# Why Rich Open Problems? Bruce Williams.

One of the major obstacles against student’s performance in mathematics is teachers constantly telling students how to “solve” problems using “strategies”. Eg. When you see the word “total” in a problem, it means addition or multiplication. (This will not work with the question of “how long did I spend at the dentist in total?”)

The alternative is for students to “make sense” of the problem and use one or more of the many problem solving strategies developed through regular experience in solving problems in the classroom.

The use of Open Problems fosters student’s inherent abilities as a problem solver from their earliest experiences. The human ability to problem solve is intrinsic and only needs to be nurtured through rich experiences.

Open Problems are problems that allow for more than one correct solution. For example, “At the pet store I saw fish tanks, each with the same number of fish. How many fish might there have been?” has a multitude of correct solution, but also has incorrect solutions too.

1. Forces students to think and understand the problem.Students need to make sense of the problem to put in numbers that make sense to develop a solution that makes sense. When this problem has been tried in class, all students obtained a correct solution.

(The similar “closed” problem to above is “At the pet store I saw 3 fish tanks, each with 5 fish. How many fish were there in total?” Students will guess to add or multiply to get the correct solution, and the correct solution is the only thing they and many teachers are concerned with. Try this in class and both solutions of 8 (3+5) or 15 (3X5) will appear in most classes. If a student correctly guesses to multiply, does this tell the teacher anything about the students mathematical understanding, clearly not.)

2. Are self differentiating problems, therefore allowing access to the problem to all students.

If I know the solution to 3X5, where is the learning? and If I don’t, I do not have access to the problem.The Open version allows struggling mathematicians to use numbers they feel confident with (2 tanks with 4 fish in each) while more confident mathematicians are able to demonstrate their abilities (15 tanks with 9 fish in each)

3. Allow multiple efficient strategies for solving the same problem.For 2X4 I could use a known fact, while 15X9 use partial products. They could be efficiently represented differently too. 3X5 with an array, 15X9 in a T chart. The sharing of the open problems then becomes interesting, engaging and a learning experience as they see other students strategies, not the “correct” teachers strategy.

4. All student have ownership of the problem as they visualized their individualized problem with their own numbers.

5. Shows the teacher how each student is working as a mathematician. Instead of how well they can follow the teachers directions in how to solve this problem. (this happens because teachers are afraid students will get the wrong answer if they don’t tell them how to solve it, the result, no thinking and little learning).

6. Develops students as real mathematicians. Students are using the tools of mathematics to develop a solution; much in the same way a builder used their tools to develop a concept. Being able to add or multiply makes me no more a mathematician as knowing how to hammer makes me a builder.

Bruce Williams. CreatingRealMathematicians.com