If you ask me whose philosophies on elementary school teaching have influenced me the most, I will answer without pause: Jane Kats. Jane is a math educator, the author of numerous books and workbooks for educators, parents and children, and a general math enthusiast. Meeting and interacting with Jane, as well as reading her books, has revolutionized my views on elementary school mathematics and how it should be taught to students. I’ve had the privilege of having many conversations with Jane and watching her teach. Recently, I’ve had the chance to interview Jane, and today I’d like to share a bit of her wisdom.
Jane’s views on the main goals and essential components of an elementary school curriculum.
Breadth of topics: In elementary school, everyone usually devotes a lot of time to arithmetic, making sure kids know their digits and how to write them without confusing right and left. But in addition to that, there is reasoning and topics such as logic and geometry that are just as important. Many topics in geometry are not given to young children whereas the kids can handle them. Then the children get to high school geometry and go “Ah, how is this so?” And the same problem occurs with a number of other topics. For example, we first tell kids that we can only subtract smaller numbers from bigger ones. But then it turns out that you can take away 8 from 5 and get -3. And the kid goes, “How is this possible? You can’t take away a bigger number from a smaller one!” because initially the kid is told one thing and then they are told the exact opposite.” Young children should get exposed to these things as well.
Understanding, not memorization: But the most significant thing is for children to realize that mathematics is not something that can be memorized but something that needs to be understood. If we’re talking about arithmetic, then math is something that should be felt with your hands, so it is important not to rush as quickly as possible to writing down the number 21 using digits, even if the kid is capable of doing it. It’s better for the kid to see it visually first, say as two sticks of 10 and 1 little cube. Then the kid understands it better. The longer a kid spends with concrete counting materials, the stronger a foundation they’ll have.
The same goes for addition and multiplication facts – they do not need to be memorized. Because when a kid knows that 6+4=10, she does not recognize 3+3+4 as 10. Because it is a different fact, one she did not learn. And 3+4+3 is yet another, completely different fact. In games like Turboschet children can see that it’s the same end result. When you memorize something, then it’s a different kind of knowing; one that works in math class but not in life. Various games are better at teaching how it is in life.
Kids can come up with the problems too: It is also important to show children that math isn’t something where the adult always knows the answer – that there exist problems that a kid can make up for an adult and they will be interesting. For example, “Here, I drew for you the view from the front, what can the tower look like from the side? Now let’s take turns making up problems for each other: you give me one, I give you one.” So that there’s not this impression that the adult is the only one who can come up with problems and the kid is the only one who can solve them.
Varied problems with multiple solutions: It is wrong when a child associates math with columns of exercises. Plus, minus, plus, minus. Not only that, but they are all exercises where only two numbers are being added. Then when such a child sees a problem with three addends they go, “How is this? This can’t be.” It is also is important for a child to encounter problems that have several correct answers and problems that do not have an answer. For example, “List all numbers that are divisible by 5 that are more than 6 and less than 30. Now list all numbers divisible by 5 that are bigger than 3 and smaller than 5. Those don’t exist? You are correct, those don’t exist.”
Hands-on exploration of math: One of Jane’s motto’s is, “Everything needs to be explored in a hands-on way.” In elementary school, students need to learn how to count and understand the base 10 system. But in addition to that, a program should involve exploring these concepts on fingers, cubes, beads, and any other concrete materials longer than is customary in most school programs. There needs to be a lot of repetition, but it needs to be in varied forms. For example Jane likes to do with kids things like when she says “Kar” you wave your hands, when she says woof you jump and when she says meow you wipe your mouth like a cat. “Now let’s do kar, kar, woof, woof, meow. How many total movements did we do?” Math needs to be explored with the whole body and not just be all abstract and virtual. In another activity, for each meow the children take a red stick, for each woof a yellow stick and for each kar a blue stick. Then they may have to create the sequence meow, meow, woof, woof, kar again and exhibit it as a picture. Children need to learn to translate between these different “languages”
Taught by math enthusiasts: Math worksheets with redundant exercises induce nausea and not the love of math. The goal should not be to force-feed the kids mathematics. The goal should be to find some areas of math that will make them go “wow”. Different teachers love different topics, and that is normal. If a teacher loves a topic they will present it to the kids in exciting ways and the kids will say, “that sounds cool, I want to do that too.” For example, Jane can spend hours cutting up a Möbius strip with the kids and won’t get bored. Every adult has their own favorite ages to work with and their own favorite topics, and that’s okay. If the adult isn’t fascinated by what she’s teaching, the kid won’t get interested either. If a child talks to different teachers, then someone will tell him about the Möbius strip, another one about geometry, a third about graphs and a fourth will show arithmetic tricks. And that is good. It is rare that a kid is equally good at all areas of math. But if they find something that they take pleasure in, that’s already good.