Tag: Scaffolding

Reducing Cognitive Load in PBL

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One of the things that I have been thinking about for a very long time is the idea of those who oppose PBL.  Namely those who prescribe to behaviorist and cognitive scientist theories of learning, which I know a great deal about because of my doctoral work.  So many teachers, parents and others have asked me about this over the past 25 years that you’d think I would have an answer.  I know I have thoughts but I do want to do more research in this area.

I do not pay lip-service to the ideas of cognitive load theory for sure and definitely respect those who follow these ideas.  I do think there is a place for thinking about this theory in PBL, but not an argument for why NOT to do it.  At its heart though, I believe the learning outcomes that are important in the different types of theory (CLT vs. constructivist learning for example) is what ends up differentiating them and also the way the knowledge is constructed.  I do believe in the importance of reducing the Cognitive Load for students so that their long-term memory can be triggered and practiced.

So I do believe there is a place for this in PBL – it just hasn’t been discussed a great deal.  There is always this us vs. them notion that one is right and the other is wrong – it comes from very strong belief systems and I totally understand where they are coming from.  However, if PBL is done well in a scaffolded, structured way, I believe that you can both reduce cognitive load and also ask students to think creatively.

Here is an image I saw from an article in the Guardian recently entitled Teachers: Your Guide to Learning Strategies that Really Work by Carl Hendrick. This graphic is describing the six ways to make your classroom best-ready for learning.

 

Positive Class room climate

 

When I was looking at this, the first thing I thought of is “This is my PBL classroom.”  However, I could tell there would have to be some discussion of the “reducing cognitive load” part.  All the other aspects, I believe you can find in some other blogpost of mine somewhere.  In a PBL classroom, the way that students get timely feedback is in so many ways (see my rubrics, journals, etc.).  The nightly homework is the scaffolding of learning and monitoring of independent practice – again when done well.  I won’t go through every one of these, but would love your takes (in the comments below) on each of them.

So then, how can we talk about reducing cognitive load in PBL – where is lecture and worked problems that the teacher does?  I would argue that the cognitive load is reduced by the scaffolding of the problems in the curriculum.  In other words, by triggering students’ prior knowledge the cognitive load is reduced in such a way that they are remembering something they have learned from the past, and then being asked to look at something new.  The “something new” goes through many phases of problems – concrete, multiple representations, all the way through to abstract – in order to slightly build up the cognitive load.  Again, this is all if it is done well and very deliberately with the idea of not to overload students’ thinking but to help to build the schemas that are needed for constructing knowledge both individually and socially.

The problems are worked by the students, yes – I will give you that.  But it is the teacher’s responsibility to make sure that the steps are correct, students get feedback on their thoughts and ideas, that on the board at the end of the discussion is a correct solution and so much more.  What this type of teaching does, in my view is both reduce cognitive load to a point, yet also allow students to gain agency and ownership of the material through their prior knowledge and experience.

Something else that Mr. Hendrick says in his article is:

“Getting students to a place where they can work independently is a hugely desired outcome, but perhaps not the best vehicle to get there. Providing worked examples and scaffolding in the short-term is a vital part of enabling students to succeed in the long-term.”

And I would ask, what does students’ success mean in this framework?  Some studies have shown that worked examples are beneficial in only some cases for student learning.  Others have shown that students that are taught with worked examples out-perform those who received individual instruction.  I could go on and on with the studies contradicting each other.  But what if they weren’t in contradiction?  What if there was a way that they could work together – both reducing cognitive load and also giving students agency and voice in the classroom?  Allowing students the freedom to become independent problem solvers but also scaffolding the learning in such a way that their cognition was not overloaded?  Maybe I’m an optimist, but I do believe there is a way to do both.

Late night thoughts on Assessing Prior Knowledge

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So it’s 11:50 pm on a Tuesday night, so what?  I can still think critically, right?  It was the last day of classes and I had an amazing day, but then all of a sudden Twitter started gearing up and lots of discussions began and my mind started racing.  I had planned on writing a blogpost about a student’s awesome inquiry project (which, it ends up, took me about 2 hours to figure out a way to make an iBook on my iPad into a video to try to post on my blog, so that will have to wait), but then I read a great post by Andrew Shauver (@hs_math_physics)

Mr. Shauver writes about the pros and cons of direct instruction vs. inquiry learning but has a great balanced viewpoint towards both of them. In this post, he is discussing the how and when teachers should or can use either method of instruction.  It is important, Shauver states to remember that “inquiry can work provided that students possess the appropriate background knowledge.”

I would totally agree, but I’m just wondering how we assess that – does it really work to lecture for a day and then say they now possess the appropriate background knowledge?  Do we lecture for two days and then give them a quiz and now we know they possess it?  I wonder how we know?  At some point, don’t we have to look at each student as an individual and think about what they are capable of bringing to a mathematical task?  We should set up the problems so that there is some sort of triggering of prior knowledge, communication between peers, resources available for them to recall the information?

Joseph Mellor makes a great point that in PBL most of the time you might plan a certain outcome from a problem, or set of problems, but the triggering didn’t work, or the kids didn’t have the prior knowledge that you thought.  He says that he is often either pleasantly surprised by their ability to move forward or surprised at how much they lack. In PBL, we depend on the students’ ability to communicate with each other, ask deep questions and take risks – often admitting when they don’t remember prior knowledge – hopefully to no suffering on their part. This can be a big hurdle to overcome and can often lead to further scaffolding, a deeper look at the writing of the problem sequence, fine tuning the awareness of their true prior knowledge (not just what the previous teacher said they “learned”) or yes, maybe a little direct instruction in some creative ways.  However, I do believe that given the opportunity a lot of students can be pleasantly surprising.  What do you think?

PBL – Students making Mathematical Connections

As someone who has used Problem-Based Learning for almost 20 years and sad to say has never been part of a full-fledged Project-Based Learning curriculum, what I know best is what I call PBL (Problem-Based Learning).  I know there is a lot of confusion out there is the blogosphere about what is what, and with which acronyms people use for each type of curriculum.  I did see that some people have been trying to use PrBL for one and PBL for the other, but I guess I don’t see how that clarifies – sorry.

So when I use the acronym PBL in my writing I mean Problem-Based Learning and my definition of Problem-Based Learning is very specific because it not only implies a type of curriculum but an intentional relational pedagogy that I believe is needed to support learning:

Problem-Based Learning (Schettino, 2011) – An approach to curriculum and pedagogy where student learning and content material are (co)-constructed by students and teachers through mostly 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.

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.  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.  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).  It really helps you to understand the pros and cons and parent and administrator concerns.

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.  For me, in PBL, I look at how their prior knowledge connects with how, why and what they are currently learning.  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 materials.  I’ve added a few more scaffolding problems (see revised materials) 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.

“Find the center of the circumscribed circle of the triangle with vertices (3,1), (1,3) and (-1,-3).”

Students can actually use any method they like – they can use the old reliable algebra by finding midpoints, opposite reciprocal slopes and write equations of lines and find the intersection points.  However, I’ve had some students just plot the points on GeoGebra and use the circumcenter tool.  The point of this problem is for them to just review the idea and recall what makes it the circumcenter.  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 “oh yeah, when we did this before we said that when it’s an acute triangle the circumcenter is inside and when it’s an obtuse triangle the circumcenter is outside.  But when it’s a right triangle, the circumcenter is on the hypotenuse.”

Of course then the kid of did the problem on geogebra will say something like, “well it’s not just on the hypotenuse it’s at the midpoint.”

 

Dicussion will ensue about how we proved that the circumcenter of a right triangle has to be at the midpoint of the hypotenuse.

A day or so later, maybe on the next page there will be a problem that says something like

“Find the radius of the smallest circle that surrounds a 5 by 12 rectangle?”

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.  When working together oftentimes a student see a right triangle in the rectangle and makes the connection with the circumcenter.

A further scaffolded problem then follows:

“The line y=x+2 intersects the circle  in two points.  Call the third quadrant point R and the first quadrant point E and find their coordinates.  Let D be the point where the line through R and the center of the circle intersects the circle again.  The chord DR is an example of a diameter.   Show that RED is a right triangle.”

Inevitably students use their prior knowledge of opposite reciprocal slope or the Pythagorean theorem.  However, there may be one or two students who remember the circumcenter concept and say, “Hey the center of the circle is on one of the sides of the triangle.  Doesn’t that mean that it has to be a right triangle?”  and the creates quite a stir (and an awesome “light bulb” affect if I may say so myself).

A few pages later, we discuss what I like to call the “Star Trek Theorem” a.k.a. the Inscribed angle theorem (I have a little extra affection for those kids who know right away why I call it the Star Trek Theorem…)

I will always attempt to revisit the “RED” triangle problem after we discuss this theorem.  If I’m lucky a student will notice and say, “Hey that’s another reason it’s a right triangle – that angle opens up to a 180 degree arc, so it has to be 90.”  and then some kid will say “whoa, there’s so many reasons why that triangle has to be a right triangle”  and I will usually ask something like, “yeah, which one do you like the best?” and we’ll have a great debate about which of the justifications of why a triangle inscribed in a circle with a side that’s a diameter has to be right.  So who are the bigger geeks, their teacher who names a theorem after Star Trek or them?

References:

Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential and inquiry-based teaching. Educational Psychologist, 41(2), 75-86.

 

Hmelo, C. E., Duncan., R.G., & Chinn, C. A. (2007). Scaffolding and achievement in problem-based and inquiry learning: A response to Kirschner, Sweller and Clark (2006). Educational Psychologist, 42(2), 99-107.