Think aloud

Think aloudGood Problem SolversA Good problem solvers become aware of what they are doing and frequently monitor, or self-assess, their progress or adjust their strategies as they encounter and solve problems.

NCTM Principles and Standards, 20003Why engage in problem solving?Good problems:Deepen our thinking about mathematicsHelp us recognize connections between topics in mathematicsForce us to confront mathematical misconceptionsWorking on good problems we learn mathematics better! Even a simple one-step or two-step story problem can be a good problem for some students.

4 The quality of the problems can serve as an index of how well a person can solve problems (Kilpatrick, 1987). In order to effectively teach with a problem-solving focus, a teacher needs to carefully plan problems that will give students the maximum opportunity to hone their skills.

This means the problems need to be complex enough to let students approach them from different angles, explore different strategies, reflect on their progress, and revise their methods.General theory postulates three types of process in problem solving (Newell & Simon 1972)1. Asking oneself questions to derive extra information pertaining to the problem and its possible solution.What do I know now, what is given, what is the problem, have I seen such a problem before?2. Aplication of solving procedures3.The phase relates the solution to the problem and check the solution for correctness or plausibility.Polya model 1. UNDERSTANDING THE PROBLEM 2. DEVISING A PLAN 3. CARRYING OUT THE PLAN 4. LOOKING BACK Polya model strategic problem solving 1973* State the problem in your own words* What are you trying to find or do? * What are the unknowns? * What information do you obtain from the problem? * What information, if any, is missing or not needed? * What are the data? What is the condition?* Detect the variable involved in the problem* Know the relationship between the variables which have ben ascertained* Understand which variable need to be through searched or answered* Draw a figure. Introduce suitable notation.1. UNDERSTANDING THE PROBLEM The following list of strategies, although not exhaustive, is very useful. * Look for a pattern. * Examine related problems, and determine if the same technique can be applied. * Examine a simpler or special case of the problem to gain insight into the solution of the original problem. * Make a table. * Make a diagram. * Write an equation. * Use guess and check. * Work backward. * Identify a sub goal.

2. DEVISING A PLAN * Implement the strategy or strategies in step 2, and perform any necessary actions or computations. * Check each step of the plan as you proceed. This may be intuitive checking or a formal proof of each step. * Keep an accurate record of your work.

3. CARRYING OUT THE PLAN * Check the results in the original problem. (In some cases this will require a proof.) * Interpret the solution in terms of the original problem. Does your answer make sense? Is it reasonable? * Determine whether there is another method of finding the solution. * If possible, determine other related or more general problems for which the techniques will work.

4. LOOKING BACK Novices Not familiar with the problem, often do not perform many orientation activitiesDo not evaluate their solution or not looking back the process.ExpertsSolve problems in order denoted by the three phases or with more detail by following the Polya ModelSpend much time on orientation activities Always check their solution.What the theory say about the differences ? a prediction about differences between novices and expert at tasks concern the process rather than the result.

The theory also does not predict that expert will find better solution.

The most important elements to success are capacity limit of working memory & perception and previously acquired knowledge. -People will use for a task by looking at relevant instruction and experience. Instructions for thinking aloud activityDo not use phrases tell me what you thinkModelling cognitive processExample:

A father, a mother and their son are 80 years old together. The father is twice as old as the son. Their mother has the same age as the father. How old is the son?protocolsStudent 1Treat the problems as mathematical equation and solve it straightforwardly Student 2Starts with a guess: the father is 30 years old.Evaluates the result by using knowledge about reasonable age differences between parents and sonForm a new estimation and evaluates it again. Both give the same answer but different solving strategy.The protocol shows where the student encounters difficulties and when he gets confused.Protocols are collected by instructing students to solve a problem while saying what goes through their head stating directly what they think

You can help students become strategic readers and problem solvers by modeling the thinking you use to make sense of a problem.

Not just telling students the steps you take to compute an answerbut teachers modeling to think aloud the steps and decisions they make when solving a problem. (Silbert, Carnine, & Stein, 1989).

As students listen to the questions you ask yourself and the ways you guide your own reading, they learn to ask themselves questions to guide their own sense-making.

Think-Aloud Paired strategy

The think-aloud process is usually introduced in four steps, gradually transferring responsibility to students:

The teacher reads a problem and stops as needed to explain her thoughts. Students listen. They all solve the problem together.

The teacher reads the problem and stops often. Students express their thoughts at each point (and often write them). The whole class, led by the teacher, solves the problem together.

The teacher reads the problem, allowing students to signal stopping points as thoughts occur to them. Students solve the problem individually, and then discuss their interpretations of it and solution strategies.

Students do this together, in pairs. They work together to solve the problem.A benefit of this strategy for students who need additional support is that they also listen to the thinking of their classmates, enabling them to learn additional strategies from each other.

Think aloud can provide data about sophisticated and less sophisticated cognitive process.

NCTM makes the point that to effectively teach problem-solving skills, teachers must themselves have the knowledge and dispositions of effective problem solvers (NCTM, Problem Solving, Teachers Role).

Try it: Use this problem with a partner to try the first and second steps in the think-aloud process.

The boys swim team and the girls swim team held a car wash. They made $210 altogether. There were twice as many girls as boys, so they decided to give the girls team twice as much money as the boys team. How much did each team get?