PBL 2 – day 2, curriculum reasons for technology! And problem solving…
Technology for teaching & technology for learning secondary school mathematics.
Lots of great discussions today! Issues of software and student accessibility – perhaps hardware concerns? definitely learning ability/disability concerns. Knowing about ‘mathematical processes’ which are present in the curriculum…but are they really understood? And are they present in our thinking and decision-making about technology? Following the PBL steps and reviewing the Learning Goals for the PBL task were key features to conversations today, and to re-shaping and forming arguments and directions for inquiry over the next few days as team members get ready to do their ‘homework’ so the team can move on and be productive in the next team meeting. (actually, very much a ‘flipped classroom’ kind of experience I think)
A question has been rumbling around in my head lately, especially today, is mathematical thinking for secondary school students the same as the mathematical thinking I believe I am hearing in the PBL groups from preservice mathematics teachers? If so, does the same definition apply or should there be a subtle shift in the articulation? This feels like it could be a similar kind of argument about problem solving as a definition – are the “steps to problem solving” the same as “mathematical thinking as problem solving”? (In my opinion, they are different… and pedagogically necessarily so…
“Pólyais likely the most familiar person associated with describing the steps of problem solving. In mathematics curricula, for example, the various National Council of Teachers of Mathematics (NCTM) principles and standards publications, and the Ontario Ministry of Education curriculum (OME, 2005), problem solving is described as a mathematics process or performance skill, a way of ‘doing’ some (but not all) mathematics, where “[i]t is considered an essential process through which students are able to achieve the expectations in mathematics” (OME, 2005, p. 12).
Problem solving is more than just a set of steps to successfully find a solution to a mathematics problem; it is also how one thinks as a problem solver. Some have explored ways one can be a problem solver, and explain what knowing mathematics is all about, for example, “knowing-to” from Mason and Spence (1999), the Harvard verbs of mathematical inquiry (Harvard, 1995), and Cuoco, Goldenberg, and Mark’s (1996) mathematical habits of mind.
It appears problem solving has two meanings, a) a noun: the way some mathematics is done, and b) a verb: the thinking and reasoning mathematically through a problem to come to a successful conclusion or solution. People who are proficient at mathematics may be unaware of how problem solving can exist as a noun and as a verb, and that this distinction can make a difference in their thinking and actions, for example, preservice teachers who are teaching others to learn mathematics and mathematics problem solving. If preservice teachers are not crystal clear that teaching mathematics as problem solving is different from teaching mathematics through problem-solving, what learning success can we expect from students?” (from Pyper, J. S. & MacGregor, S. (2018). Problem-solving: How preservice teachers understand it during their preservice learning. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.). Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 19-26). Umeå, Sweden: PME.)
References from above text.
Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behaviour, 15, 375-402.
Harvard. (1995). Assessing mathematical understanding and skills effectively. An interim report of the Harvard Group, Balanced Assessment Project. Harvard Graduate School of Education. Retrieved from http://hgse.balancedassessment.org/amuse.html
Mason, J., & Spence, M. (1999). Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment. Educational Studies in Mathematics, 38(1/3), 135-161.
OME. (2005 (revised)). The Ontario Curriculum: Grades 9 and 10 Mathematics. Toronto, Ontario: Queen’s Printer for Ontario Retrieved from http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf.