### A parallel world

I hope you enjoy this (mathematically) inspirational piece that Alexander Givental sent to me. It relates to the discussion about the need of answer keys and a certain Russian geometry book.

Dear Homeschoolers,

I'd like to thank Maria Miller for initiating the discussion on answer keys and passing some of my remarks on to you. Yet I feel the need to communicate something directly.

The question emerged in connection with "Kiselev's Geometry" which happens to come in a very "unusual" form: with no teachers' scripts, parents' guides, workbooks, answer keys, solution manuals.

In fact, I don't see anything wrong with answer keys. While for the most of 600 problems in "Kiselev's Geometry", there is no any answer, as those problems ask to prove or construct something, there are also several dozens of computation problems there, and when I have a spare

day I might post a page with answers to such problems on the web. However, the whole discussion of the need and use of answer keys misses a key point, and I am writing to let you hear it.

The point is: there is a parallel world out there. In that world, people use the same words like math, geometry, textbooks, problems, etc. but these words have different meaning. You may judge for yourself how different it is.

In that world, a half of all children say they prefer math over languages and humanities (quote-unquote) *because in math you don't need to memorize anything*. In that world, the question "What can be done to make high school geometry less of a pain?" that HomeschoolMath experts are wrestling with, sounds an extreme form of absurd, because most students there refer to their geometry course as the most enjoyable learning experience. Authors of modern textbooks in that world warn their readers (I am quoting from memory): "even if you have not been good in math so far this may very well change because geometry is a different kind of math where your imagination and creativity matter most." Neither the authors nor their readers have ever heard of such oxymorons as "two-column proofs", "formal proofs", "informal proofs." Likewise, they cannot imagine that mathematics can be taught "without proofs". For, in that world math seeks deep, hidden relationships between seemingly unrelated matters, and, being a science, it is also concerned with *why* these relationships hold true.

Respectively, textbooks in that parallel world are not 700-pages long, 5 pound heavy, watery monsters prescribing daily diets for SAT-fitness, but ordinary books easy to read and carry around whose primary goal is to expose the subject (e.g. a math theory) as clearly and concisely as possible. "Teacher's editions" there are identical to student's ones, and the idea that for reading one book one may need another one that explains how to read the first one, would pass for a "postmaster" joke.

People in that world are not terrified with "word problems" and cannot imagine why anyone would be, as mathematics for them makes sense, and is a tool to make complicated matters simple, and not the other way around. Their teachers there understand the difference between concepts

and mere terms, and their students are embarrassed to answer questions, asking for mere rephrasing of definitions. The euphemism "strategies" for mechanical routines is not in use by the "parallel" people. They do prefer however the real problems which require thinking (not "a little" of it, but just as much as needed to find out what is true and why).

The question "How long are you ready to ponder over a math problem?" is replaced in that parallel world with "How little time may a math problem take to be worth your attention?" As a lower estimate, consider that getting into a college includes a written math test with 4 hours given for 5 problems. Respectively, it is not unusual - in that world - to spend the whole evening trying to solve 2-3 homework problems, or to fall asleep at midnight attempting the last one of them, and to wake up in the morning with a ready solution.

This picture of two parallel worlds may seem to you as surreal as Harry Potter's worlds of muggles and wizards, but there are two essential differences. One is that this parallel world is real, and exists not only in Eastern Europe, Russia, China, and in some past, but also to some extent here and now. The second difference is that you and your children *are welcome* in this parallel world. Those who choose to try moving there will get help; some will learn to fly (no broomsticks!), but most will at least learn how to walk without crutches.

However, if you plan to try yourself in the parallel world, you should be ready to live by its rules. The requests for teachers' scripts, parents' guides, solution manuals, and other crutches are understandable, but they may not be granted. Your children need to learn how to learn from ordinary books which expose scientific theories.

There is nothing inherently wrong with solution books to collections of math problems, they are found in the parallel world when needed, and people there know when and how to use them. "Kiselev's Geometry" has been in active use without such a solution book for over a century, which proves that in the parallel world the solution book for it is not needed. However in order to see this for yourself you need to read the book and solve the 600 problems. The choice whether to do this or not is of course yours.

Sincerely,

Alexander Givental

Tags: math, mathematics, teaching, geometry, philosophy

Dear Homeschoolers,

I'd like to thank Maria Miller for initiating the discussion on answer keys and passing some of my remarks on to you. Yet I feel the need to communicate something directly.

The question emerged in connection with "Kiselev's Geometry" which happens to come in a very "unusual" form: with no teachers' scripts, parents' guides, workbooks, answer keys, solution manuals.

In fact, I don't see anything wrong with answer keys. While for the most of 600 problems in "Kiselev's Geometry", there is no any answer, as those problems ask to prove or construct something, there are also several dozens of computation problems there, and when I have a spare

day I might post a page with answers to such problems on the web. However, the whole discussion of the need and use of answer keys misses a key point, and I am writing to let you hear it.

The point is: there is a parallel world out there. In that world, people use the same words like math, geometry, textbooks, problems, etc. but these words have different meaning. You may judge for yourself how different it is.

In that world, a half of all children say they prefer math over languages and humanities (quote-unquote) *because in math you don't need to memorize anything*. In that world, the question "What can be done to make high school geometry less of a pain?" that HomeschoolMath experts are wrestling with, sounds an extreme form of absurd, because most students there refer to their geometry course as the most enjoyable learning experience. Authors of modern textbooks in that world warn their readers (I am quoting from memory): "even if you have not been good in math so far this may very well change because geometry is a different kind of math where your imagination and creativity matter most." Neither the authors nor their readers have ever heard of such oxymorons as "two-column proofs", "formal proofs", "informal proofs." Likewise, they cannot imagine that mathematics can be taught "without proofs". For, in that world math seeks deep, hidden relationships between seemingly unrelated matters, and, being a science, it is also concerned with *why* these relationships hold true.

Respectively, textbooks in that parallel world are not 700-pages long, 5 pound heavy, watery monsters prescribing daily diets for SAT-fitness, but ordinary books easy to read and carry around whose primary goal is to expose the subject (e.g. a math theory) as clearly and concisely as possible. "Teacher's editions" there are identical to student's ones, and the idea that for reading one book one may need another one that explains how to read the first one, would pass for a "postmaster" joke.

People in that world are not terrified with "word problems" and cannot imagine why anyone would be, as mathematics for them makes sense, and is a tool to make complicated matters simple, and not the other way around. Their teachers there understand the difference between concepts

and mere terms, and their students are embarrassed to answer questions, asking for mere rephrasing of definitions. The euphemism "strategies" for mechanical routines is not in use by the "parallel" people. They do prefer however the real problems which require thinking (not "a little" of it, but just as much as needed to find out what is true and why).

The question "How long are you ready to ponder over a math problem?" is replaced in that parallel world with "How little time may a math problem take to be worth your attention?" As a lower estimate, consider that getting into a college includes a written math test with 4 hours given for 5 problems. Respectively, it is not unusual - in that world - to spend the whole evening trying to solve 2-3 homework problems, or to fall asleep at midnight attempting the last one of them, and to wake up in the morning with a ready solution.

This picture of two parallel worlds may seem to you as surreal as Harry Potter's worlds of muggles and wizards, but there are two essential differences. One is that this parallel world is real, and exists not only in Eastern Europe, Russia, China, and in some past, but also to some extent here and now. The second difference is that you and your children *are welcome* in this parallel world. Those who choose to try moving there will get help; some will learn to fly (no broomsticks!), but most will at least learn how to walk without crutches.

However, if you plan to try yourself in the parallel world, you should be ready to live by its rules. The requests for teachers' scripts, parents' guides, solution manuals, and other crutches are understandable, but they may not be granted. Your children need to learn how to learn from ordinary books which expose scientific theories.

There is nothing inherently wrong with solution books to collections of math problems, they are found in the parallel world when needed, and people there know when and how to use them. "Kiselev's Geometry" has been in active use without such a solution book for over a century, which proves that in the parallel world the solution book for it is not needed. However in order to see this for yourself you need to read the book and solve the 600 problems. The choice whether to do this or not is of course yours.

Sincerely,

Alexander Givental

Tags: math, mathematics, teaching, geometry, philosophy