I recently was fortunate to watch the great math educator Constance Kamii leading professional development with teachers, many of them new. Dr. Kamii is the author of many articles and books about math education for young children. She studied for 15 years under Jean Piaget, and currently teaches teachers at the University of Alabama.
Her focus is teaching math to young children, and her constructivist approach runs counter to most mainstream math instruction found in US schools today. Her research with real children in real classrooms is impeccable and widely cited, her video casework is undeniable, and she’s read and studied worldwide – especially in Japan and Brazil.
The big idea of this post – questioning assumptions
Dr. Kamii makes me question everything I think I know about math and how children learn. What I do know is that the way that most children are taught math here in the US is successful for far too few people. But people seem to feel that it’s the only way. Here in the US, it’s perfectly acceptable to say, “I’m not good at math” and acknowledge that fractions mystify most college freshmen. Yet still we cling to our beliefs that somehow, if we just keep reciting times tables, handing out more worksheets, and doing “math minutes”, eventually, the kids will get it.
It seems obvious to me that we have to question assumptions about teaching and learning math — especially in the face of overwhelming evidence that something is very very wrong.
What follows is my understanding of what Dr. Kamii shared with these teachers about teaching math to young children. These are not direct quotes, and I hope I got the gist of what she meant while discussing a complex subject.
Starting off – memory, perception, and representation
Dr. Kamii started with some basic thoughts about memory, perception, and representation.
- In young children, memory is a reconstruction of their understanding. It’s not a storage/retrieval mechanism. So for math, kids at a lower level of number concept will have a hard time remembering “facts”.
- Perception is a personal construction of what’s out there, so it will change as a child grows.
- Representation when talking about how children learn math is not the same as using the word “represent” as in a metaphor. The mind represents things – it’s a verb. The nature of representation is not fully understood, but if you work with children, and listen and watch them, you can see when these understandings happen. Teachers have to carefully watch how children represent their understanding of the world, and with math in particular, and be careful not to try to simply impose an outside representation on a child.
- You can’t force them. If a child has no number sense, and you introduce numbers, there is no perception in the mind to connect it to. You can try to make them memorize it and practice it, but that just delays what needs to happen and confuses children.
By telling a child how to solve problems and memorize facts, we are trying to shortcut a developmental process that just can’t be shortcut – and in the end, will actually delay progress and make it less likely that they will develop the deeper understanding they will need for lifelong mathematical ability.
Too much emphasis on symbols and equations
Kamii feels that in most traditional classrooms there is too much emphasis on writing equations. She teaches that there should be very limited exposure to equations in early grades.
Introducing symbols too early creates confusion where children feel the way to solve a problem is to choose the right symbol. Children may come to kindergarten able to answer questions like “If you have two apples and I take away one, how many do you have?” — but after a few months will respond, “Do I plus or minus?”
Many teachers try to “help” children with word problems by providing them with decoding strategies. For example, they tell children to look for certain words that signal the operation they are supposed to use to solve the problem — if you see the word “more” you add, if you see the word “per” you divide. This unfortunately encourages children to not really try to understand the problem, and often adds to a child’s uncertainty of “what to do.” Instead of thinking, they guess.
Throughout the day, the teachers are asking questions, about how do you know what to do, how do you see these signs in children. Stephanie, who runs the school, says – it takes years to get from the place where you say, “this might be right” to the place where you say, “I know how to teach like this.” She assures them that everyone will be working together all year long to make sense of this.
Subtraction is not backwards addition
Dr. Kamii moves on to addition and subtraction. First, she talks about how most math education moves through these stages – add, subtract, multiply, divide. However, she says, this is not a natural progression. Subtraction is not a natural follow on to addition — multiplication is. But not as in “memorize the times tables” kind of multiplication. The kind of natural multiplication that students will do as they become more confident with addition. Things like creating groups, counting by fives, etc.
Example: Even as adults, people don’t normally express things in terms of subtraction. When you go somewhere, you say you are getting closer to your goal, you don’t say you are getting further away from your starting point.
If you are not successful in addition, you can’t move on to subtraction. Conversely, if kids know addition very well, they will move easily to subtraction. Dr. Kamii says, “I wouldn’t bother teaching subtraction in first grade.”
Other points she makes:
- Don’t teach kids to count backwards to subtract. Too hard to keep track of the first count. (10-3 = 10, 9, 8…is the answer 8?). Similarly, number lines can be very confusing.
- Manipulatives should not be used, it’s an abstraction of an abstraction. Let students draw their own representations of a problem.
- One of the worst things to teach is to “count on” (which means that teachers encourage students to add by saying the first number, and “counting” out the second number.) The problem is that it’s a trick and not natural for kids. For example, if you watch kids play games where they move a game piece by counting, they always try to count the square they are on as part of their move.
- When it’s time to move to bigger numbers, the issue of “borrowing” comes up. If kids are introduced to this concept too early, they believe that borrowing creates a number that is bigger than original number. But some kids will be able to do the borrowing trick without understanding what the place value means. It looks like they are catching on, but they are just mimicking the trick. You have to constantly test their understanding by talking to them about real problems that use numbers.
So what do you do instead?
Her methods are to use a constant supply of word problems and games, not worksheets. She plays some games with the teachers and shows them several articles and books she has written about games and puzzles. She shows them assessment strategies that seem like games to students but reveal various types of mathematical sense in children.
Games that assess number sense
Word problems are crucial, she says. But let children solve them their own way, without imposing symbols or pre-canned strategies. They should work both together and in groups, and the class can discuss the solutions together. Children should be allowed to talk through their solutions to the group and will often come to see their own errors in ways that are much better learning experiences than just being told “you’re wrong”. (More about this later.)
The teacher’s job should be to move the group discussion along and try not to impose their own understandings and ways to solve problems on any student. It’s very important for every child to feel that there are multiple ways to solve a problem (not necessarily multiple solutions) and they are free to express their own solution. That way, if they have a logical flaw in their thinking, they will often hear it as they try to explain it to the class.
The teachers chime in
Some of the teachers are now bringing up their experiences from previous years. That it’s hard to get the parents to understand this teaching methodology. The teachers say that parents teach algorithms and tricks to kids, and when this happens, it shows up right away that the kids are less willing to tackle problems because they feel they don’t know the right trick. A teacher tells a story about having to help kids when they have been messed up by memorizing algorithms. They lose the confidence to invent good solutions when they encounter new problems.
A teacher asks – why is subtraction is more difficult? Dr. Kamii responds — It’s human nature. First we construct the positive aspect of action (this is a basic concept from Piaget). The negative aspect follows only after you’ve done the positive aspect. You don’t look at a picture and say, “oh, that’s not green” or you don’t say, “hand me the thing that’s not a fork.”
Another example is sorting – ask kids to sort things into two groups. Then ask them to name the groups. 4 year olds will just list everything in each group. Then comes a time when they can label some things, like “these are flowers and these are fruit.” It’s very rare that a child will create groups like “flowers” and “not flowers”. The negative is a much later, secondary phase.
A teacher asks – How do you introduce subtraction? Dr. Kamii responds — A great way to encourage subtraction is to play games where kids can roll dice and use the numbers either added or subtracted. Early second grade is a good time to introduce these games. Some children will stick to only addition until they see that subtraction is better, then they will subtract. Let them come to that conclusion.
One game is called Sneaky Snake – played by two students. There are two pictures of snakes with the numbers 1 – 12 on them. You roll two dice and you can cover up a number on your snake, either the sum or the difference of the dice. The one who covers up all the numbers first wins.
Word problems that include subtraction are fine. Let them draw, cross the drawings out, but don’t introduce the subtraction symbol.
Teacher – how do you know what kinds of problems to give them? You have to play games with kids and watch them. This is authentic assessment. Watch and see if they are just counting everything or trying new things. Listen to them. This is the primary role of the teacher – to understand the mathematical ability each child has by paying close attention as they solve problems. You have to have a good supply of games and puzzles that challenge children at different levels. It takes time to develop.
Video of a first grade problem-of-the-day session
After this we all watch a video of a first grade class working on a problem of the day. The problem is: if you have 62 cents, and the school store is selling erasers for 5 cents each, how many erasers can you buy? In the video, each child works out their own solution on a blank piece of paper, while the teacher circulates and listens to solutions. She never says anyone is right or wrong, just encourages them to explain their solution. One puts 62 marks and circles twelve groups. Another makes boxes that represent 5 and explains their answer. Another writes down 5, 10, 15… and counts those numbers, but when she gets to 62 she counts that as well – so comes up with 13. One draws nickels and counts them.
After a while the teacher in the video starts a class discussion and asks for all the answers – she writes about five different answers volunteered by the class on the board and asks for explanations. After a short time, there is only one girl left who is still sure the answer is 13, while the rest of the class has settled on 12. She maintains her answer for a while while other students explain that you can’t count the last two cents because it won’t buy a whole eraser. They use several different arguments and work hard to explain to her what she did wrong, and she defends herself until she changes her mind. Suddenly, she says made a mistake by counting the last two cents. During this, the teacher never says anyone is wrong (or right for that matter).
The big idea
Dr. Kamii explains – Kids should hang on to their beliefs, it creates autonomy. They shouldn’t give in to pressure until convinced internally that there is a better solution (not that they are wrong). They should move from one solution they own to a better solution they believe in.
I can certainly see that in most math education, teachers and parents end up pressuring students to adopt a solution or methodology they don’t really understand. Some kids will actually “get it”, some kids will cave and learn the trick, some kids will resist to the very end. Of course some will be successful, but many, many children are left feeling that they are “bad” at math.
It’s very difficult to let go of beliefs, even in the face of evidence that they are wrong. As a society, we believe that the way we were taught math is the “right” way, and the only way. Having to give memorization-based tests to children unfortunately reinforces this assumption.
But I hope that some of you out there are open to thinking about a different way to teach math, to question assumptions, to ask questions, and take a look at evidence that there are other ways that might be more successful in developing life-long math ability in children.
For more information
Recent article – Teachers Need More Knowledge of How Children Learn Mathematics
Recent published research – Teaching arithmetic to low-performing, low-SES first graders
Young Children Reinvent Arithmetic: Implications of Piaget’s Theory – Series of three books for first, second and third grade. The books take you from theory to practice in a real classroom. Each contains many games and ideas for teaching math in a constructivist way. (Amazon) (Teachers College Press)
Other Kamii books and videos at Teachers College Press
Amazon – other books – some of Constance Kamii books and videos are available from Amazon.com (look at the used ones, you can get some of these for just a few dollars)