Homeschooling Corner: A Mathematician’s Lament, by Paul Lockhart

Paul Lockhart’s A Mathematician’s Lament: How School Cheats us of our Most Fascinating and Imaginative Artform is a short but valuable book, easily finished in an afternoon.

Lockhart’s basic take is that most of us have math backwards. We approach (and thus teach) it as useful but not fun–something to be slogged through, memorized, and then avoided as much as possible. By contrast, Lockhart sees math as more fun than useful.

I do not mean that Lockhart denies the utility of balancing your checkbook or calculating how much power your electrical grid can handle, but most of the math actual mathematicians do isn’t practical. They do it because they enjoy it; they love making patterns with numbers and shapes. Just because paint has a very practical use in covering houses doesn’t mean we shouldn’t encourage kids to enjoy painting pictures; similarly, Lockhart wants kids to see mathematics as fun.

But wait, you say, what if this loosey-goosey, free-form, new math approach results in kids who spend a lot of time trying to re-derive pi from first principles but never really learning algebra? Lockhart would probably counter that most kids never truly master algebra anyway, so why make them hate it in the process? Should we only let kids who can paint like the Masters take art class?

If you and your kids already enjoy math, Lockhart may just reinforce what you already know, but if you’re struggling or math is a bore and a chore, Lockhart’s perspective may be just what you need to turn things around and make math fun.

For example: There are multiple ways to group the numbers during double-digit multiplication, all equally “correct”; the method you chose is generally influenced by things like your familiarity with double-digit multiplication and the difficulty of the problem. When I observed one of my kids making errors in multiplication because of incorrect regrouping, I showed them how to use a more expanded way of writing out the numbers to make the math clearer–promptly eliciting protests that I was “doing it wrong.” Inspired by Lockhart, I explained that “There is no one way to do math. Math is the art of figuring out answers, and there are many ways to get from here to there.” Learning how to use a particular approach—“Put the numbers here, here, and here and then add them”–is useful, but should not be elevated above using whatever approach best helps the child understand the numbers and calculate the correct answers.

The only difficulty with Lockhart’s approach is figuring out what to actually do when you sit down at the kitchen table with your kids, pencil and paper in hand. The book has a couple of sample lessons but isn’t a full k-12 curriculum. It’s easy to say, “I’m going to do a free-form curriculum that requires going to the library every day and uses every experience as a learning opportunity,” and rather harder to actually do it. With a set curriculum, you at least know, “Here’s what we’re going to do today.”

My own personal philosophy is that school time should be about 50% formal instruction and 50% open-ended exploration. Kids need someone to explain how the alphabet works and what these funny symbols on the math worksheet mean; they also need time to read fun books and play with numbers. They should memorize their times tables, but a good game can make times tables fun. In short, I think kids should have both a formal, straightforward curriculum or set of workbooks (I have not read enough math textbooks to recommend any particular ones,) and a set of math enrichment activities, like tangrams, pattern blocks, reading about Penrose the Mathematical Cat, or watching Numberphile on YouTube.

(Speaking of Penrose, I thought the chapter on binary went right over my kids’ heads, but yesterday they returned all of their answers in math class in binary, so I guess they picked up more than I gave them credit for.)

YouCubed.org is an interesting website I recently discovered. So far we’ve only done two of the activities, but they were cute and I suspect the website will make a useful addition to our lessons. If you’ve used it, I’d love to hear your thoughts on it.

That’s all for now. Happy learning!

Homeschooling Corner: Math ideas and manipulatives for younger grades

Archimedes

When you love a subject and your kids love it, too, it’s easy to teach. When you’re really not sure how to approach the subject or your kids hate it, it gets a lot trickier. (See: spelling.)

So I thought I’d make a list of some of our favorite math related materials–but please remember, all you really need for teaching math is a paper and pencil. (Or less–Archimedes did math with a stick and some sand!)

Countables

Little ones who are just learning to count and add benefit from having something concrete they can hold, touch, and move around when thinking about concepts like “two more” or “two less.”

You can count almost anything–pebbles, shells, acorns, pennies, Monopoly money, fingers–but having a box of dedicated, fun, colorful countables on hand is useful. My favorites:

Abacus. The abacus has the lovely advantage that all of its counters are on rods and so don’t get scattered around the room, stepped on and lost. I made my own abacus (inspired by commenter Dave‘s abacus) out of a shoe box, plastic beads, pipecleaners, and tape. You can count, add, subtract, multiply, divide, etc., on an abacus, but for your purposes you’ll just need to learn addition and subtraction.

Different abaci have different numbers and arrangements of beads. If your kids are still learning to count/mastering addition and subtraction up to ten (standard kindergarten goals,) I’d use an abacus with 9 beads per string. (Just like writing numbers, after you get to nine on the “ones” string, you raise up one bead on the “10” string.)

We adults tend to take place value for granted (“it’s obvious that we use the decimal system!”) but switching from column to column can be confusing for young kids. There’s no intuitive reason why 11 doesn’t = 2. The abacus helps increase awareness of place value (typically taught in first grade) because you simply run out of beads after 9 and have to switch to the next row.

Once kids have the basic idea, you can switch to a more advanced abacus like the Soroban. The top bead on the Soroban is worth 5, so students count 1-2-3-4, then click the 5 bead and clear the unit beads, then add unit beads to the five to count 6-7-8-9, then click one bead in the tens column and clear all of the beads in the unit and five column. My apologies if it sounds complicated; it really isn’t, it’s just a little tricky to put into words.

You can get abacus workbooks; I have not used any so I cannot review them but they look fun. Rather, I just use the abacus as a complement to the other math problems we are already doing. (I have read Mr. Green’s How to Use a Chinese Abacus, which was the only book my library had on the subject. It is a very good introduction aimed at adults.)

Counting Penguins

There is nothing magical about penguins; I just happen to like them. The set has 100 penguins in ten sets (distinguished by color) plus ten “ice bars” that hold ten penguins each. (Besides addition and subtraction,) I find these useful for introducing and visualizing multiplication , eg, 3 rows of 5 penguins = 3×5.

Counting Cubes

For bigger numbers, we have a bag of 1,000 interlocking cubes. Kids will want to just plain build with them, like Legos, which is fine–a fun treat after hard work. You can easily use these to represent 1s, 10s, and 100s (it takes a while to assemble a full 1,000 cube,) and to represent operations like 3x3x3, helping bridge both place value and multiplication. Legos work for this, too, though you’ll probably want to sort out ones that are all the same size and shape.

 

Pattern Blocks

(I think I’ve been incorrectly calling these tangrams, though the principles are similar.)

These pattern blocks are a family heirloom, sent to me by my grandmother upon the birth of my first child. I played with them when I was a child; my siblings played with them; now my children play with them. Someday I will pass them on to my grandchildren… but you can also get them on Amazon. (We use these with a book of pattern block activities that hails from the 80s; I am sure there are many good books of a similar nature published within the past couple of decades.

Apparently there are workbooks with pattern block activities aimed all the way up to 8th grade, but I have not read them and cannot comment on them.

Cuisenaire rods

We didn’t use cuisenaire rods when I was young, but I think I would have liked them. Similar to the tangrams pattern blocks, there are lots of interesting workbooks, games, and other activities you can do with these.

Building toys

Open-ended building toys (Legos, Tinker Toys, blocks, magnetic tiles) come in almost endless forms and can be used to build all sorts of geometric shapes.

Fractions

Fraction blocks and fraction circles are both handy.

Games

Almost any kids’ board game can be transformed into a math game by adding cards with math problems to be solved before completing a turn or using math dice. Your local games shop can help you find dice with numbers higher than six, or you can just tape paper onto an existing cube to make a custom die of your liking (like an + and – die). There are also tons of fun logic games; I pull these out whenever kids start getting restless.

Books

There are so many great math books, from Sir Cumference to Penrose, that I can’t hope to list them all. I encourage you to check out your library’s selection. Here are a few of my favorites:

The Adventures of Penrose the Mathematical Cat (plus sequels) makes a very pleasant enrichment portion of our daily maths. Each day we read one of Penrose’s stories (on subjects like Fibonacci numbers, primes, operations, etc) and do a short, related math activity.

Penrose is probably most appropriate for kids in mid to late elementary, not little ones just learning to count and add. (Note: the first story in the book was about binary, which flew over my kids’ heads.) Sir Cumference is more appropriate for younger learners.

Mathematicians are People, Too: biographies of great mathematicians. I’m not keen on the title, but my kids liked the chapter on Archimedes.

Balance Benders These workbooks come in different levels, from beginner to expert. Each puzzle presents students with a drawing of a balance with shapes on either side, and asks them to figure out, from a choice of answers, which statements about the shapes are true, eg “One circle equals two squares” after viewing a balance with two circles and four squares. (We also do logic puzzles and picture sudoku.)

 

Textbooks

I am not recommending any textbooks because I don’t have any idea which is the best. We don’t use a pre-packaged curriculum, because they tend to be expensive–instead I’ve just picked up a whole bunch of different math texts at the second hand shop and been gifted some lovely hand-me-downs from relatives. At this point I might have too many math books… I use 3 or 4 interchangably, depending on exactly which concepts we’re covering and whether I think the kids need more practice or not. I recently lucked into a volume of the “What your X Grader Needs to Know” series, and it gives a very nice overview of grade-level math expectations (among other things.)

Incidentally, the local public school math expectations appear to be:

Kindergarten: Reliably add and subtract the numbers 0-10; add small numbers to numbers between 10 and 20; be able to write all of the numbers from 0-20; count to 100.

1st grade: Place value; add and subtract one and two digit numbers with no regrouping.

2nd grade: Add and subtract multiple two an three-digit numbers.

I think they only explain regrouping in third grade.

In my experience, kids can do a lot more than that. These aren’t the standards I use in my classroom. But if you’re struggling to get your kindergartener to concentrate on their math worksheets, just remember: professional teachers don’t actually expect all that much at these ages. (And my kids don’t like doing a bunch of worksheet problems, either.)

Don’t sweat it. Do a few problems every day, if you can. Try teaching the same material from different angles, if necessary. Don’t be afraid to pull out pencil and paper and just make up a few problems and work through them together. Make patterns. Play games. Relax and have fun, because math at these ages really is beautiful.

Homeschooling Corner

Welcome! Highly unscientific polling has revealed an interest in a regular or semi-regular feature focused on homeschooling.

Note that I am NOT some homeschooling guru with years of experience. We are just beginning, so I want some other people to discuss things with. I don’t have a curriculum picked out nor a coherent “philosophy,” but I am SO EXCITED about all of the things I have to teach I couldn’t even list them all.

I was thinking of starting with just a focus on what has been successful this week–which books/websites/projects we liked–and perhaps what was unsuccessful. I invite all of you to come and share your thoughts, ideas, questions, philosophies, recommendations, etc. Parents whose kids are attending regular schools but want to talk about learning materials are also welcome.

One request: Please no knee-jerk bashing of public schools or teachers. (I just find this really annoying.) Thoughtful, well-reasoned critique of mainstream schooling are fine, but let’s try to focus on the homeschooling.

This week’s successes:

DK Workbooks: Coding with Scratch (workbook) has been an amazing success.

Like many parents, I thought it’d be useful to learn some basic coding, but have no idea where to start. I once read HTML for dummies, but I don’t know my CSS from Perl, much less what’s best for kids.

After a bit of searching, I decided to try the the DK Coding with Scratch series. (This particular workbook is aimed at kids 6-9 yrs old, but there are others in the series.)

Scratch is a free, simple, child-friendly coding program available online at https://scratch.mit.edu/. You don’t need the workbook to use Scratch, (it’s just a helpful supplement.) There are also lots of helpful Youtube videos for the enterprising young coder.

Note: my kids really want to code because they want to make their own video games.

In general, I have found that toys and games that claim they will teach your kids to code actually won’t. (Eg, Robot Turtles.) Some of these games are a ton of fun anyway, I just wouldn’t expect to become a great coder that way.

Professor Astro Cat’s Frontiers of Space is as good as it looks. Target market is 8-11 years old. There’s a lot of information per page, so we’re reading and discussing a few pages each day.

There are two other books in the series, Professor Astro Cat’s Intergalactic Activity Book, which I’m hoping will make a good companion to this one, and Astro Cat’s Atomic Adventure, which looks like it fills the desperately needed “quantum physics for kids” niche.)

I’m still trying to figure out how to do hands-on science activities without spending a bundle. Most of the “little labs” type science kits look fun, but don’t pack a lot of educational bang for your buck. For example, today we built a compass (it cost $10 at the toy store, not the $205 someone is trying charge on Amazon.) This was fun and I really like the little model, but it also took about 5 minutes to snap the pieces together and we can’t actually carry it around to use it like a real compass.

Plus, most of these labs are basically single-use items. I like toys with a sciency-theme, but they’re too expensive to run the whole science curriculum off of.

Oh, sure, I hand them a page of math problems and they start squawking at me like chickens. But bedtime rolls around and they’re like, “Where’s our Bedtime Math? Can’t we do one more page? One more problem? Please?”

There are only three math problems every other page (though this does add up to over 100 problems,) the presentation is fun, and the kids like the book better than going to sleep.

The book offers easy, medium, and hard problems in each section, so it works for kids between the ages of about 4 and 10.

There’s an inherent tension in education between emphasizing subjects that kids are already good at and working on the ones they’re bad at. The former gives kids a chance to excel, build confidence, and of course actually get good at something, while the latter is often an annoying pain in the butt but nevertheless necessary.

 

Since we’ve just started and are still getting in the swing of things, I’m trying to focus primarily on the things they’re good at and enjoy and have just a little daily focus on the things they’re weak at.

I’d like to find a good typing tutor (I’ll probably be trying several out soon) because watching the kids hunt-and-peck at the keyboard makes my hair stand on end. I’d also like to find a good way to hold up workbooks next to the computer to make using the DK books easier.

That’s about it, so I’ll open the floor to you guys.