Points on math education

I've been lazy lately when it comes to blogging and I'm sorry for that. I've been sort of taking time off from computer work and painting some windows since it just needs done at our house. I view it sort of as "therapy", since it's so different from the computer work and I just take my time and I don't have to think that hard. I like painting.

Anyways, I foud something really nice at Mathmom's. She's written, I feel, an excellent piece about problems in math education in elementary grades. Her point of view is of classroom instruction, but it's still really relevant even for homeschoolers. Some excerpts:

On calculators:

But to be honest, as much as I hate calculator use in school, in this age of calculators and computers, efficiency at hand computation is not, IMO, the most critical math skill for kids to learn. I am NOT saying that it should be ignored, or that kids should be allowed to skip it, and just use calculators in class (see rant linked above). But it is not, IMO, the be all and end all of math education, nor is it a prerequisite, IMO, for studying anything else.

What I consider even more important is a strong sense of number. I want kids who know immediately when the answer they got (either by hand computation or with a calculator) is way off. I want kids who have an instinctive understanding of the distributive law before it is ever formally taught or named (12 sevens is obviously the same as 10 sevens and 2 more sevens). I want kids who know when the amount of change handed to them makes no sense. I would rather have a kid who can multiply 64 x 25 mentally (by halving 64 twice and doubling 25 twice, to see that it's equal to 16 x 100 = 1600) than a kid who can sit down and carry out the long multiplication with pencil and paper, by rote.

I feel that we need to consider several things when it comes to calculators. It's best when the kids can do mental calculations and do paper-and-pencil methods, including understanding why they work. Calculators should be used judiciously, but used. Like she mentions, number sense is of paramount importance so that kids can estimate their answers and tell if the calculator "got it wrong" (e.g. they punched wrong buttons).

On spiraling curricula:

Steve is right that a spiral curriculum can lead to a lax attitude of "it's ok if they don't master this now, because they'll see it again later" that goes on ad infinitum, and the kid never masters anything. This is clearly no good. But the solution isn't necessarily to take away the spiraling for those who need it, IMO. The solution is to have limits - for example, it's ok if they don't completely "get" long multiplication when it's previewed in 3rd grade, or even when it's introduced more formally in 4th, but they have to get it when it's reviewed in 5th, or they shouldn't move on.

This sounds like a really sane approach.

She also talks about including non-routine problems for ALL students to solve. I definitely recommend this practice and have written about it before! MathMom gives several good reasons for this:

1. First, it provides a fabulous way of helping students to appreciate the uses of the procedures and skills they have learned or are learning.

2. Second, this is the kind of thing that "real mathematicians" do! ... A "mathematician" does not sit down and solve 25 ratio and percent word problems, knowing exactly which skills are required to perform the computations. Instead, she investigates "puzzles", looks for interesting patterns makes new discoveries, generalizes results.

3. Third, it develops self esteem and confidence.

4. Fourth, it builds transferrable problem solving skills.

Read it all at Ramblings of a math mom: Math wars.

High school geometry - a review

It's done! Finally! Took me some time to finish this review, perhaps because it involved three products:

• The book Geometry: A Guided Inquiry. As the name suggests, this book is based on letting students learn about theorems and their proofs in the setting of "guided inquiries" or interesting problems. It is quite unique in its approach.


• A Home Study Companion which includes solutions and about 300 interactive demonstrations


• Geometer's Sketchpad - dynamic geometry software.

This review isn't just what you typically find on the web; someone called it an "exquisite in-depth review". It's fairly long... with sample pages and other pictures, examples, and more.

I encourage you to read it even if you don't need a high school geometry book right now... because you'll get valuable insight just HOW GOOD geometry instruction can be, how the book handles proof, or what to think about an axiomatic vs. discovery based geometry text.

Review of Geometry: A Guided Inquiry with Geometer's Sketchpad and Home Study Companion.

Free math videos...

...for this week only!

MathTV.com is letting us test-drive (public beta) their new website for FREE for one week only:

http://www2.mathtv.com/

Hundreds of videos on pre-algebra, algebra, trig, and calculus topics. Even in Spanish.

Brain rules

This was an interesting website, and can give you some insight into how our brains work -- which is always a good thing to know, to enhance learning!

It is based on a book, which explores the 12 brain rules in detail. These rules or principles have to do for example with exercise, sleep, stress, how our memories work, and so on.

On the website you can read and watch little videos about each. Warning: the video about "attention" is illustrated by an immoral "attention getter". But you will glean some good ideas and learn pretty interesting stuff in the others; I enjoyed most of it:

Brainrules.net/the-rules

It's been quiet...

at my blog lately and I'm sorry for that. But, I have something in the works.

I'm working on a review of a "three-combo" geometry curriculum for high school geometry:

• The textbook Geometry: A Guided Inquiry


• Geometer's Sketchpad - a dynamic geometry software


• A Home Study Companion to accompany the book, which includes solutions and dynamic illustrations with Sketchpad

I enjoy doing geometry. This book has problems that "evolve" from one to the next, leading to important conclusions. Then it has really interesting projects, and many of them can naturally be constructed with Geometer's Sketchpad, which is lots of fun - for me anyway.

The problem is, I cannot afford to take the time to study and solve all these projects and interesting problems, but just to pick and choose one here, another there.

Anyhow, I hope to complete this review within a week.

Backwards math

This is just a really cute story of a 3rd grader who on her own figured out a way to do "backwards math"...

Backwards math

Kindergarten math

People occasionally ask me about kindergarten math, and if I'm going to write kindergarten level books for the Math Mammoth series. The answer is no, I don't feel there's any need for me to write books, because there already exist plenty of good materials for these very basic and easy concepts.

BUT I did write a comprehensive article about what you can do in kindergarten math, including many games you can play, and what basic concepts should be covered in order to prepare for 1st grade.

Problem solving & math as an art

Continuing on a litlte bit more with my thoughts concerning Lochart's Lament.

Lockhart starts out his lament with a comparison: WHAT IF music teaching only consisted of learning to write music, write notes on paper, and only after high school level would students be allowed to actually hear and make music? WHAT IT art instruction would consist of "paint by numbers" until high school, which is when they'd actually start applying paint...

Lockhart remarks that if he wanted to destroy a child's natural curiosity and love of pattern-making, "...I simply wouldn't have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education."

He calls school mathematics "pseudo-mathematics", where emphasis is on the accurate yet mindless manipulation of symbols.

These are, of course, very strong words. I don't fully agree... I don't feel all that's done at school would be pseudo-mathematics or mindless manipulation of symbols. Really, you CAN teach addition so that it makes sense, and we need to learn the symbols for it (5 + 6 = 11).

Lockhart uses the triangle example that I blogged about before to illustrate how mathematics is an ART:

"To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion -- not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive."

I can see where he's coming from... yet I wouldn't put down so harshly all that is done within school mathematics. There IS a place for drills, for computation practice, for studying algorithms, measuring units, and so on.

Lockhart says, "The main problem with school mathematics is that there are no problems." Here I agree. We should add GOOD PROBLEMS - or true problem solving to our lessons. I don't mean "exercises"...

"But a problem, a genuine honest-to-goodness natural human question -- that's another thing.
How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind's engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them).

A good problem is something you don't know how to solve. That's what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions. A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?"

In other words, good problems do not simply practice a technique or idea that the student just saw used in an example. Notice: a good problem is such that you don't initially know how to solve it.

Now, this does not mean kids shouldn't solve "routine" problems or exercises, because ONLY by having KNOWLEDGE of techniques and concepts can new, non-routine problems be tackled.

But I'd recommend you take some time, perhaps a day every two weeks, where your math lesson consists of problem solving so that your students CAN experience this kind of problem solving process that leads them to conjecture, to investigate, to THINK hard, fail first but persevere, to justify their thoughts -- to come up with proofs.

It is of course even better if a teacher can lead the teaching with such good questions or problems even more often, such as starting the lesson with an interesting problem that leads to a new concept. But I realize not all of us can do that, and that it may take more time than direct instruction. If you can give them at least a glimpse of it sometimes, feel proud, because then you've done better than many.

... For good problems, check some resources here.

Carnival of Homeschooling - the April fool's day edition

Carnival of Homeschooling is up at Why Homeschool. This one is the April fool's day edition, and Henry has sprinkled the carnival with juicy stories of past April fool's day jokes from BBC or others.

Check also Denise's entry for subtracting mixed numbers.

Kindergarten question

How to teach subtraction in the kindergarten with the tens place values and in horizontal sequence, for eg, 20-5=? They know to count their fingers, from the smaller number to the bigger number, but when the fingers are not enough to count, then what?

I'm not sure kindergartners are necessary ready for this. I would ONLY do these types of problems with concrete help.

Get a 100-bead abacus. Instruct them to first "make" 20 or show 20 on the abacus, and then move 5 away. Then "see" how many are left.

This online abacus is also really good for illustrating such.

Once they've done 20 - 5 and many other problems, you can ask if they notice a similarity in these problems:

10 - 5

20 - 5

50 - 5

70 - 5

etc.

But if some don't, wait till 1st grade. Practicing problems that "cross the ten" without manipulatives, such as 23 - 5 or 71 - 9, can wait even till 2nd. I realize kids might be able to do them by counting down, but to learn effective strategies for solving these kind of problems, children need a good foundation of place value (tens and ones), and that doesn't come immediately.

I'd use addition problems first, and a little easier ones, such as

20 + 7

44 + 2

62 + 7

and so on. These can help cement the place value concept as well, if you approach them right, such as, "62 has 6 tens and 2 ones, and then you add 7 more ones. Can you do that on the abacus, or with your ten-bundles and sticks, etc.?

With subtraction, try problems that "stay" within the same ten, such as 65 − 3. The idea is to use concrete aids and help them see they don't need to "touch" the whole tens when subtracting, so it's really similar to doing 5 − 3.

But still, I feel even this is not essential for kindergarten but can wait till first grade.