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Published on February 13, 2014;
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Published on February 13, 2014 by Mario Pineda, Ph.D.
at http://drpineda.ca/mathematics/2014/02/13/Mathematical-thinking.html

*This post was published as part of an assignment in EDSE 338 in the Faculty of Education at the University of Alberta.*

Mathematical thinking is not the same as doing mathematics, at least not in the context of how mathematics typically is taught in contemporary school systems where the key to success is for students to think inside the box, rote memorization and the applications of algorithms to staged problems (McGarvey, 2014; Rogness, 2014). In contrast, in mathematical thinking the focus is on thinking outside the box, understanding over memorization and to learn to think in a certain way to solve real world problems arising in science, engineering, in our society and in the students' own lives.

The focus on doing mathematics is clearly visible in text books where students are asked to solve highly stereotypical mathematical problems. One of the more common text book problem goes something like this:

Farmer Bob has 210m of fencing to enclose his pig pen on all 4 sides. What dimensions should his pig pen be to enclose the largest possible area?

While this problem serves as a useful foundational exercise for *doing math* by practising the formulation and solving of quadratic equations it provides little opportunity for going beyond doing mathematics to develop mathematical thinking skills. The main limitations of this and similar types of problems is that students primarily rely on the lower levels of Bloom's taxonomy when working with these problems (Ben-Hur, 2006), e.g. identifying that the problem can be solved using quadratic functions and the recalling and the instrumental application of the quadratic formula (the remembering and application level in Bloom's taxonomy) (Skemp, 2006). It is interesting to note that while identifying quadratic functions as one possible approach for solving this problem requires a higher level in Bloom's taxonomy (the analysing level), the context of the problem, i.e. it is located in the chapter on quadratic functions in the text book, provides an obvious give away to students as how they are expected to solve the problem thus reducing the task to the analysing level.

The levels of Bloom's taxonomy that are notably absent in these types of problems are evaluating, i.e. the student makes decisions based on in-depth reflections, criticism and assessment, and creating, i.e. the creation of new ideas and information using what has been previously learned. Ultimately mathematical thinking is about extending mathematics from simply doing mathematics by recalling and applying algorithms to generalizing concepts by evaluating and creating new mathematical solutions. The differences between the levels in Bloom's taxonomy that these type of stereotypical mathematical problems versus more open ended problem solving involve translates directly into an ability (or lack thereof) for students to develop mathematical thinking skills, or in Vygotsky's words "learning precedes development" (McLeod, 2013). In other words, in order to develop mathematical thinking skill students have to be pushed out of their comfort zone of what they know (e.g. learning by solving stereotypical problems) into the Zone of Proximal Development (ZPD) where problems are open ended and where learning takes places by evaluating and creating. By providing scaffolding in the ZPD students are able to work at higher levels of Bloom's taxonomy allowing them to develop their mathematical thinking skills rather than just doing mathematics within their comfort zone (Ben-Hur, 2006).

In the example of Farmer Bob's pig pen one possible rephrasing to break the stereotypical mould, eliminate the staging and allow students to utilize the highest levels in Bloom's taxonomy could be:

Farmer Bob has a certain amount of fencing to enclose his pig pen. Provide the best solution for maximizing the area of the pig pen given the available amount of fencing.The rephrased problem has been generalized as it does not mention a specific amount of fencing nor foes it provide clues as to what shape the pig pen should have. Ideally the rephrased problem would be asked out of the context of the chapter on quadratic equations allowing students to think outside the box and utilize some of their background knowledge, e.g. how does the area of pig pens with a give perimeter relate to their shape.

In the The Math of Infectious Diseases set of lessons students are introduced progressively to the topic of using mathematical models in biology, from the relative comfort of watching video clips and engaging in whole-class discussions on the utility of mathematical models in biology (Lesson 1), to the introduction of new mathematics in the context of their previous knowledge (Lesson 2) and to the Zone of Proximal Development where students create and evaluate their own mathematical models of biological systems (Lesson 3 and 4). Over the course of the week students are scaffolded from *what is known* towards *what is not known* through the Zone of Proximal Development where new instruction takes place through the use of authentic non-staged problems culminating on the last day when students apply their new knowledge and insights by exploring a computer simulation model of infectious diseases (Lesson 5) (Ben-Hur, 2006). The focus of these lessons enable students to generalize their pre-existing mathematical knowledge in contextualized authentic real world problems that students and anyone ever having a bout of the flu or cold would be able to relate to.

The aim of The Math of Infectious Diseases set of lessons is to go beyond the application of formulas and the instrumental manipulation of symbols by allowing students to brainstorm about mathematical approaches for solving authentic problems. The emphasis is on understanding the science and mathematics and in particular the larger picture of applied mathematics without the distractions of symbolic manipulation and solving equations. The goal is that this elective component would complement the school curriculum, which usually focuses on symbolic manipulation and solution techniques.

What does, however, the term mathematical thinking skills mean? Although it encompasses some aspects of expertise, competence, knowledge and understanding of mathematics - all of which require cognitive thinking - its meaning is much broader than that. The five strands of what Kirkpatrick's (2001) refers to as mathematical proficiency captures the breath and depth of what mathematical thinking represents: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. All of these strands are critical not only for creating well-rounded learners but also for real-world problem solving. Fostering the five strands of mathematical thinking is one important aspects of the teaching and learning process. It enables the teacher to contextualize the instructions making the learning authentic and relevant to the students' own lives. A mathematically thinking classroom has the potential for becoming a self-full filling prophesy. By providing high level contextualized mathematics instructions students develop their mathematical thinking skills. Once students have develop their mathematical thinking skills they become more proficient at applying their mathematical thinking to open ended real world problems at high levels in Bloom's taxonomy. This will further contextualize their learning enabling students to continue to develop their mathematical thinking. Ultimately the end result is that mathematical thinking begets mathematical thinking.

Are there any potential disadvantages of fostering mathematical thinking in high school classrooms? From the perspective of the teacher the chief "disadvantage" is that the teacher has to be able to let go and give students more intellectual freedom without looking for a single right answer. Open ended problems, in particular problems drawn from the real world, often do not have one single right answer. During the class presentation of my assignment (The math of infectious diseases), when I asked the class to formulate compartment models of the various infectious disease scenarios it soon became apparent that groups working on the same disease formulated widely divergent models. None of the models were necessarily wrong, they just represented different assumptions and simplifications that groups made. As a teacher you need to be able to reconcile this and view this as a "teachable moment". Some of the questions that the class can discuss after their presentations in Lesson 4 are:

How can models of the same system be different?

What is a wrong model?

What is a right model?

What is a useful model?

What is a mathematical model?

Another potential "disadvantage" of fostering mathematical thinking in high school classrooms, at least as it may be perceived by students, is that solving problems at the highest levels of Bloom's taxonomy in the Zone of Proximal Development is difficult work requiring strong work ethic, or as Jonathan Rogness, a mathematics professor at University of Minnesota, deftly puts it in the syllabus for one his mathematics classes:

Occasionally somebody will ask me how to do well in a math class. After twenty some years of math classes, I can tell you that there is only one surefire method: do lots of math problems. This is why we assign homework; the only way to learn math is by doing math! If it makes you feel any better, I realize this isn't the most welcome advice in the world. Sometimes I don't follow it myself in my graduate courses. But it really is true. You can read something in a book and think you understand it, but you'll only know for sure if you work out some of the exercises.While it initially may appear that Rogness is talking about

If we compare mathematics with the automotive world, school math corresponds to driving. In the automotive equivalent to university mathematics, in contrast, you learn how a car works, how to maintain and repair it and, if you pursue the subject far enough, how to design and build your own car.My aim with the The Math of Infectious Diseases set of lessons is to provide an opportunity for high school students to go beyond the "driver's seat" giving them a taste of how to "maintain, repair and build their own vehicle" by utilizing and developing their mathematical thinking.

Kirkpatrick, J. 2010. The Strands of Mathematical Proficiency. In: Jeremy Kilpatrick, Jane Swafford, Bradford Findell (eds). Adding It Up: Helping Children Learn Mathematics. URL: http://www.nap.edu/catalog.php?record_id=9822 (Accessed on February 12, 2014)

Ben-Hur, M. 2006. Concept-rich mathematics: Building a strong foundation for reasoning and problem solving.

Devlin, K., 2014. Introduction to Mathematical Thinking with Keith Devlin. URL: https://www.youtube.com/watch?v=0xCRl54AjX0#t=120 (Accessed on February 13, 2014)

McGarvey, L. 2014. URL: http://www.cbc.ca/player/Radio/Local+Shows/Alberta/Alberta+at+Noon/ID/2431634960/ (Accessed on February 12, 2014)

McLeod, S., 2013. Lev Vygotsky. URL: http://www.simplypsychology.org/vygotsky.html (Accessed on February 12, 2014)

Rogness, J., 2003. Excursions in mathematics. URL: http://www.math.umn.edu/~rogness/math1001/ (Accessed on February 12, 2014)

Skemp, R.R., 2006. Relational understanding and instrumental understanding. Mathematics Teaching in the Middle School, 12(2):88-95.

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