This article provides a brief review of problem solving and describes open-start problems in detail. The article then considers how open-start problems could address some important concerns in the teaching and assessment of mathematics and raises issues with regard to the future use of open-start problems in assessment.
Summary
This article offers an alternative to traditional word problems that are used to teach and test a students ability to use problem solving techniques. There are two types of problems that are the focus of problem solving: open-ended problems and as the authors propose open-start problems. An open-ended problem has no definite answer. There are multiple ways to solve the problem and a student can come to a different result based upon the method of solution. These types of problems and this type of thinking prove to be difficult to assess and evaluate. If educators want students to be problem solvers and if they want to be able to test that ability on high-stakes test, then another type of problem needs to be posed. Open-start problems have a definite correct answer, but the means of starting the problem is open. A open-start problem could have an algebraic, geometric, or a hybrid approach (various strategies used within each of these approaches). Open-start problems are problems where the solution and the means of finding the solution is not immediately apparent. They are problems that the students have not seen before.Open-start problems depend upon a mastery of the prior knowledge needed to apply to the solution. This is where open-start problems become difficult, because they require a mastery and because of the demands to cover a large curriculum, many teachers do not have time to teach all students to mastery, let alone have them apply that mastery to creative thinking in problem solving. Open-start problems are best assessed using an authentic assessment, looking at the process, not just the correct answer. A challenge in using open-start problems is in the ability to write a problem and then consider the multiply ways in which a student can come to a solution. Teachers need to be trained to create problems, know which level of problems are correct for each student, and how to best assess in an authentic manner. Students need to have a level of insight so as to see what type of math can be used to begin the problem and how potentially several types of math can be put together to come to a solution. Students who are problem solvers is the goal and open-start problems are one way that educators can assess a students ability to problem solve.
Evaluation
The authors begin with posing the question: What is the use of a students learning mathematics if the cannot use it to solve problems? This statement rings true for every teacher of math as they work to develop problem solvers, those who can creatively approach a problem or situation they have not seen before and then apply what they already know to find the solution. I have seen both types of problems: open ended and open start, and both of them are a challenge for students. It is not a natural thing for students to work through a challenge, or to apply what they know to a different situation. In my experience, most students close up and shut down when they are confronted with a challenging problem, or more specifically a problem that challenges them. I like the idea of open-start problems versus open-ended question because, as the authors described, they are more easily assessed. Another thing that is attractive to open-start problems is that when a student completes to problem, the teacher can challenge the student to consider another approach. In contrast, open-ended problems have the same ability to challenge greater depth, because in open-ended problems there are many depths to the “solution” and there is not necessarily a final solution. Both of these problem types lend themselves to challenge students in their creative thinking, and they both require a mastery of prior knowledge needed to begin the solution process. As a teacher of mathematics the biggest barrier to problem solving is the willingness and the stick-to-it-ness of students to not give up when they are faced with a problem that challenges them, that does not have a defined beginning.
Reflection
As I reflect on this article I realize that my students are expected to do open-start problems more often then I realize. They are preparing for the SAT and in the math portion, much of what we have done so far this year are open-start problems that are based upon material they “should” have mastered. I have been repeatedly telling them in class that they have the prior knowledge, now they need to learn how to apply it to new situations. Lately it has been repetition as my means of teaching problem solving, but I know that is not an effective way to teach. The big question I have as I reflect on this is how does a teacher teach problem solving? What are the techniques that have been used to improve students abilities to problem solve, as describe in this article? I believe all students can be problem solvers if giving the right tools. What are those tools, and where is the “Home Depot” where I can get them for my students?
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