More about the process standards:

This is my comments on each of the process standards, with additional notes on how these show up in the Wheels Problem.

Problem Solving in the NCTM standards has these sub-goals:

* build new mathematical knowledge through problem solving;
* solve problems that arise in mathematics and in other contexts;
* apply and adapt a variety of appropriate strategies to solve problems;
* monitor and reflect on the process of mathematical problem solving.

* build new mathematical knowledge through problem solving;

This is the most subtle and often the most powerful of the ways in which problem solving is incorporated into math teaching. To help children build new knowledge through problem solving, the teacher must pose a problem whose solution helps one understand important math ideas, and it has to be at the right level. It can't be too easy (students can do it without having to think about it) or too hard (they get stuck, and can't proceed without detailed instructions). The Wheels Problem is a good example. It was a problem that was easy enough that children could figure out how to solve it without too much hand holding, but it was hard enought that the children had to put significant thought into solving it. The teacher (Ms Gregory) explains that the goal of this problem is to help the children build understanding for addition.

Also note that when children were stuck, what she did was to make sure they really understood the problem, and then she gave them the idea that they could "maybe use all cars"--so she made sure they understood the problem, and related it back to a previous problem they had solved to help them get started, but she wasn't prescriptive about how to solve the problem.

Helping children learn new ideas through figuring out how to solve problems works really well for a lot of students, but there are also times when some students are either too advanced or not ready yet for your problem

So, it's doesn't all work like clockwork, and it's not always easy to get a problem that's on just the right level. This one is really good, and it still has a few bugs...

* solve problems that arise in mathematics and in other contexts;

This is the easiest way to think about problem solving, and it is the one most emphasized by state standards. We want children to be able to recognize how to use mathematics to solve appropriate problems. Children are doing this to some extent in the Wheels Problem because they are using modeling, counting, and addition to solve a problem about vehicles. I want you to notice, however, that this isn't the first wheels problem these children have done--they have done several versions of this same problem over the course of several days. When people first learn something new, their knowledge is not very flexible, but in order to move from inflexible to flexible knowledge, one has to spend time working on narrowly defined problems. After people understand a variety of narrow problems well, then they have the background needed to make genereralizations and be more flexible.

* apply and adapt a variety of appropriate strategies to solve problems;

One way of teaching about problem solving is to teach different strategies for solving problems. This is a tricky thing, because those strategies are also understood inflexibly at first, and children will sometimes have a hard time figuring out how to apply a strategy. Another way of introducing strategies is by having children share the ways that they solve problems with the class, and build strategies out of things that the children have figured out themselves. Probably there needs to be a balance in this: if a child comes up with a strategy him or herself, then he/she will be better able to understand and generalize it. On the other hand, some strategies are well suited to certain types of problems, and should probably be taught explicitly. (We'll look at an example later of good explicit teaching of a strategy.) In the wheels problem, children share their solutions and how they found them. Ms Gregory notes that the students learn ways of solving these problems from listening to each other. In this way children are sharing and learning problem solving strategies from each other. There are two clearly different strategies that students at the ends of the scale for which this is an appropriate problem: making 24 objects, and apportioning them out into vehicles, and coming up with numbers of wheels on vehicles and adding them. There are also several students using strategies that are something like a mixture of these two strategies.

* monitor and reflect on the process of mathematical problem solving.

This is the process of talking to oneself as one solves a problem and saying "is what I'm doing working? How can I tell that the answer I'm getting is correct? Is there another way I could look at this problem?" These are practices that can help people be more aware of, and learn more from the problem solving process.

Additionally, you should be aware that problem solving problems are considered to be separate from exercises. A problem is something where, although you understand what is being asked for, you have to put some thought into how you will find the solution. An exercise is something where you are following a set of prescribed steps to reach a solution. Whether something is a problem or an exercise depends somewhat on the sophistication of the problem solver. Some traits that are associated with problems are: being open ended; having more than one solution; having more than one solution strategy. These are neither necessary nor sufficient for something to be a problem, but you may find them cited as evidence that something is a genuine problem.

The Wheels Problem has many things to support a claim that it is a problem: It has many solutions; there isn't a prescribed way of solving it that the children have practiced; it is similar to, but not just like problems the children have done before; the children in the class solved it in many different ways; it creates a situation in which children would see how addition works and applies to a problem, and thus helps build a deeper understanding of addition.

Reasoning and Proof:

* recognize reasoning and proof as fundamental aspects of mathematics;
* make and investigate mathematical conjectures;
* develop and evaluate mathematical arguments and proofs;
* select and use various types of reasoning and methods of proof.

* recognize reasoning and proof as fundamental aspects of mathematics;

The way to think about reasoning and proof at the elementary level, is to interpret it as understanding and explaining how things work, why they make sense, and how you know they are correct. It's valuable for children to see that understanding and explaining why math makes sense is an important part of math

* make and investigate mathematical conjectures;

Children should be encouraged to come up with mathematical ideas and investigate and share them. One way to encourage this is to ask questions like: "what happens if you add two odd numbers together?" A question that is simple enough to understand, accessible to investigate, and interesting to discuss "why does that happen?"

* develop and evaluate mathematical arguments and proofs;

Developing and evaulating mathematical arguments (explanations) is something that you see in the Wheels Problem. An important part of the lesson is where children explain their answers to their peers, and show how they got the answers. Their work shows their mathematical argument/proof that their answer is correct. Children are encouraged to listen to each other and to learn from each other. Ms Gregory notes that the children have learned strategies from each other through these discussions. Thinking about someone elses explanation, figuring out for yourself that it makes sense, and using it is a form of evaluating that argument.

* select and use various types of reasoning and methods of proof.

This is relevant in a formal way in high school (and college), and is relevant in an informal way, as children learn from practice and listening to each other (and the teacher) ways of explaining and proving that an answer is correct and makes sense.

Communication

* organize and consolidate their mathematical thinking through communication;
* communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
* analyze and evaluate the mathematical thinking and strategies of others;
* use the language of mathematics to express mathematical ideas precisely.

* organize and consolidate their mathematical thinking through communication;

When you explain your ideas and strategies to someone else, you revisit them, and the ideas become more clear to you too. Listening to children is often an effective way of helping them learn. It's nice when someone asks you "how did you figure that out" and they are interested in listening to your answer. By figuring out how to tell them , you are also telling yourself. The teacher in the Wheels Problem notes that a lot of the learning in these problems happens when the children are sharing their solutions and how they got them with their peers. Some of that learning happens for the children who are listening, and some of that learning happens as the child who is explaining thinks aloud to explain how they solved the problem.

* communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

A worthy goal, and one that takes practice.

* analyze and evaluate the mathematical thinking and strategies of others;

This should sound very much like one of the Reasoning and Proof goals. Children can learn a lot from listening to each other, and they should be encouraged to discuss the ideas: the first step is to understand what the other person is saying, and the second step is to figure out if it works and makes sense

* use the language of mathematics to express mathematical ideas precisely.

Here we have the grammar of mathematical explanation. Definitions are important in math. If you are fuzzy about what math words mean, then you're more likely to get lost somewhere down the line, so while one should be realistic about what children are ready for right now, you should also try to keep the definitions straight and the explanations clear and correct.

In the Wheels Problem, children are communicating their solutions, their strategy for getting their solution, and how they know that their solution is a correct answer to the problem. This helps the students explaining because they are clarifying their thinking to themselves as they explain it to the class, and it helps the others in the class to see other solution strategies that work for that problem.

Connections

* recognize and use connections among mathematical ideas;
* understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
* recognize and apply mathematics in contexts outside of mathematics.

* recognize and use connections among mathematical ideas;
* understand how mathematical ideas interconnect and build on one another to produce a coherent whole;

These first two really go together in my mind. They are all about making connections between math ideas. The goal is for everything to fit with everything else, so that understanding one piece of math can help you understand other pieces. Math does fit together really tightly, with things building on other things: multiplication can be thought of as repeated addition and addition is a particular application of counting. Building those connections into your teaching and understanding really is valuable for understanding why math makes sense.

* recognize and apply mathematics in contexts outside of mathematics.

We also want children to be able to apply math--we want math users who can look at at a problem or a situation and say: I can see how math describes what's going on here. Again, this isn't something you get at first, but by building solid understandings of applications in some contexts, we give students the tools to be able to work toward more flexible knowledge, so that over time they are able to see the same patterns in new contexts.

In the Wheels Problem, there is a little bit of both of these types of connections. If you look closely, several of the children have work that shows both a picture with the wheels where you could actually count up the wheels to verify that there are 24, and they also have an addition sentence, so these children are making the connection between counting and addition, and making that connection more concrete and well understood for themselves. .

There is also the contextual connection, that students are building a strategy for understanding how numbers can tell you something about all the wheels in the parking lot, and how you can get from how many cars, motorcycles, etc, to how many cars total. By understanding this problem well, the teacher is laying the groundwork for students to be able to look at a similar problem and say "that's just like the Wheels Problem"

Representation:

* create and use representations to organize, record, and communicate mathematical ideas;
* select, apply, and translate among mathematical representations to solve problems;
* use representations to model and interpret physical, social, and mathematical phenomena.

* create and use representations to organize, record, and communicate mathematical ideas;

In the Wheels Problem, children are creating a representation when they write down how they solved the problem. They used these representations to communicate their solutions, and also to organize their solutions. There is a lot of variation in the representations, but they all show what the children did, and how it solved the problem. It isn't necessary that representations be newly invented on the spot, but a good representation should be effective for organizing the problem, recording the how it was solved so that the representation could be used to show the solution to someone else, or prove the correctness of the solution to someone else.

* select, apply, and translate among mathematical representations to solve problems;

Often the representation that a problem comes in originally isn't very helpful for solving the problem, and a key step is to find another representation that will be more helpful for solving the problem. The new representation might be a table, a picture or an equation, and different representations will be "best" for differen sorts of problems. In the Wheels Problem, children took their understanding of the written problem, and translated that understanding to a representation by balls of clay, stickers or numbers.

* use representations to model and interpret physical, social, and mathematical phenomena.

A model is a mathematical way of describing something. For example, when we looked at the Algebra standard there was a problem about the cost of sandwiches, and for that problem the equation: C=3S was a mathematical model for the cost of the sandwiches. It doesn't have to be an equation to be a mathematical model, it can be anything which represents the thing being modeled, and makes it easier to solve some problems about those things.