The Key to Mathematics: Comprehending the Data

I found this website on solving word problems and thought I would share.  It is found at http://www.hawaii.edu/suremath/intro_algebra.html.

On this page, there's a block of information on problem solving.  Have students read through this block and determine the important bits of information.  How can this information help make solving math problems easier?



Only identify what is asked for. Ignore all the ifs, ands and buts....
A problem is understood by solving it not by pondering it.

So, let's take a sample chapter and see how we can use the information given in order to come up with a solution.....
(from http://www.hawaii.edu/suremath/theBook/introduction.html)

The process of problem solving 1/1
INTRODUCTION
The goal	The goal is to make it possible for you to solve problems that use mathematics in a reliable, confident manner. The problem solving process that you will come to understand is applicable to all subjects that use mathematics for problem solving.
	Though our goal is to learn to solve problems which require use of mathematics, it is necessary to understand the properties and solution methods of more general problems. We will start this by exploring some simple routine problems that we encounter in daily life. Doing so will bring out the essential features of problems and their solutions.
Some perspective. Problem Solving as a natural process.	Easily the most important thing to understand about problem solving is that problem solving is a natural, indeed an innate, process. We are born problem solvers. To appreciate your built-in problem solving abilities you need only to look at the many problems you routinely solve daily. Such problems as brushing your teeth, riding a bicycle, driving a car, going to a store are examples of routine daily problems which serve as informative examples of the problem solving process. Observing and thinking about solving such problems brings out quite basic problem solving principles. First you will observe that the goal, the problem to be solved, is known. It is the conditions that need to be satisfied, the constraints, that make attaining the goal a problem. These constraints also bring life to the solution process.
Definition of a problem	These observations make it possible to have an operational definition of the word problem. A problem is a request for a result subject to conditions that must simultaneously be satisfied. To understand the meaning of this definition and the manner in which it specifies how to solve problems, we will explore the rather routine problem of leaving your home and going to the mall.
Respond to what is asked for.	Having chosen your trip to the mall as our problem, we first consider the resources available to accomplish the goal. There are choices. You could realize this goal by driving a car, riding a bicycle, walking or a variety of other means. Specifically, we must respond to the request, "how to get to the mall?"
Request Response	Let's make the likely choice - driving. This response immediately generates a request; "Where are the keys?" With the keys in hand you go to the car, unlock the door, place the key in the ignition, start the car, put it in gear (forward or reverse according to existing constraints) and so forth. What we see here is an ordered process of Request-Response. Each response is a consequence of the previous request and a precursor to the next request. (We put the key in before turning it, for example.
Request Response Result	You complete the trip to the mall by responding to requests as they arise: stopping at red lights, making left turns or right turns as needed, accommodating to an unexpected detour, deciding to pick up your dry cleaning and so forth. Thus solving the problem of driving to the mall consists of responding to requests in a logically sound sequence to generate the result.
	There are usually a number of different solution paths for a given problem, generated by choosing differently at places where options present themselves. The diagram at the left illustrates this idea. At the start of the trip-to-the-mall problem there was a choice of type of transportation. This is illustrated by the different logical paths arising out of the starting point. Conditions may arise that encourage change to a different solution path as depicted in the diagram. It is seen that solving a problem is a natural process carried out in sensible steps. The steps are dictated by the current state of the solution. Essentially, problems solve themselves. You need to supply knowledge (location of the mall) and skills (how to drive a car).
Knowledge Resources Tools Skills	The trip-to-the-mall problem illustrates that you do not need to figure-out how to solve a problem before you start. How to solve a problm is well defined by the problem. The solution steps themselves will lead you to the result. The difficulty arises in knowing how to respond to a request. This requires knowledge. You may not possess or be able to acquire the knowledge needed to respond to a request. If we limit the problems of interest to types which are within your current knowledge field, the next thing needed is appropriate resources. One resource immediately available is your brain. You can think out the response to a request. Another resource that is widely available is pencil and paper. This is a powerful supplement to the brain. Other resources include dictionaries, encyclopedia, textbooks, reference books, calculators, computers, the world wide web, teachers, friends, parents and many more. Problems of any kind require various tools to implement the steps of the solution. The basic tools for mathematical problem solving are the tools available in algebra. These consist of such things as translating words to equations, manipulating equations, algebraic substitution, factoring, expanding, graphing functions and so forth. The mathematical tools available are virtually endless and the degree to which you might need more advanced tools, such as those available in the calculus, will depend on your professional objectives. However, algebra is the cornerstone of mathematical problem solving. Finally, skills are needed in order to use the tools efficiently. Skills are developed through practice
To summarize Request Response Result	This introduction serves to establish the main idea involved in problem solving. Problem solving consists of a natural step by step process .Each step is a consequence of the previous step and a precursor to the next step. In the next chapter we will see this idea at work.

We need to first understand the various elements in this message.  Are there terms/words wwith which the students are unfamilar?  How will be introduce these before asking students to read?  How do I help students who find reading too difficult?  Is there a way I can break this information into more reasonable units?

One trick I like to use when reading informational texts is to have students complete a 3x5 card with the following information:

1.  Title of reading (& page # if appropriate).
2.  Three important ideas, facts, concepts.
3.  Any words I don't understand completely.
4. One or more questions I have about the reading.

Depending on the levels that your students are reading, you may wish to take a segment such as the example above and break it into managable parts.  (The information is alread subdivided into ten parts--you may wish to start with this.) Have students paraphrase the information in each part--this will make the overall information more accessible.

Hold a class discussion on the segment of information.  The most common problem for students is that they do not understand what they are being asked to do--and some of this problem is due to comprehension problems.  Before you can help a student with math problems, you must make sure they have comprehension skills that will allow them to read the information provided BEFORE you ask them to solve the problem.

So what do you do if you have students in one classroom who are performing on various levels of comprehension?

First, you must subdivide students into groups according to their levels--you may choose two, three, or four different levels per class depending on the group you have.  Give students a pre-test to determine what they know.  Ask them not to guess--it hurts them when they guess correctly but then are expected to be able to do the work!

Second, you have to prepare plans for each different level of student.  For this reason, it is a good idea to start out with only two or three levels in your classroom.  One method that usually works well is to divide the class into "Non-Comprehenders" and "Comprehenders."

•  The "Comprehenders" understand how to read the problems and know how to place the information in order to come up with a solution. (Of course, you may end up with subgroups within this one group, but for simplicity we will not subdivide them at this time.)  This group can work in teams of three to four students on practice problems until everyone in the group can easily go up to the board and explain the steps in the solution.  The team will work together until everyone can do this.
• The "Non-Comprehenders" do not understand how to read and comprehend the information you have given to them.  Or perhaps they can comprehend the reading material but do not know how to use the information to solve the problems. While the other group is working in teams to understand how to explain each step in the solution, this group will be instructed by the teacher on ways to better comprehend the information.  Such instruction as mentioned above will help students take each piece and fit it together so it makes total sense to them.

During the last fifteen minutes of class, the first group presents problems and shows how to come up with the solutions.  Every member of the group should be able to take a new problem from the teacher and to figure out the steps to come up with the correct solution.  If this first group is larger than the second group, you can use a timer and give each student 30 seconds to explain a part before turning over the marker to the next student.  Every student in the first group should be able to pick up the explanation at any time and continue without interruption.