Question

Dear Mitch,

My cousin said she heard that all of the Adler-n-subtract.com products are completely or partially handmade by former students of yours who are still in high school or college and need some income, or teachers who are retired or on maternity-leave or something like that.

Is that true?

Sincerely,

Janet Smithwick

Answer

Dear Janet,

Yes, it is true. We try not to make a big deal out of who makes the products or how exactly they are made because we prefer to keep the focus on the effectiveness of the products for "Painless learning and teaching." Although there has been the occasional criticism that -- like all hand-made products -- there is some variation from item to item, as when one Mom bought three copies of ** FISH IN SNEAKERS **and found the colors on a couple of pages differed from book to book. (How could we have known they were for her triplets who MUST have the same exact EVERYTHING!?!)

So we are working to bring a little more consistency to our products, but we will stick to our mission statement (which we never officially shared with the public), which includes the fact that the majority of our products are made from recycled materials; we do this not merely to keep the cost down (which is not always the case), but because we firmly believe in it. Any thinking person doesn't need to do much mathematics to realize that it is better to **subtract fewer trees** from the world's forests to manufacture more and more products, as these products are likely to eventually be **added to the world's pile** of items to be discarded or recycled yet again. Even recycling, in most of its forms, require the use of electricity and oil and the gasoline burned by large trucks to bring materials to and from recycling plants.

As many people have mentioned in the letters we've received, they first read about my approach to teaching math and developing math products when I taught in small schools with limited budgets and realized that it was as easy to develop and demonstrate math concepts with items pulled from any classroom's trash can as it is with the surprisingly expensive products designed for that purpose and purchased by most school districts in the United States from educational supply companies. And I soon discovered that there were distinct advantages to using everyday throwaway things to prefabricated math "manipulatives". Overall, materials made from the things people see every day in their 'real lives' makes it clearer – actually, makes it obvious and irrefutable – that MATH IS REAL. When I first began as a teacher of math, and then, a short time later, as a person directing and managing a large math department, I observed that many students (and a surprising number of teachers), believed that standard, store-bought math tools, 'manipulatives', and kits of such products, were necessary to the teaching of math. They are not necessary. In fact, there is a quiet form of damage that often occurs to a math program when such products are expected by students and relied upon by teachers. That 'damage' comes from the unstated message that math is so artificial or – at the least – separate – from everyday life and everything we use in life -- that to 'work' out perfectly one needs to use products that are produced for the sole purpose of proving that math is real. But what's 'real' about plastic cubes or rods or caps that are seen nowhere outside a classroom and serve no purpose other than to prove that plastic cubes and rods and caps designed to demonstrate math actually can demonstrate math?

The goal in teaching and learning mathematical concepts is to enable everyone to understand the workings of numbers and shapes and their relationships and patterns so clearly and organically that they naturally generalize their new knowledge and see how it applies everywhere – in the way stores divide up their merchandise among their aisles, in the way products are packaged, in the way people develop the pace at which they walk based upon the length of their legs and the time they have to get from place to place, etc. By using real items from everyday life – the packages, bottle caps, and family photos that students can collect to calculate proportions -- the teacher has successfully skipped the first artificial step along the way to getting students to do this generalizing. Additionally, using materials such as milk cartons and plastic water bottles typically requires some adapting of them for lessons, and it is almost always the case that students – and teachers—learn at least as much (but usually more) during the 'making' and preparing stage as they do during the official 'lesson' using the finished products. Why? Well, first of all, it generally requires more thinking, exploring, revising and accurate work to construct something that can then be used to demonstrate the truth of math concepts than it does just to use the product. But that time and energy involved in the construction actually IS the use, so it is the opposite of the waste of time some might consider it upon first observation. If classroom time is so limited that the students cannot make the products and then use them, and a choice has to be made, in almost all cases I would go with the making of the products, and forgo their final use. I would (and have) require the class to make the materials for another class's lesson, and by the time the products were complete the majority of the students involved in the manufacturing of the products would know exactly how the concepts work. Even with the most old-fashioned, least imaginative, and least engaging of all math learning tools, the dreaded flashcards, the students who use store-bought cards that come already printed are missing out on the main learning opportunity cards can provide; typically, the students who make their own with blank cards find they learn them faster, and have to study them less, because (whether they realize it or not, and they usually do not), they are spending more time with the cards and are required to do the hardest part themselves. For a student to study the cards he makes for twenty minutes, first he has to spend about twenty minutes making them, which of course combines to forty minutes, though almost never does anyone consider the time making them to be part of the studying of them. Yet even the least careful student will make a more concerted effort to be accurate when making the cards than when studying them, because NO ONE wants to waste his time studying cards that aren't correct. This is especially true for the student who would rather not study at all. And, optimally, after each study period the cards are discarded and new ones are made each time studying begins. (And since more senses are required to make cards, the memory of writing all the digits, the checking to make sure they are correct and that the maker has not repeated or missed making any that he is expected to know, that learning time is more productive than the easier –but-less effective part in which they are stared at and lips are moved to drum their sounds into one's memory bank. To guarantee that time is maximized during this phase, a parent or teacher is wise to obtain blank cards in assorted colors, so that each day or two a new color is given to the students and is easily distinguished as the new blanks to be made into a correct set. The old ones can be discarded; their role is diminished with their fading effectiveness.

** QUESTION**: WHAT IS THE ADVANTAGE OF A THREE-LEGGED STOOL OVER

A FOUR-LEGGED STOOL??

**GIVE UP?**

** ANSWER**: A THREE-LEGGED STOOL CAN NEVER WOBBLE.

** NEVER**,

EVEN IF ONE LEG IS SUBSTANTIALLY SHORTER THAN THE OTHER TWO.

(WHICH IS WHY PHOTOGRAPHERS' CAMERA STANDS ARE TRIPODS, OR THREE-LEGGED.)

** WHY**? Well, mathematically, because three (non-co-linear) points define a plane.

** QUESTION**: Which math product demonstrates that in a way that gives it a real-life feel?

** ANSWER**: I DON'T KNOW. I really cannot recall offhand, though surely I would be able to locate something in a math teaching catalog...

** QUESTION**: What about with everyday trash?

** ANSWER**: EASY; reach into the class' trash can after a pizza party and you will probably luck into one of those miniature plastic three-legged stools that pizzerias often place in the center of the pie to prevent the box from sinking down onto the pie and compromising the beauty of the meal.

But if there is no three-legged stool in any nearby pizza boxes, three toothpicks or pins or matchsticks stuck into a small piece of cardboard will make the point just as clearly.

** QUESTION**: But what about that 'non-co-linear' part? Where the heck is anyone going to find that without an illustration in a math book or fancy gadget in an educational catalog?

** ANSWER**: "NONCOLINEAR" simply means that the three points are not lined up in a way that they can be connected with a ruler to make a single straight line. "Non" means NOT, "Co" is a route in English and almost a whole word in Spanish (CON) that means with, as in "cohort" or "coworker", and LINEAR means having to do with a straight line.

"But," a student will comment, "I don't get it. Why can't three points in a row form a plane?"

** Answer**: Because three such points can lie in more than one plane. In fact, they will lie in an infinite number of planes.

NOW, THE REAL WORLD DEMONSTRATING THE REALITY OF MATHEMATICAL CONCEPTS IN ALL OF THE REAL STUFF OF LIFE:

POINT TO THE DOOR LEADING FROM THE CLASS TO THE HALL, OR A DOOR THAT OPENS TO A CLOSET. CHANCES ARE THE DOOR HAS THREE HINGES ON IT, NOT JUST TWO.

SO?

THOSE HINGES REPRESENT POINTS ON A PLANE, AND THEY HAPPEN TO BE LINED UP, SO THEY ARE COLINEAR, __NOT__ **NON**COLINEAR...

SLOWLY OPEN THE DOOR OR CLOSE IT, AND WATCH HOW THE DOOR'S ARC GOES THROUGH A THREE-DIMENSIONAL SEMICIRCLE OF MICRO-THIN PLANES ALONG THAT JOURNEY. Which is why hinges need to be put on carefully, aligned in a *straight line*. Which is why if you remove one of those hinges and install it in another edge of the door, such as the top or bottom or handle-side, the door will be locked into a single plane.

And that's that.

Any student who feels he would like to take some supplies home to review the concept need only study a door in his own house. That should do it.

One final thing: Janet, your letter asked about two things: 1) the recycled material construction that adler-n-subtract.com tries to use wherever possible, and 2) the fact that many of our products are "homemade" by students and teachers. I addressed the recycled part of your question, and now I will address the "homemade by students and teachers" part with a picture. The picture was taken after a recent contest conceived and carried out by some of the students we employ to make our products. What you will see are some of the results, and you will probably be struck by the amount of variation we welcome. The guidelines, which we keep posted above the door to our

**"Prime Factor-y**"

are simple:

1) The aesthetics (visual qualities, such as decorations or paint color) cannot be of the type that is likely to distract a student from its use as a learning tool;

2) The product must be constructed well, so that it functions as it is supposed to and will last longer than consumers have come to expect products to last, and

3) Since it is a fact that an invention can always be improved upon, and since the ideas for many of the most important improvements arise during the actual construction phase, changes that are improvements are welcome.

All we ask is that the person making the product and stumbling upon that potential improvement pauses to share his idea with someone who puts his name on each and every product (me), to convince me that the change is an improvement. (Change just for the sake of change is not the goal, but true improvements can range from sturdier construction to ways of producing the same item for less cost that can be passed down to the consumer in the form of a lower price. )

One of our first products, which continues to be one of our most popular, is PERFECTOUT.

In the recent contest, students were asked to make one that was eye-catching yet sturdy.

And in the process, three important new developments arose, each of which I consider to be a substantial improvement:

1) By using a similar but different type of commercially available lock, a way was discovered to finally bring the very high cost down for each unit, while maintaining the important feature of having the device reprogrammable after each use to assure that recall of a set of combinations would not diminish the item's usefulness for studying and learning for standardized tests. And, AMAZINGLY, in finding this new set of locks, there came an unexpected improvement: Each PERFECTOUT now has an increased maximum number of questions that can be required learning for each "release". Until now, each PEERFECTOUT could "test" students' thinking for 28 questions. That has been raised to 30.

2) By sealing the inside edges with weather-stripping, the new PERFECTOUT'S are not only even closer to tamperproof, but provide the treat of springing open upon the last lock's release.

3) (AND THIS IS THE IMPROVEMENT WHICH HAS PROVEN THE MOST READILY APPARENT AS A SUBSTANTIAL ADDED FEATURE):

The "prize" inside the center window is substantially more secure from students who become prematurely frustrated and attempt to tamper with the device to obtain the reward he or she has not yet been able to earn through careful thinking and the development of all the necessary problem-solving skills that serve people best as they have to sit down for standardized tests and —even more importantly – stand up to face the challenges that life in the real world may present.

The additional reinforcement is constructed of galvanized steel crosshatching, lining the inside of both front and back panel window. Its horizontal and vertical bars leave half-inch square panes that enable the prize to be admired just as easily throughout the adventure, yet make it clear that honesty and hard work are the most assured way of gaining the rewards of success.

Hope this helps,

Mitch

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