During my practicum, I observed a teacher implementing some of the techniques Dave Hewitt talked about in terms of allowing students to derive understandings from a given problem or exercise. In the activity this teacher put forth, she drew a central inequality (e.g. 2 < 5), circled it, and had lines coming out of the circle labelled "divided by 2," "multiplied by -4," "+ 2" etc. and had her students fill in the answers at the end of those lines on their own. Afterwards, she had them call out their answers and asked if anyone noticed a pattern. A student responded but I knew most of them had reached that conclusion as I walked around to observe them and listen to the conversations they had with their neighbours about their findings. I think it was a great activity and would like to do something similar in my class.
The examples in figures 9 and 10 on page 8 are also good examples of this. During a math workshop one of my university classes held in a high school, we had a set list of word problems that we proposed to students which included something similar to this example but on a higher level which involved factoring - I think it was something like, "A rectangle has side lengths (x + 7) and (x - 3) with an area of 39. What are the two side lengths?" However, this group of students hadn't covered their factoring chapter yet so we hadn't expected anyone to answer it. Instead, a lot of students ended up with the right answer because they did some guessing and checking. Some happened upon the answer by systematically plugging numbers in for x until they got the desired area, but others recognized that the side lengths were 10 units apart so they looked for factors of 39 that were 10 apart and found 3 and 13. Thus, they skipped right over finding x and went straight to answering the question. Other students found that x had to be bigger than 3 and less than 39, but because numbers closer to 39 would multiple to make very high numbers, they figured it would be closer to less than 15, and narrowed their scope from there until they got x = 6 and side lengths 3 and 13. So for those of us who are used to seeing these types of questions, we would have jumped straight to solving for x in whichever method we preferred, but for these students, they had to develop a method to solve the question, and used important mathematical thinking and reasoning along the way. I think another important note here was to have students verbally explain their process to us and their peers, because it forced them to reflect on their process instead of solving and moving on.
A computer science prof I once had used to have us try problems in class on topics we hadn't yet covered, and would walk around to observe our work. His reasoning for this was, "If you're not struggling, you're not learning," which I believe is a similar sentiment to "If I'm having to remember ..., then I'm not working on mathematics." We were presented with a problem and no pre-determined algorithm for how to solve it so we often took unique paths to solve the same question, much like the students in the math workshop I talked about. I would like to attempt this with my students as well by presenting them with warm-up puzzles (possibly from websites like the ones Asha shared with us) or exploration exercises like the teacher whose class I observed. Another idea I had was to maybe put math terms with their definitions on the board (the "arbitrary" aspects of math), give students a problem to work on in groups (the "necessary" aspect), and ask them if they are able to incorporate those terms in their explanation to me/the class (bridging the two). Some of the possible hurdles to this would be students looking the terms up on their phones and finding the answer to the problem by copying an existing example of an algorithm, or not engaging with the activity for a variety of reasons. However, hopefully all it takes is some adjustments such as asking an extending problem that they can't look up, offering some hints so the problem becomes within reach etc. to realign the students to a state of "flow" and so they gain some useful understanding through the exercise.
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