Lateral thinking in mathematics

Question

Especially in mathematics, we give a set of definitions and rules, and ask our students to prove a particular statement or to solve equations or inequalities.

By this kind of system we limit students to a particular frame and mainly train their minds for vertical thinking which not only discourages improving their creativity and think differently but also it doesn't enhance exploring skills in mathematics. I think lateral thinking is what the most important way to overcome practical challenges in real life too. My issue is how we have to focus on this matter in a class room and what kind of activities we can do with students while preparing the students to face exams in relevant curriculum?

Here I suggest one of my approaches where you can allow students to use unconventional and nontraditional techniques in their suggestions.
Bring the puzzle set Towers of Hanoi of 10 discs and 3 poles to the classroom

1. Allow students to think about it for about 15 minutes in their own way and express their views,
2. Ask students to suggest a game to use it,
3. Ask students to give a suitable name for the puzzle,
4. Demonstrate the normal way of playing the puzzle and check whether who can accept the challenge to solve the puzzle in the case of 64 discs,
5. Give a month to prepare full list of issues related to the puzzle without mentioning any guidelines.
   Finally you can allow them to share their suggestions and recommend any amendments if needed.

I suggest specifically this puzzle because here students have many areas to improve such as by studying the history of the puzzle, mathematics concepts can be applied,observing the patterns, different versions, how to make the challenge much difficult and availability of this puzzle set etc.

Is this kind of activity(also can be used as a project)useful to improve lateral thinking for grade 10 or above in high school and advanced level or what you would suggest more useful for them?

Answer 1

A relevant term in didactics of mathematics is 'problem solving', which means facing a problem where one does not know the means of solving it already. You may want to check the literature for this. Your particular activity also reminds of landscapes of investigation (undersøgelseslandskab).

There are other terms like open tasks and rich mathematical tasks. In Norwegian we also speak of LIST-tasks (low threshold, high ceiling).

For concrete tasks in English, you might want to check out Youcubed: https://www.youcubed.org/tasks/

For literature, I would recommend:

• BOALER, Jo. Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. John Wiley & Sons, 2015. Chapter 5.
• KARLSEN, Lisbeth. Tenk det!: utforskning, forståelse og samarbeid - elever som tenker sjæl i matematikk: ungdomstrinnet. Cappelen Damm akademisk, 2014.
• SKOVSMOSE, Ole. Landscapes of investigation. Zentralblatt für Didaktik der Mathematik, 2001, 33: 123-132.
• WÆGE, K.; NOSRATI, M. Motivasjon i matematikk (2. utg.). Oslo: Universitetsforlaget, 2019. Especially chapter 6.

Answer 2

Another problem that accommodates the "lateral thinking" the OP espouses is discovering and understanding Euler's polyhedron formula, $V-E+F=2$.

Discovery is aided by polydrons or equivalent manipulatives.

And there is a complex history behind the formula detailed in Imre Lakatos's classic book:

Finally, this is more than just a puzzle. It is a significant and deep theorem.

In my experience, this topic is quite accessible to high-school students in the U.S.