In the first dialogue/dance, Ali discussed her concern that teaching to standards or objectives has the same inherent problems associated with teaching to the test. Most teachers say they want to support learning but as the pressure on them mounts to ensure their students can meet standards measured by achievement tests, they tend to focus on "recipes" to teach those standards. Often times, this is at the expense of providing a thought-provoking learning environment (see note 1). The short-term goal of doing well on the test may be met, but the long-term consequence can be negative. For example, math computations can be made by memorizing algorithms but the math relationships involved may not be understood. If children do not understand early number relationships, they are not likely to understand more complex mathematical ideas presented to them later. In addition, they may learn to dislike math. To read the first part, click here. So what is a teacher to do? The second dialogue/dance presents some ideas. We hope you will enter in with your ideas too. -----------------------------------------------------

Hillel:	Okay, Ali, so what are some things that a teacher might do in order to move away from this formulaic approach to teaching math?
Ali:	Well, One idea that constructivist teachers use to support learning is to encourage children to think, to mentally act to get answers. That doesn't sound the least bit unreasonable but many teachers ignore this idea when they ask children to do rote math activities. Any time children do math without thoughtful mental engagement, it is a lost opportunity for the children to build math logic.
Hillel:	Can you give me an example about how to engage with children mentally?
Ali:	Remember the way you were taught to add 17 + 15? Remember the rule we memorized? Add the 5 and the 7 to make 12, put down the 2 and carry the 1. Then 1 + 1 + 1 equals 3. The answer is 32. Sound familiar? It hasn't really changed too much over the past thirty or forty years. Some teachers tell the children to say "put down the 2 and carry the 10." But children still are asked to follow rules to solve that simple addition problem. Math logic is not constructed when children are told to remember and follow a rule to get the answer.
Hillel:	So how might a constructivist teacher talk with children about that problem?
Ali:	When constructivist teachers ask children to "make sense" of that problem without specific instruction, children respond in a variety of ways. For example, some children will say, "I know that 10 and 10 are 20 and that 5 and 5 are 10 plus 2 more are 12. 20 and 12 are 32." Another child might do it differently. For example, she might say, "I know that 10 and 10 are 20, and 5 and 7 are 12 which is another 10 for 30 and 2 more are 32." When children share their various ways of making sense of this problem, not only do they get ideas from one another, but they learn that math makes sense. Knowing that they "get it" leads to confidence, which, in turn, leads to liking math.
Hillel:	But this seems that it might take so much time - and still the kids might not get the right answer or even gain some understanding of mathematical thinking. Maybe not all cultures or all groups within a culture value this thinking aspect. Many groups value getting it "right"!
Ali:	Well, we all value getting math right. I'm talking about the path to getting it right. If children are thinking and discussing their thinking along this path, they will ultimately both be right and have the logic for why it's right. This is very empowering. I really don't think it takes more time, but the time is definitely spent differently. I remember doing exercises of 25 math problems by myself. What I've described is fewer problems but with lots of shared insight.
Hillel:	I like your idea about not necessarily more time spent, but just finding other ways of using time. And also, even if it does take more time, it still may be worth it.
Ali:	I saw a video of a math lesson in Japan which included this kind of problem solving and shared insight. Was this not typical? I thought you told me that there's a big push on in Japan now for developing more creative and independent thinking. Doesn't that mean there's a move away from just memorizing formulas?
Hillel:	Yes, that's the rhetoric, but I wonder if it's really what people want or know how to get, even if they do want it. I read it a lot in the papers and hear educators and media people talking a lot about it, but I can't see the kind of classroom environments which encourage such development. But since I'm not a math teacher and not at the elementary level, I really hope that some other people in Japan - teachers, parents, students - will join our dance and tell us how their experiences are.
Ali:	Yes, that would be great!
Hillel:	So in the meantime, let's keep on dancing!!! What's happening in the U. S.? In Japan, people imagine that American classrooms are so creative and open.
Ali:	If we stay with mathematics teaching to think about this question, I think we're still seeing a lot of teacher directed drill lessons. When children must remember a rule to lead them to the correct answer, there is no underlying logic to give them a sense of whether their answer is correct. Instead of developing their own number sense, students depend on the teacher to tell them if their work makes sense. Everyone does the problem the same way. There are no unique or clever solutions offered to surprise and challenge students to think differently. Rules are tricks that only require memory work, the lowest rung of Bloom's taxonomy. (See Note 2)
Hillel:	Well, I think you've touched on some key points for me here. One is that there is a system of dependency created through the idea that this "teacher knows the correct answer and how to get it". This is not a dialog or even a collaboration of learners. What's more, there is no surprise or mystery and no sense of wonder is created. Sadly, it isn't just about mathematics, but nearly everything that we can observe throughout the educational system.
Ali:	In the United States, if children care about doing well in school, they will endeavor to do what the teacher requires. In the short term, they get their good grade, but in the long term, they may lose an opportunity to develop real understanding. The children who are not motivated to do well in school are the ones who I worry about the most. If they don't care about getting good grades, they certainly aren't going to do those dull exercises that have no relevance to their lives. They fall further and further behind. They feel dumb. Whether children are or are not motivated to learn these math computation rules, the sad consequence of using rules instead of logic is that children won't gain confidence in their ability to do math. Math becomes magical to many children, outside of their ability to affect or control. Even worse, when children know they don't "get it," they often learn to dislike math.

Hillel:	Ali, it might be interesting to talk a little about math education in Japan here. You know that in Japan, Singapore and Korea, the children consistently outperform students from North American and Europe on some standardized math tests, and many Western educators often idealize the math education in these countries.
Ali:	Yes, I know this. Of course it's really important to be aware of what these standardized tests are really measuring. Is it computational speed, correct completion of problems with memorized formulas, or is it really looking for deeper understanding of mathematical thinking?
Hillel:	I'm not sure what these tests measure. But also there are some home studies and after school programs for math education that are very popular in Japan. I don't know what they focus on, but ONE of the most popular home study programs is run by Benesse Corporation, the company who is supporting the foundation of Child Research Network (CRN) who are publishing this article and website.
Ali:	It would be interesting if they could respond to this and say something about the purpose of their mathematical home study program.
Hillel:	Yes, I think that would really help our dance! ...... You know, you mentioned earlier about the importance of children developing confidence in their mathematical ability and not turning off. I think that some case may be made for highly motivated students doing a lot of drill and practice on what are called "basic mathematical facts" as one way of gaining this kind of confidence.
Ali:	Yes, MOST math educators who agree that it is helpful to know easy addition and multiplication problems so that one can compute more difficult problems. And SOME STILL ADVOCATE LEARNING THESE EASY "FACTS" THROUGH DRILL AND PRACTICE.
Hillel:	So what are some constructivist alternatives?
Ali:	There are alternatives to rote math work sheets or flash cards. Consider game play. Children have to think about adding two cards together to play double war. (See Note 3) They play it again and again and thus "practice" figuring out these early number relationships (i.e., the so-called math facts) that are the basis of so many early math objectives. Game play is a natural situation in which children must think to play. If the thinking is about number relationships as it is when you roll two dice and add them to send your playing piece around the game board, then it serves both the objective to remember the "math facts," as well as the constructivist idea of offering meaningful mental engagement. Card and dice game play isn't used as a strategy or method just to meet academic objectives. It is used to encourage children to think, to mentally act to get answers. There is a big difference between the two approaches. When teachers design teaching strategies without consideration for how learning occurs, the responsibility for student learning remains with the teacher. When teachers design teaching strategies that serve student thinking, the responsibility for learning necessarily and naturally transfers to the student.
Hillel:	Maybe this is a good place to stop our dancing for a bit and hope for some interesting response from the readers.

---------------------- NOTES ------------------------ Note 1: It's important to understand that the provoking may be initially stimulated by the teacher, but the learners themselves are providing their own stimulation through being engaged in learning that is what a child in Boston called "hard fun" (as reported by Seymour Papert). (back to text) Note 2: for more information about Bloom's Taxonomy, see various homepages, such as: http://www.wested.org/tie/dlrn/blooms.html http://a41064.west.asu.edu/students/dfields/96-598/b.bloom.html (back to text) Note 3: "a game where each player turns over two cards, and the player with the highest SUM wins all the cards turned over. (When a player runs out of cards to turn over, he or she picks up the cards s/he has won and uses them. Each player keeps doing this until one player has all the cards.)" This description of double war is from the homepage, "Teaching Math to Young Children" by Rick Garlikov http://www.garlikov.com/math/TeachingMath.html (back to text)

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