Reducing Cognitive Load in PBL

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.

Looking at PBL Practice from a Thematic Perspective

So I’m here down in Florida – loving it (all sing-songy like Oprah would say).  I’ve been to so many talks that have been great learning experiences so far.  The weather is beautiful – I went for a very long walk and tried to think about what my talk was missing.  I did a bunch of edits and now I think I’m ready to post it.

Here’s the powerpoint of the talk:

Here’s the document that I handed out with some “threads” of themed topics:

Three Threads Document

Please contact me with any questions, comments or concerns – I love talking to people about PBL and my work.

How do you justify the time that PBL takes?

I just wanted to respond to a really great question that someone asked on Twitter the other day.

This is a common concern of teachers starting out with the idea of PBL. What does “Class Discussion” mean, first of all? I would agree that discussion does “eat up valuable” time in class on a daily basis, for sure. But what is actually happening in that discussion where something else would be normally happening in the math classroom? What does the discussion replace?

In my mind the discussion itself replaces the lecture, teachers ‘doing of problems” for the kids to then repeat, then kids often sitting on their own or in pairs doing problems that were just like the ones the teacher showed them how to do. The importance of the class discussion (which honestly is the main idea of PBL) is for students to share their ideas of prior knowledge, connections between problems, where they are confused and see where others were not confused and what prior knowledge and experience they brought to the problem.

Here’s a diagram that I use when doing PD work with PBL teachers to help explain all of what is supposed to be happening during class (it’s a lot!)


The student presentations are really just a jumping-off point. It is not just for students to explain “how they did a problem” – as they say – or they think what they’re supposed to do. The steps of Hmelo-Silver’s “process of learning in PBL” diagram that I’ve circled in pink is what students would/should do for homework. However, the part that is circled in blue is actually the learning process that happens in the class discussion – so is this time that has been “eaten up” in class or is it actually a very necessary part of the important learning, reflection and self-regulation of the process that needs to happen?

Is this harder for students? Heck, Yeah. There is so much more focus, listening, questioning and reflection that is needed in order for this process to be successful and productive. But there are ways to make it easier for students and that’s what the “class discussion” time is for. It takes a lot of practice and mastery on the teachers’ part to realize what is needed. Making mathematics discussion productive is a very important part of teaching in PBL and definitely not a part that should be seen as subtle, intuitive or straightforward.  There is so much more to this that I can not put in a single blog entry, but it’s definitely worth beginning the discussion.  Would love to hear others’ thoughts.

I’ve looked at life from both sides now…

This past July, I spent a few days at the MAA Mathfest in Chicago for the first time. The main reason I went was because the Academy of Inquiry-Based Learning was having a Conference within the Mathfest with the theme of “Diversity in IBL.”  IBL is generally what college faculty call the type of teaching and learning that many of us at the secondary level has been calling PBL for years.  I was so interested to hear many mathematics professors talking about the struggles of writing curriculum, dealing with facilitating discussion, using writing – all of the same parts of this type of teaching that we may have been talking about for so any years.  I highly suggest that we could benefit from talking to each other.  If you would like to get involved with this movement, Stan Yoshinobu, the Director of the Academy of Inquiry-Based Learning, has put forth some challenges for his community.  Check them out.

One of the most interesting talks that I attended was by a professor from Denison University, Lew Ludwig, titled, “Applying Cognitive Psychology to the Mathematics Classroom.”  As a devout social constructivist, I generally like going to talks where I can learn more about other views of education.  Seeing both sides definitely helps me understand many of the views of my colleagues and see if evidence supports my own perspective. Ludwig had published a review of another article that was titled, “Inexpensive techniques to improve education:  Applying cognitive psychology to enhance educational practice”(Roediger and Pye, 2012).

Basically, the presentation summarized three simple techniques that cognitive psychology had evidence helped student learning. The three techniques were called

  1. The distribution and interleaving of material and practice during learning.
  2. Frequent assessment of learning (test-enhanced learning, continual assessment)
  3. Explanatory questioning (elaborative interrogation and self explanation; having students ask themselves questions and provide answers or to explain to themselves why certain points are true).

In the original article, the authors write:

“Repetition of information improves learning and memory. No
surprise there. However, how information is repeated determines
the amount of improvement. If information is repeated back to back
(massed or blocked presentation), it is often learned quickly but
not very securely (i.e., the knowledge fades fast). If information is
repeated in a distributed fashion or spaced over time, it is learned
more slowly but is retained for much longer”

When this was reported, I was first in shock.  I couldn’t believe I was hearing something in a presentation about Cognitive Psychology that was actually supported by the definition of PBL that I use.  The curriculum I use takes the idea of looking at topics and teaching them over a longer time span, but distributed among other topics.  I have called this decompartmentalization of topics, which helps students see the connectedness of mathematics.

The second idea, consistent assessment, is based on the concept that testing is not really a great measure of how much a student has learned, but it actually solidifies the learning that has occurred.  So three groups of students were given different ways of learning by reading a passage of information. The first group read a passage four times. The second group read the passage three times and had test.  The third group read the passage once and was tested three times.  Their performances on tests on the information in the passage 5 minutes later and then one week later.

Diagram of retention testing research

from Roediger & Pye (2012) p.245

So if we connect the idea that testing is not the best method of seeing how much students have learned and the fact that consistent assessment actually helps students retain their knowledge, what I do in my PBL classrooms, is not only “test” but do all sort of forms of assessment (writing, oral assessment, hand-in homework with feedback, labs, quizzes, problem sets, self-assessment, etc.) alternately throughout the term.  There is probably not a week where students are not assessed in at least 2 ways. I feel that this has led students to have good retention of material and the assessments are strong measurements of their learning.

The last one was the one I was most excited to hear about – explanatory questioning.  This seemed to give students so much more responsibility for their own learning than traditional cognitive psychology as I had understood it.  The authors of this study claim that explanatory questioning can be broken in to two areas:

Elaborative Interrogation – students generating plausible explanations to statements while they are studying and learning.  This speaks directly to the idea of mathematical discussion and how students generate explanations when they ask themselves “why?”

Self-Explanation – students monitoring their learning and describing, either aloud or silently some features of their learning.  This idea can be found all over the PBL classroom but in mine, it’s generally found most in metacognitive journaling where students use self-explanation the most.

“Obviously, the elaborative interrogation and self explanation are related because both strategies encourage or even require students to be active learners, explaining the information to themselves (perhaps rephrasing in language they understand better) or asking themselves why the information is true.”

I honestly couldn’t believe what I was reading – this is an article on educational methods based on cognitive psychology that is suggesting that we require students to be active learners and “explain the infomation to themselves”?  This is lunacy.  I have been teaching for 25 years where students have been complaining to their parents that they have had to explain things to themselves – who would’ve known that I was applying cognitive psychology?

My guess is that these ideas are only enhanced by the social aspect of the classroom and other constructivist ideas – clearly the constructivitst classroom in enhanced by or agrees with some of these cognitive psychology methods as well.  Listening to both sides of the theories is actually helpful and I’m seriously going to continue doing this! Although I never thought that there might be strong connections between cognitive psychology theories and PBL, I do know that it’s life’s illusions I recall and I “really don’t know life, at all.”

PBL and second language learners

As I am not going to be in the classroom next year, I have been going through some old boxes from my study and as many people who have been teaching for a long time have, I have boxes and bags full of cards from past students.  I spent the afternoon one day going through these, reminiscing about so many great kids that I remember.  One of them I had a card from the beginning of her freshman year and also one from the end of her senior year.  Crazy!!

I don’t claim to be an expert in emergent English language learners and mathematics at all.  I did have 10 years of teaching experience at a school (Emma Willard) where they had an ESL program and many students came into my mathematics classes who were not proficient in the English language.  I do think those girls knew what they were getting themselves into and were up to the challenge, but some of them were very frightened.

Since she has now been out of college for a while, I would assume it’s ok for me to share this on my blog.  Here is the card she gave me as a new student in 2001:

Jinsup's card from freshman year

Jinsup’s card from freshman year

This card was written with the voice of a student who was used to a very structured, repetitive mathematics class and I believe she knew that coming into the U.S. things would be different, but possibly not as different as they were in my class.  When she said, “I’m so nervous that you will let me to talk a lot in the class” I’m sure she was saying that she was nervous that I would expect her to contribute to the class discussion.  What I did with many of those students, including Jinsup, was I focused in the beginning on letting them listen and write.  I gave them lots of feedback on their journals and made sure they had the correct vocabulary and that their grammar in their writing made sense.  I allowed them to ask more questions initially than to present their ideas until their confidence became stronger.  Jinsup, as most Korean and Japanese students did, had excellent skills, as that was what their math education had focused on since elementary school.  However, she was not very good at reasoning, sense-making or critical thinking on her own.  It was almost as if she had not been asked to communicate about mathematics, as she was trying to say in her note to me.

However, she ended up doing very well in that first class and then I taught her again in precalculus (which we called Advanced Math) and then in BC Calculus her senior year.  Her excellent background allowed her to focus on the reasoning aspects of all of these courses and in the end, I was very impressed with her growth.  She really got the best of both worlds – the skills from her Asian mathematics education and the collaboration, communication and reasoning skills from the PBL here.

This is the note she wrote me at the end of her senior year:

Jinsup's card at end of her senior year

Jinsup’s card at end of her senior year

Although I know this is only an anecdote and I don’t really have research evidence that PBL totally works with ELLs I do have confidence that with the right environment and patience, it is actually a great way of teaching for many of these second language learners.  It allows them to find their voice in a language that is already new to them but at the same time have some practice in terminology that they may have heard before.  I think this might be my next interesting research project – if anyone has some thoughts on this I’d love to hear them.

Yours, Mine and Ours

Yesterday we had a speaker in our faculty meeting who came to talk to us about decision-making process in our school.  He spoke about the way some colleges, universities, independent schools are very different from businesses, the military, and other governing bodies that have to make decisions because we are made up of “loosely-coupled systems.” These are relationships that are not well-defined and don’t necessarily have a “chain of command” or know where the top or bottom may be.  They also don’t necessarily have a “go-to” person where, when a problem arises, the solution resides in that location.  The speaker said that this actually allows for more creativity and generally more interesting solution methods.

About mid-way through his presentation he said something that just resonated with me fully as he was talking about the way these systems come to a decision cooperatively.

“The difference between mine and ours is the difference between the absence and presence of process.”

Wow, I thought, he’s talking about PBL.  Right here in faculty meeting.  I wonder if anyone else can see this.  He’s talking about the difference between ownership of knowledge in PBL and the passive acceptance of the material in a direct instruction classroom.

Part of my own research had to do with how girls felt empowered by the ownership that occurred through the process of sharing ideas, becoming a community of learners and allowing themselves to see others’ vulnerability in the risk-taking that occurred in the problem solving.  The presence of the process in the learning for these students was a huge part of their enjoyment, empowerment and increase in their own agency in learning.

I think it was Tim Rowland who wrote about pronoun use in mathematics class (I think Pimm originally called it the Mathematics Register). The idea of using the inclusive “our” instead of “your” might seem like a good idea, but instead students sometimes think that “our” implies the people who wrote the textbook, or the “our” who are the people who are allowed to use mathematics – not “your” the actual kids in the room.  If the kids use “our” then they are including themselves.  If the teacher is talking, the teacher should talk about the mathematics like the are including the students with “your” or including the students and the teacher with “our”, but making sure to use “our” by making a hand gesture around the classroom.  These might seem like silly actions, but could really make a difference in the process.

Anyway,  I really liked that quote and made me feel like somehow making the process present was validated in a huge way!

Repost: Always Striving for the Perfect Pose

Back in 2010, I wrote an blogpost comparing teaching with PBL to doing yoga. Since I have been doing Bikram Yoga for almost a year now and still can’t do “standing head to knee pose” *at all* – I thought I would repost this one just to give myself some perspective, and possibly many of you out there who might need a little encouragement at this beginning of the year time. I know that every year when students begin a year in a PBL math class the obstacles return. Parents are questioning “why isn’t the teacher teaching?” Students are questioning “why is my homework taking so long?” Teachers new to the practice are questioning “When is this going to get easier?” and “why aren’t they seeing why this is good for them like I do?” The best thing to remember is that it is a process and to understand how truly different and hard it is for students who are used to a very traditional way of learning mathematics. Give them time, have patience for them and yourself and most of all reiterate all of what you value in their work – making mistakes, taking risks, their ideas (good and bad) and be true to the pedagogy.

Here’s the original blogpost I wrote:

I don’t think my professor, Carol Rodgers, would mind me borrowing her yoga metaphor and adapting it to PBL. I use it often when talking to teachers who are nervous about falling short of their ideal classroom situation or teaching behaviors. I think this can happen often, especially when learning best practices for a new technique like facilitating PBL. There are so many things to remember to try to practice at your best. Be cognizant of how much time you are talking, try to scaffold instead of tell, encourage student to student interaction, turn the questions back onto the students, etc. It really can be a bit overwhelming to expect yourself to live up to the ideal PBL facilitator.

However, it is at these times that I turn to Carol’s yoga metaphor. She says that in the practice of yoga there are all of these ideal poses that you are supposed to be able to attain. You strive to get your arms, legs and back in just the right position, just the right breathing rhythm, just the right posture. But in reality, that’s what you’re really doing – just trying. The ideal is this goal that you’re aiming for. Just like our ideal classroom. I go in everyday with the picture in my head of what I would want to happen – have the students construct the knowledge as a social community without hierarchy in the authority where everyone’s voice is heard. Does that happen for me every day? Heck no. I move the conversation in that direction, I do everything in my power for that to happen, but sometimes those poses just don’t come. Maybe I just wasn’t flexible enough that day, or maybe the students weren’t flexible enough, maybe we didn’t warm up enough, or the breathing wasn’t right. It just wasn’t meant to be. I have exercises to help me attain the goal and I get closer with experience. That’s all I can hope for.

So I tell my colleagues who are just starting out – give yourself a break, be happy for the days you do a nearly perfect downward facing dog, but be kind to yourself on the days when you just fall on your butt from tree pose. We are all just trying to reach that ideal, and we keep it in mind all the time.

Getting Kids to Drive the Learning

It doesn’t always work this way, but it would be awesome if it did.  When PBL is perfect or ideal, the students are the ones who make the natural connections or at least see the need or motivation for the problems that we are doing.  Yeah, some of them are just really interesting problems and the get pulled in by their own curiosity, but as all math teachers know, we have a responsibility to make sure that students learn a certain amount of topics, it is quite that simple.  If students from my geometry class are going into an algebra II class with trigonometry the next year where their teacher will expect them to know certain topics, I better do my job and make sure they have learned it.

So how do I, as a PBL teacher, foster the values for the problem-based learning that I have while at the same being true to the curriculum that I know I have a responsibility to?  This is probably one of the biggest dilemmas I face on a daily basis.  Where’s the balance between the time that I can spend allowing the students to struggle, explore, enjoy, move through difficulty, etc. – all that stuff that I know is good for them – while at the same being sure that that darn “coverage” is also happening?

So here’s a little story – I have a colleague sitting in on my classes just to see how I teach – because he is interested in creating an atmosphere like I have in my classes in his.  We have just introduced and worked on problems relating to the tangent function in right triangle trigonometry in the past week and now it was time to introduce inverse tangent.  I do this with a problem from our curriculum that hopefully allows students to realize that the tangent function only is useful when you know the angle.FullSizeRender (3)

So as students realize they can’t get the angle from their calculator nor can they get it exactly from the measurement on their protractor (students had values ranging from 35 to 38 degrees when we compared), one of the students in my class says, “Ms. Schettino, wouldn’t it be great if there was a way to undo the tangent?” and the other kids are kind of interested in what she said. She continues, “Yeah, like if the calculator could just give us the angle if we put in the slope.  That’s what we want.”  I stood there in amazement because that was exactly what I wanted someone to crave or see the need for.  It was one of those “holy crap, this is working” moments where you can see that the kids are taking over the learning.  I turned to the kids and just said, “yeah, that would be awesome, wouldn’t it?  Why don’t you keep working on the next problem?” and that had them try to figure out what the inverse tangent button did on their calculator.  They ended up pressing this magical button and taking inverse tangent of 0.75 (without telling them why they were using 0.75 from the previous problem) to see if they could recognize the connection between what they had just done and what they were doing.

At the end of the class, the colleague who was observing came up to me and said, “How did you do that?” and I said, “What do you mean?” and he said, “How did you get the kids to want to learn about inverse tangent? I mean they just fell right into the thing you wanted them to learn about.  That was crazy.”  I really had to think about that.  I didn’t feel like I did anything honestly, the kids did it all.  I mean what made them all of a sudden care about getting the angle?  Why were they invested?   It doesn’t always happen in my classroom that’s for sure.  This is not a perfect science – there’s no recipe for it to work – take a great curriculum, interested kids, an open, respectful learning environment and mix well?

I do think however that a huge part of it is the culture that has been created throughout the year and the investment that they have made in their ownership and authorship in their own learning. We have valued their ideas so much that they have come to realize that it is their ideas and not mine that can end up driving the learning – and yes, I do end up feeling a little guilty because I do have a plan.  I do have something that I want them to learn, but somehow have created enough interest, excitement and curiosity that they feel like they did it.  It is pretty crazy.

Considering Inclusion in PBL

It’s always refreshing when someone can put into words so eloquently what you have been thinking inside your head and believing for so long.  That’s what Darryl Yong did in his recent blogpost entitled Explanatory Power of the Hierarchy of Student Needs.  I feel like while I was reading that blogpost I was reading everything that I had been thinking for so long but had been unable to articulate (probably because of being a full time secondary teacher, living in a dorm with 16 teenage boys, being a mother of two teenagers of my own and all the other things I’m doing, I guess I just didn’t have the time, but no excuses).  Darryl had already been my “inclusive math idol” from a previous post he wrote about radical inclusivity in the math classroom, but this one really spoke to a specific framework for inclusion in the classroom and how in math it is necessary.

 

In my dissertation research, I took this idea from the perspective of adolescent girls (which, as I think towards further research could perhaps be generalized to many marginalized groups in mathematics education) and how they may feel excluded in the math classroom.  These girls were in a PBL classroom that was being taught with a relational pedagogy which focuses on the many types of relationships in the classroom (relationship between ideas, people, concepts, etc.)  – I did not look at it from the perspective of Maslow’s Hierarchy of Student Needs and this is really a great tool.

Interestingly,  I came up with many of the same results. My RPBL framework includes the following (full article in press):

  1. Connected Curriculum– a curriculum with scaffolded problems that are decompartmentalized such that students can appreciate the connected nature of mathematics
  2. Ownership of Knowledge – encouragement of individual and group ownership by use of journals, student presentation, teacher wait time, revoicing and other discourse moves
  3. Justification not Prescription– focus on the “why” in solutions, foster inquiry with interesting questions, value curiosity, assess creativity
  4. Shared Authority – dissolution of authoritarian hierarchy with deliberate discourse moves to improve equity, send message of valuing risk-taking and all students’ ideas

These four main tenets were what came out of the girls’ stories.  Sure many classrooms have one or two of these ideas.  Many teachers try to do these in student-centered or inquiry-based classrooms.  But it was the combination of all four that made them feel safe enough and valued enough to actually enjoy learning mathematics and that their voice was heard. These four are just a mere outline and there is so much more to go into detail about like the types of assessment (like Darryl was talking about in his post and have lots of blogposts about) the ways in which you have students work and speak to each other – how do you get them to share that authority when they want to work on a problem together or when one kid thinks they are always right?

The most important thing to remember in PBL is that if we do not consider inclusion in PBL then honestly, there is little benefit in it over a traditional classroom, in my view. The roles of inequity in our society can easily be perpetuated in the PBL classroom and without deliberate thought given to discussion and encouragement given to student voice and agency, students without the practice will not know what to do.  If we do consider inclusion in the PBL classroom, it opens up a wondrous world of mathematical learning with the freedom of creativity that many students have not experienced before and could truly change the way they view themselves and math in general.