What is the point of an answer key? A quite interesting question I had never really considered before. Maria Miller of the Homeschool Math blog has been exchanging emails with a translator of a Russian geometry book. Here is an excerpt I found most interesting:
Teacher’s guides are intended to save a teacher, clueless about the subject he teaches, from the embarrassment in front of the class. They don’t make him less clueless for, if they did, then teacher’s guides would be used in place of textbooks. With or without teacher’s guides, a clueless instructor–teacher or parent–is of little help to the student. Conclusion: teacher’s guides are obstructions to learning–for the instructor, and therefore, for the student.
This reminds me of how I was taught in Germany. We had no text books in any subject. We came to class and the instructor, largely through discussion and question & answer, guided us through the principles of what we were to learn. I did very well in math, which was strange considering how poorly I did with it in the US. At first, I chalked it up to the fact that the concreteness of the numbers and signs made it the one subject I could follow despite the language barrier. But there was more to it than that. In the US, I had always sat with the problem, looked at the sample problem, and tried to copy the steps to arrive at an answer that I then checked in the back of the book. I learned math by rote memorization and drill, without ever really comprehending what I was actually doing.
I couldn’t do that in Germany. I didn’t have a text book and I didn’t have an answer key. I had a thin sort of problem book which had problems for each day, but there were no examples or solutions. I had to struggle through with what I knew of mathematics to try to come up with an answer. In class, we discussed these. I never knew it was possible to discuss so much about mathematics. And, completely foreign to me, we were NEVER given the answer. After discussion, we reworked the problem and could discuss it more. The teacher might point out an error, but left us to solve it on our own. We could conceivably spend a two hour block on two or three math problems. I have never looked at mathematics quite the same way since.
It also reminds me of one of the goals of the Principle Approach…to free us through personal mastery of the subject matter rather than dependence on others for watered down materials, such as a textbook, workbook or teacher’s guide. This is the most daunting aspect of beginning…that there is no text. No neat curriculum package that tells what to cover each day. Just you, your child, your bible and your knowledge of what you are about to teach. I do not act as an instructor in an American public school, who delivers instruction the child is expected to consume. Instead, I attempt to lead, guide, direct and, above all, inspire.
To infuse into the mind; as, to inspire with new life.
Through my guidance and modelling, I seek to make a connection to my child’s heart so that she will discover the principles of each subject for herself. With this mastery, she can then judge right from wrong, whether that be in her daily choices, her math lessons or any other subject she tackles. For in life, there is no answer key. If you judge by someone else’s standard without being able to judge the rightness for yourself, you will always be dependent on others, even for your ideas.
Related Tags: math, teaching math, Principle Approach, mathematics, teaching, pedagogy, homeschooling, home school
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Very interesting! When did all this take place? Do you know if they still teach that way in Germany?
Unfortunately, even many teachers are not knowledgeable enough about math to be able to teach it that way, let alone parents. But it sure sounds inspiring!
This was in the early 90s. I would suspect this only applies to the Gesamtschule or the Gymnasium. Germanny has a three tiered education system, with the Gymnasium at the top. These youth are prepared for university…and, really, upon completion of the Gymnasium program they have the equivalent of a two year degree here.
The Realschule only educates to the 10th grade and the Hauptschule to the 8th grade, with no chance of higher education. The Gesamtschule is/was a relatively new concept, very rare, but gave “promising” realschule students the opportunity to take coursework to prepare for the Abitur and hence university work.
I’m sure our honors and AP math classes are taught differently than our “fundamental” classes!
My kids always have complete access to the answer keys. If they can’t work forward, they will work backward to figure out how to figure it out. They don’t seem to have the concept of using it as a crutch or to provide “the” answer; it is another tool for developing mastery of the material. Their goal is understanding, not completing a certain number of problems with an answer that is deemed correct by anyone else.
So, answer books work well around here, adding to the intellectual vitality.
That’s how I always did math when I was a kid, too. I think that is how a lot of us do it. I just found the exchange Maria had with the Russian translator interesting and it related to my experience in Germany. I look at things a bit differently since then.