When intuition fails us as scientists, we fall back on experimentation. This post is about how to build individual skills for students to experiment in a group setting. These skills are important for every course, even Language Arts, so don't go anywhere, English teachers.

The scientific method is something like this:1

1. Choose a question.
2. Try to intuit "the answer".
Continue when there are multiple possible answers.
3. Choose good examples to test the possible answers.
4. Study each example, performing tests.
5. Try to make a general explanation of your results.

In reality, every step above has loopbacks on failure. For instance, in the last step, you may realize you chose bad examples and have to go back to testing. Or maybe you notice another possible answer you forgot to find examples to check against. In the worst case, there is no good answer at all because you worded the question poorly. The objective of this post is to help you be able:

• to teach the process of dealing with these loops and stumbling blocks to your students, by giving clear and precise verbage
• to construct lesson plans that automatically balance students getting discouraged and shutting down with students having a lack of experience with failure

## What am I doing wrong?

#### General advice for students, every step of the way.

Since students should first experience complete freedom in the final steps, I'll discuss the problem areas in reverse. I'll also present everything as I would to an adult because that's how I present things for my students.

(Step 5.)(adv-step-5) I can't seem to make any sense of these results. How do I conclude anything?

• If the analysis is too specific, then perhaps the results aren't being connected and interpolated. Consider two examples, and everything in between them. Can a guess be made? How wrong might it be?

• If the analysis is too specific, then perhaps the results aren't being extrapolated. Build an induction step: given the furthest example so far, make a prediction for a further example.

• If the analysis is too general, then either some results are being brushed under the rug, or some good examples were never studied. Consider a missing example.

• If no analysis comes to mind, then the results do not match any of the "answers" that were considered initially, and no new "answers" come to mind. Put the problem down and look at it later with fresh eyes. Discuss it with colleagues. Improvement could reside with any step: a better question, a new answer, an example you forgot to study, or an example you studied incorrectly will eventually come to mind.

(Step 4.)(adv-step-4) I don't know how to study this one example. One of my results must be wrong, but I can't find the error.

• If one particular example is causing problems, then it is raising a subquestion. Try to break it down with potential "answers" and ways to test them. Discuss it with colleagues, since one may have already encountered it before.

• If many examples are causing problems, then you're tackling too much at once. Choose one or two "simplest" examples to consider as subquestions.

(Resolving issues with Step 4 is what most education focuses on nowadays. Grade school and undergraduate study tends to use direct instruction, while some undergrad and all graduate study forgoes instruction because of practicality. The above tips are for the situations lacking instruction. More on this in a future post.)

(Step 3.)(adv-step-3) What examples are there? I know one when I see it, but I can't think of any. How do I choose a good one? I can't find an example separating two of the possible answers.

• If examples of an object don't come to mind, then probably the definition of the object isn't solidly motivated in your mind. Discuss it with your colleagues, both what the object is and why it is being studied.

• If the "good" examples don't stand out, then randomly pick some to study. It's a win-win: if the ones you pick aren't good, then it will become apparent why as you stumble in Step 4, and then you will know how to pick better examples.

• If none of the examples differentiate two of your answer choices as you move to Steps 4 & 5, even as you keep returning to Step 3 to "fix" it, then perhaps the answer choices are the same. Ask a subquestion of "Are these two explanations equivalent?" and try to show that they are. You might discover a property of the space you're working in that makes them equivalent or a more general space you could work in where they could be different.

(Step 2.)(adv-step-2) There are too many/few answer choices. I can only think of options I know are wrong.

• Honestly, the best advice here is always to move on to Step 3 to try to find inspiration:

• Too many answer choices isn't a problem; either choose a few to test at a time, or categorize them and test the differences between the categories.

• Too few means you need to get a better feel for what's going on, and examples will help with that.

• Only thinking of answers that are wrong isn't a problem, just carefully prove every single one wrong until you notice something you were missing the whole time.

• If examples don't help, discuss it with your colleagues.

(Step 1.)(adv-step-1) How can the question be wrong? How do I notice problems with it? How do I fix it?

• If the question is too general, then the answer will be too vague. Make the question more specific.

• If the question is too specific, then the answer will be trivial. Make the question more general.

• If the question made false assumptions, then you will notice inconsistencies in the answer. Correct the question.

• If the question is not provable, then there will be no way to get to the answer. Swap to the meta-question: "Is the original question provable?" Or perhaps study specific cases instead.

#### Producing Encouraging Experiences with Failure

The biggest problem with teaching this process is it gives students a lot of freedom, and every poor choice one makes along the way results in a lot of (what feels like) wasted effort. That makes students anxious, so the trick is to restrict their freedom, preferably giving them as much choice as possible in that restriction.

My favorite way of giving a choice of restriction is in a Who Wants to be a Millionaire kind of way, with lifelines. For example, say I choose a question, intuit some possible answers, and am leaving the last three steps for the students to solve. I'll tell them that I am their lifeline, and that each student may use me to help them directly with at most one of those three steps: I can help them choose examples to test, help them study the examples they chose, or help them turn their results into explanations. But I'll only help with one. This "lifeline" choice of phrasing is important — saying "I'll help you with one." feels a lot better than "You must do two." while also framing my interaction as assistance rather than doing work for them.

Note. My "lifeline" help goes beyond my normal aid, which I offer for every student working on any step to encourage them to ask questions. For my normal kind of help, I always reserve the right to answer quickly with a cryptic question while moving on to help the next student. Then, by presenting failure in trying to interpret my responses, students "earn" more direct answers. In contrast, my "lifeline" aid is guaranteed to be straightforward instructions with guidance through the entire process.