Problem-Based Learning (Schettino, 2011)<\/span><\/a> – An approach to curriculum and pedagogy where student learning and content material are (co)-constructed by students and teachers through mostly <\/span>contextually-based problems in a discussion-based classroom where student voice, experience, and prior knowledge are valued in a non-hierarchical environment utilizing a relational pedagogy.<\/p>\n Educational Psychologist and Cognitive Psychologists like Hmelo-Silver at Rutgers University have done a lot of research on how students learn through this type of scaffolded problem-based curriculum dependent on tapping into and accessing prior knowledge in order to move on and construct new knowledge.\u00a0 There was a great pair of articles back in 2006\/2007 where Kirschner, Sweller & Clark spoke out against problem- and inquiry-based methods of instruction and Hmelo, Duncan and Chinn responded in favor.\u00a0 I highly recommend reading these research reports for anyone who is thinking of using PBL or any type of inquiry-based instruction (in math or any discipline).\u00a0 It really helps you to understand the pros and cons and parent and administrator concerns.<\/p>\n However, after you are prepared and know the score, teachers always go back to their gut and know what works for their intuitive feeling on student learning as well.\u00a0 For me, in PBL, I look at how their prior knowledge connects with how, why and what they are currently learning.\u00a0 One of the best examples of this for me is a sequence of problems in the curriculum that I use which is an adaption from the Phillips Exeter Academy Math 2<\/a> materials.\u00a0 I\u2019ve added a few more scaffolding problems (see revised materials<\/a>) in there in order to make some of the topics a bit fuller, but they did a wonderful job (which I was lucky enough to help with)and keep adding and editing every year. The sequence starts with a problem that could be any circumcenter problem in any textbook where students use their prior knowledge of how to find a circumcenter using perpendicular bisectors.<\/p>\n \u201cFind the center of the circumscribed circle of the triangle with vertices (3,1), (1,3) and (-1,-3).\u201d<\/p>\n Students can actually use any method they like \u2013 they can use the old reliable algebra by finding midpoints, opposite reciprocal slopes and write equations of lines and find the intersection points.\u00a0 However, I\u2019ve had some students just plot the points on GeoGebra and use the circumcenter tool.\u00a0 The point of this problem is for them to just review the idea and recall what makes it the circumcenter.\u00a0 In the discussion of this problem at least one students (usually more than one) notices that the triangle is a right triangle and says something like \u201coh yeah, when we did this before we said that when it\u2019s an acute triangle the circumcenter is inside and when it\u2019s an obtuse triangle the circumcenter is outside.\u00a0 But when it\u2019s a right triangle, the circumcenter is on the hypotenuse.\u201d<\/p>\n Of course then the kid of did the problem on geogebra will say something like, \u201cwell it\u2019s not just on the hypotenuse it\u2019s at the midpoint.\u201d<\/p>\n \u00a0 Dicussion will ensue about how we proved that the circumcenter of a right triangle has to be at the midpoint of the hypotenuse.<\/p>\n A day or so later, maybe on the next page there will be a problem that says something like<\/p>\n \u201cFind the radius of the smallest circle that surrounds a 5 by 12 rectangle?\u201d<\/p>\n Here the kids are puzzled because there is no mention of a circumcenter or triangle or coordinates, but many kids start by drawing a picture and thinking out loud about putting a circle around the rectangle and seeing they can find out how small a circle they can make and where the radius would be.\u00a0 When working together oftentimes a student see a right triangle in the rectangle and makes the connection with the circumcenter.<\/p>\n A further scaffolded problem then follows:<\/p>\n \u201cThe line y=x+2 intersects the circle \u00a0in two points.\u00a0 Call the third quadrant point R and the first quadrant point E and find their coordinates.\u00a0 Let D be the point where the line through R and the center of the circle intersects the circle again.\u00a0 The chord DR is an example of a diameter<\/em>.\u00a0\u00a0 Show that RED is a right triangle.\u201d<\/p>\n![\"\"](\"http:\/\/www.carmelschettino.com\/wp\/wp-content\/uploads\/2013\/05\/geogebra-pic-for-blogpost-1024x762.png\" "\\"geogebra")
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