I just read an interesting article in the New York Times called "Brain Calisthenics for Abstract Ideas". Here are two quotes from the article that stood out to me:
"For years school curriculums have emphasized top-down instruction, especially for topics like math and science. Learn the rules first—the theorems, the order of operations, Newton’s laws—then make a run at the problem list at the end of the chapter."
"Now, a small group of cognitive scientists is arguing that schools and students could take far more advantage of … [the] bottom-up ability, called perceptual learning. The brain is a pattern-recognition machine, after all, and when focused properly, it can quickly deepen a person’s grasp of a principle, new studies suggest."
This article really hit home with me for a few reasons. One point the article made is that the brain is attracted to pattern discovery; it begins scanning for patterns before we’re even aware of it happening. The brain is not necessarily attracted to rules or theorems or lists of steps and procedures. The second principle that jumped out at me is the value of giving children real materials to work with as they discover for themselves the workings of a problem or as they distill the rules we wanted to teach them in the beginning.
As I read the part of the article about the value in letting students explore the basics of a new concept before giving them the rules and theory behind it, I started thinking about what this might look like on a practical level. Following are some examples I came up with.
An example of how to create a bottom-up lesson for early computation:
For the most part, young children are taught to count objects on the page in order to add them up. Some kids are still counting on their fingers in senior high math classes. Giving children plenty of practice with counters and dot patterns will over time give their brains visual snapshots of what numbers mean—the “how many” of them—and they will be able to do computation quickly in their heads. Their instincts for how many will be sharply developed.
Here is an example of homemade dot cards that show 5s in various arrays:
Make a stack of dot cards (with 1-5 dots per card). Throw one down on the table and have the child quickly guess how many dots are on the card. Have him do all this without counting the dots…just guessing. The ones he guesses right, he gets to keep.
The cards above all show 5 in different arrangements. They not only show “how many” five is, but they imply various number combos that make a 5, such as 1+4 and 2+3. They also show what an odd number looks like.
Next, try tossing three cards down—two 5s and a 4 or 2 5s and a 6. Have the child guess very quickly which card doesn’t belong.
An example of how to create a bottom-up lesson for phonics/sound spellings:
Use cards like these:
Note they are four different sound spellings—words with AY, with AI, OY, and OI. If you spread them out on a table, ask the child to figure out a great way to group them. He can group them however he wants. Give him time to play around with the cards. Once he’s decided how to classify the cards, ask him to tell you why he grouped them the way he did. (He may come up with four groups (AI, AY, OY, and OI). Or he may choose two groups (AY with AI, etc.)
Read through the words casually…pointing out that the red letters signify something special. You have two sounds in red: OY and long sound of A. Give the child time to see if there is a pattern he notices in where the letters are placed in the words. He may figure out that the spellings with an I are found inside the words while the sounds that end in Y (AY and OY) are at the end of the word or syllable (such as in loy-al or may-be).
Once he’s played this game, chances are good he will run into some of these word spellings as he’s reading, and having sorted and classified for himself, he will remember the sounds, their letters, and where they come in the word far better than if he’d just heard a rule.
An example of how to create a bottom-up lesson for multiplication:
I love showing children the answers to multiplication tables in 5-grids because the most wonderful patterns emerge. I also love seeing how children are drawn to the grids because they do see patterns in the numbers.
One of my favs is the 9x table. And my immediate reaction to seeing this times table is to start pointing out patterns. (Okay, I will just start. In the 10s place, you count up 0-9, then because it is the 9s table, you say 9 twice, then keep on going counting up until you get to 13. Rows 1-2 use the same numbers but with their digits reversed. Rows 1 and 3 have some similarities in the 1s place. If there were a 4th row, it would look similar to row 2 in the 1s place as well. Notice patterns in the #x and the answers.)
Once the child has looked for patterns and shared them with you, see if he notices the “rules” for 9s, which is that the numbers in each answer have to add up to a 9, that the number in the 10s place in rows 1-2 is one digit less than the #x. In row 3, the ##x is 2 digits larger than the first two digits of the answer and the number in the 1s is whatever you need to make the whole thing add up to a 9. For instance, with 13 x 9, the answer will start with 11 because it is the number 2 numbers smaller than a 9. Then you put a 7 in the 1s place because 1 + 1 + 7 =9. Don’t you just love it? We have a book on multiplication and division that encourages pattern discovery, and uses stories, visuals, and hands-on activities to make learning multiplication and division a blast (and avoid memorization in the process).
Those are just a few ideas for helping children learn from the bottom up so that the ideas sink in deeply and create a strong and lasting foundation for future learning. What about you all? What have you done with your students to build a strong visual and hands-on base for abstract ideas?