It is the only selection above that works for any number of carts --as well as any number of bottle caps!

Nesting becomes real with hands-on instruction, as do many, many other important concepts!

Hope this helps,

Mitch