Exercises to guide students in emulating how experts think when solving problems
by Linda L. Davis and Bill Rose
(Many of the ideas presented are built upon those discussed earlier in the day by Jen Sablock, Peter Lea, Jimm Myers, Z. Demet Kirbulut, Ji-Sook Han, Sister Gertrude Hennessey, and Dexter Perkins)
An unscripted effort first
I think many of us value the conclusions that students come up with on their own without our guidance, particularly when we ask them to work together. I often wonder when I give instructions how the exact wording of my instructions truly affects they way the students approach a problem.
Yet, from this workshop, in particular, from Karl Wirth's presentation entitled A Metacurriculum on Metacognition (PowerPoint 16.9MB Nov20 08), where work by Schoenfeld (1987) was illustrated, many of us agreed that many students would benefit greatly by modeling Expert Problem Solvers behaviors.
A compromise: assuming one is practicing problem-based learning, in a series of problems where the level of difficulty increases and the amount of guided instruction decreases through the series, one starts by presenting a problem (perhaps one that is difficult to very difficult) to the students and having them solve it cold. This problem then will be revisited at the end of the term, after you have stepped them through a series of problems where they learn about the cyclical way in which expert geoscientists solve problems.
Then, an expert example
The next step is to bring in an expert who is capable of telling the students how they go about solving a problem and preferably one who can talk out loud about their thought processes.
Follow with a series of problems that increase in difficulty, but decrease in guidance
-I think that many students have a severe problem with reading comprehension. On top of that many students then have trouble figuring out exactly what the problem is in a given situation.
- Reading comprehension exercises could be assigned where maybe Just-in-Time Teaching (JiTT) or pair-share and class discussion and agreement is used. These can be very short sentences or paragraphs. I love Shirley Yu's exercise which clearly showed that the reading instructions are critical, and that if the students can relate what they are reading to things they already know, comprehension or recall is better.
- So it might help if they practice very short exercises where they practice making reasonable analogies so that they have some context or familiarity to use for understanding. Maybe just a brief exposure to the vocabulary before class or before the problem is given would also help in the "decoding" of the problem, if new terminology is used and critical to the problem.
- If reading comprehension is targeted and dealt with, then the time spent in the first step used by Expert Problem Solvers is lessened, and I think also, analysis of the problem will be easier.
Analysis: This step can mean different things to different minds.
- Do the students need to analyze the problem? What is the problem? In the beginning of a semester of using Problem-Based Learning (PBL) to teach students how experts problem solve, one could state the problems explicitly, and as the semester (or even year if you have two classes tied together, e.g., mineralogy and petrology) wears on, the definition of the problem can be murkier so that they figure it out.
- Others may look at this step as analyzing the data. Terminology means different things.- A deeper level here is to analyze what will be needed to solve the problem. Give them practice by using problems of all kinds and ask them what the problem is. It could be a calculus problem, or what I learned as a "study problem." Just short little 1 minute exercises, on a par with the short reading comprehension exercises.
To analyze then, either they have to determine the factors to consider or they begin to evaluate the data given to them.
- So as an instructor per certain problems, you give them the data to work with, but give them extraneous data, so that they have to winnow out the chaff, and figure out which data is the data to use.
- If you do not give them the data, and they have to first determine what data to use, analyzing the problem at hand takes on more meaning. And the students then have to search out the data after figuring out which data to use.
Plan and Implement
Once the factors are determined, one sets up a plan to solve the problem then implement the plan.
- At times, depending up on the problem, the size of the data set for each variable has to be considered – when is the amount of data statistically significant? Somehow one has to get across the idea that for a particular variable the size of the data set has to be chosen....it may not be enough, but one can't know this until they test it. An example, to make this concrete would be trying to predict something like the eruption of Old Faithful given a range of data for as many days as one wishes (c.f. Carol Ormand's exercise...) (This may be one of the beautiful things to use later in this progression about teaching them to stop and assess what they did, e.g., the exploration of ideas that comes later part of this cyclic style of problem-solving. Was the data set large enough?)
- Then they either set up an equation or group of functions or set up a way to test their ideas.
- The plan has to be explicitly stated, particularly in group work because they so often lose sight of what they are trying to do. "What are we going to do to solve this problem, and how are we going to attack or implement the plan?" This is one way for the instructor to evaluate the exercises early on, or to intervene and help: did they keep with the plan and did they attack it in the way they said they would?
This step seems "squishy" but can be defined many ways. I "hate" when students ask me, "am I doing this right" or "is this right," because I think that this inhibits the learning process and I want them to take the plunge down a pathway that is unsure to them. Yet, I understand not wanting to waste time (or maybe appear stupid). Ignoring that, one could use the questions that Dave Mogk posited this workshop, for example: "Is this a reasonable answer?" "Does this fit with what we/you already know about the world/the situation/ the problem?" "If not, it's a 'flier,' and is the 'flier' significant or is it to be tossed?"
If they are flat out wrong, early in the semester, the instructor could here step in and let them know that the answer is incorrect.
The cyclical nature begins: Go back to analyzing the problem
Rethink this thing! What did they do wrong? Can they restate the problem? Did they go down the wrong path, or just not include all variables?
- Explore these new ways to analyze the problem. Test the water by trying out whether the new ideas or variables dreamed up after "verifying" might make a difference.
- Gain an idea of whether or not the new information is going to take them anywhere.
- Ok, decide that it will, and re do the plan and implement.
The instructor in the beginning should likely query the students here or have a check. This new plan should not only be written down, but compared and contrasted to the first plan.
I want to make sure that there is a change of plan. I have had students try the same approach to solving a problem for ten hours (or so they tell me). I think that this is someone's definition of insanity. I want it to become second nature to them to figure out when a plan is not working and that something has to be changed in order to guarantee success.
The second time around "planning and implementing" should take less time, unless new variables in fact make the problem even more difficult. Why? Well, maybe one could encourage "back of the envelope" implementation of the plan just to test it. If it is immediately clear that this is again the wrong path, they go back to more in depth analysis.
I think that depending upon the problem, there needs to be evidence or documentation as to how the plan was implemented. This is also great practice for them: if you write down what you did, explicitly and well, you can go back and check for mistakes, or areas where clues are provided so that one can proceed in the iterative process of problem solving by analyzing, pausing to think, double-checking what you have done.
I think that I need to have a "real" way to check that they verified this, and then have them evaluate that verification in a way that can be improved by bringing it into the limelight. For example, on lower levels, how many of the students read over their essays, or double-check their answers on Scantron-type tests? The number of silly mistakes could be greatly decreased just by proofing their work, but they don't. This is a great life-learning exercise to make second-nature.
Practice Practice Practice
I always have good intentions, but often drop the ball by the end of the semester. So, I have to INTENTIONALLY create these problems and work more and more of them into my detailed syllabus, so that I give them the practice necessary to really master this way of problem solving.
Take away the "scaffolding" slowly yet surely, so that by the end of the time you have with them, they need little help in setting up a way to approach solving a problem, and so that it becomes first nature to pause and think about the problem; first nature to not give up but to change the plan of action.
Does this expert way of thinking make a difference?
Go back to the original "difficult" problem (making sure you never solved it for them), and have them "re-solve" it after learning these new tools.