I love The Discovery Method from www.mathinspirations.com. I'll admit, though, that when I first heard the idea, I thought, "That is ridiculous. We have perfectly good ways of doing math that have been around for a long time. Why make the kids figure them out on their own?" However, as I learned more, I changed my mind. Part of the reason was that this is completely organized and provides a system for making sure they learn to do every kind of math K-8 (I hope they expand to high school), which I had never seen a "Directed Discovery" method offer. More than that, though, was a simple question they asked: "What does it mean to divide 2/3 by 4/5?" I had to spend a while thinking about it before I realized, and now I'm sure I won't forget the algorithm for dividing by a fraction, because I understand what I'm really doing. It was also tremendously exciting when I figured it out! And I realized, I want my kids to have that same experience. So I was sold.
This is going to be my official "curriculum" when my kids are old enough. However, this is not really an EL-friendly curriculum. Even EL kids are going to need quite a bit of logic and problem-solving ability, and my EL five year old is definitely not there yet. Maybe some of yours are, but not mine.
Until about age 7 or 8, Emily Dyke, the founder of Math Inspirations, advocates nothing but games for math. Some of her 100 games that you get with the course reinforce basic operations (addition, subtraction, multiplication, and division), so you will have some experience with those, but the main focus is on leaning the skills and mindset necessary for problem-solving. My daughter and I are currently playing a variant on the commercial game "Mastermind" with our blocks. One person hides three blocks; the other picks up groups of three and asks how many are right until they know which three are hidden. It's been interesting to watch my daughter, who was not quite 5 when we started. At first she had no idea what to do, so she would hide the blocks and I would guess while narrating my logic. After a week or so she started guessing herself. She'll still often get two blocks and then get really frustrated trying to find the third. Or she'll get confused about what she really knows. But she's definitely learning, and I can see how this and other games will prepare her for the curriculum.
I tried to do the first unit with her just for the purpose of the review. She doesn't have the persistence necessary yet for the way this curriculum is set up, but here's how it went. (The activity I did, while based on Emily Dyke's ideas, is not in the Basic Operations manual I own; I assume something similar is in the Number Sense book, but that one's not out yet. So I feel that it's legitimate to share this.)
The curriculum is all about questions; the parent is basically never to tell the kid anything. So we started with "What is a number?" "It's how we count," was a perfectly good answer. The next question is, "What is a digit?" and she has no idea. (I don't use it often - oops.) So I wrote some one-, two-, and three-digit numbers out and told her how many digits were in each. Then we built those numbers with blocks. Then I asked her to identify one digit, and she could do that. But she couldn't articulate what a digit would tell her, which was the goal of the lesson. She got frustrated and withdrew. Like I said, she needs more problem-solving experience and maybe simply more maturity.
In general, the goal of the curriculum is to get a 10 year old to be able to work completely independently for about half an hour a day, and the parent comes and talks with them for fifteen minutes twice a week. In addition, however, everyone in the family should be working on logic games or puzzles every day, so she recommends playing games daily as a family for about fifteen minutes. And younger kids or those new to the curriculum will need a lot more hands-on help.
For each concept, you start with defining terms, using something like my digit activity above. The kid writes or dictates a definition, which you then compare with some definitions from math dictionaries to make sure all the important things are covered. Then the final form of the definition is written in the child's Book of Math. Then there are very concrete, hands-on story problems using the skill (like the ubiquitous "You have five candies and your grandma gives you four more; how many do you have now?"). The idea is for the kid model and solve the problem in some way. The parent is not to explain how to model it, but if the kid is really stuck you can ask leading questions until they start trying something. With little kids, you'd be sitting with them, but the goal is for them to eventually do this on their own. When you work with them twice a week, your job is to say "Prove it" on a handful of problems they've done (anything you can see they've done wrong, of course, but also things they did right) so that they practice defending their ideas.
When the kid has solved a sufficient number (parent's discretion), they collect all their data (they need help with this) and look for patterns. For example; Multiplying two teens always gets 100 plus the sum of the ones digits in the tens place plus the product of the ones digits in the ones place. (That is a pattern I realized after playing algebra last week and it made me happy.) They come up with a hypothesis of a rule they think will work in every case. With younger kids, the parent pretends that they don't know this math, and there are cool exercises where the kid teaches the parent to help them craft their hypothesis. Then they do more problems, doing each problem twice, once with modeling and once with the hypothesis, to make sure it always works. Once it's solid, they write their new Theorem in their Book of Math. You can also discuss other algorithms with them at this point, for comparison's sake, and they might incorporate other ideas.
It's a much slower process than just telling them the rule and then letting them practice applying it. Each theorem might take weeks to develop. And it requires a lot of persistence and logical thinking; they have to try many different things until one works, and it helps to be able to see patterns and manipulate them. That's why my five year old can't do it. But I love that it makes Directed Discovery into a concrete program, instead of the fuzzy "let them play and they'll learn things" that's all I've ever understood it to mean. And since they spend at least half their time working on problems they have no idea how to do, they get a lot of confidence in solving unfamiliar problems. And if they forget something and don't have their Book of Math handy, they know they can figure it out again; after all, they did it once already.
The training was excellent and worth the $120 I spent on it (it was on sale). It really helped me understand how I can direct their discovery with questions and clear objectives. I don't agree with everything Emily Dyke says (most obviously her stance on calculators), but it's a great resource. It doesn't work with my little kids; persistence in the face of frustration and skills to solve difficult problems are not what we're working on with my five-and-under crowd in academic achievement. But they are excellent skills that I will be investing in when they're actually elementary-school age, and I love how this curriculum teaches them.
So we're just playing blocks right now, a la Crewton Ramone, and we're starting problem-solving games with my oldest (she struggles with perfectionism and is easily frustrated, so we're taking it slowly). When my kids are old enough to struggle with math, this is what we're going to do. We'll identify the vocabulary and define it through logic puzzles, then act out story problems to learn how to do the problems. We'll write our own rules for how to do math (I want to do this myself!) that we are much less likely to forget. We will probably need to supplement or else be really good about the review games to get to mastery of basic things like the addends and multiplication tables, but other than that, I love this program.
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