Phil Daro discusses the reasons against Answer-Getting HERE. Daro's 18-minutes talk gets to an important aspect about teaching. In this talk, Daro suggests that in the U.S. we (math teachers) focus on helping students get answers. On the other hand in countries like Japan, teachers focus on the mathematics that can be learned from problems, this simple, fundamental difference leads to vastly different outcomes and perspectives about teaching and learning. When the focus is on mathematics and not just Answer Getting, then students can engage in doing the kinds of things that mathematicians believe is real mathematics.
Additionally Daro discusses the notion that mistakes and answers are part of the process of learning mathematics. They are not the ultimate goals. Answers, while still essential, are only a part of a larger endeavor, and not a signal that there is nothing left to do.
Mistakes should also be valued as useful discoveries. When we discover a method that does not work, it needs to be fleshed out so that we can be sure that we can learn as much as possible about the related mathematics. Such a process is not usually part of the standard method of instruction.
IBL instruction is consistent with these ideas. Instructors in college-level IBL courses use a well-crafted set of problems to provide the context for the learning experiences. (Course materials for some college-level courses can be found at The Journal of Inquiry Based Learning in Mathematics.) Students work on these problems without being shown solutions ahead of time. Part of class time is used by students to present solutions or ideas to the rest of the class, and the audience peer-reviews these ideas. Logic and reason are used to determine if solutions are correct, and mistakes (discoveries) are used as opportunities for further investigations.
It's a wonderful notion to view mistakes as important discoveries. This sets up the framework for class discussions in a positive, scientific setting. Mistakes then become identified as useful explicitly in the running of the class, and this is where diverging from Answer-Getting becomes fundamentally different than doing mathematics. If the goal is getting answers, then by definition getting non-answers isn't getting us to our goal. While we mathematicians view mistakes from a healthy perspective (at least when we are doing math), our views and attitudes are divergent from the way some students look at mathematics. They "FOIL it" or "Butterfly it" or "Cross Cancel/Multiply."
A related point is what new IBLers often say. A common statement is, "I'm surprised that students have so much trouble with these concept questions." These concept questions may be true-false questions or questions that ask students to apply an idea or method to a slightly novel (to students) problem. The results are usually discouraging, and college instructors wonder why this is the case.
Daro's talk sheds some light on this phenomenon. In the segment where students apply the "butterfly algorithm" to add fractions, it is noted that the trick doesn't generalize (easily or obviously to students) to adding three fractions, and U.S. students perform especially poorly on adding $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$ compared to their international peers. Daro suggests that it is because we spend too much time on tricks and on Answer-Getting, often at the expense of doing the underlying mathematics (in this case equivalence, equivalent fractions, and common denominators). By the time these students get to college, their limited experiences with logic, problem solving, and higher-level thinking in mathematics leaves them underprepared for rigorous thinking.
Daro's talk sheds some light on this phenomenon. In the segment where students apply the "butterfly algorithm" to add fractions, it is noted that the trick doesn't generalize (easily or obviously to students) to adding three fractions, and U.S. students perform especially poorly on adding $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$ compared to their international peers. Daro suggests that it is because we spend too much time on tricks and on Answer-Getting, often at the expense of doing the underlying mathematics (in this case equivalence, equivalent fractions, and common denominators). By the time these students get to college, their limited experiences with logic, problem solving, and higher-level thinking in mathematics leaves them underprepared for rigorous thinking.
What can we do in college? Focus on mathematical discoveries of all types, and let students inquire together about the meaning of mathematics. Each new idea is a discovery and we can provide supportive classroom experiences, high-quality tasks, and effective coaching/mentoring to move students towards successful habits of minds and attitudes.
One can start a course by sharing prepared common mistakes and use them as the first experience in learning from mistakes and how the course will view mistakes as important discoveries. Instructors can state something along the lines of... "What discoveries have you made about this problem? Please work with your partner to write these down and be ready to share your discoveries."
One can start a course by sharing prepared common mistakes and use them as the first experience in learning from mistakes and how the course will view mistakes as important discoveries. Instructors can state something along the lines of... "What discoveries have you made about this problem? Please work with your partner to write these down and be ready to share your discoveries."
