Building Metacognition and Embracing Challenge Through Problem Sets

This post is one I have been meaning to write for a while, at least 2 years in fact, I have wanted to share it as it has had a dramatic impact on my own classes and also on that of other classes in my school, I guess I just never have gotten around to writing it down.

​Teachers at our school have made the decision to not to use textbooks to teach mathematics in the middle school.  I think that this move has been a very positive one for our school.  It is not that I feel that textbooks are evil, in fact I feel their quality has improved in recent years. However I feel there are still some parts of their design that impact on how people with engage with them, staff and students alike. 

When making the decision to no longer use a textbook for maths class I needed to find an appropriate replacement.
Textbooks are good at providing lots of questions for students to practice the skills they have learnt, and this practice is important. However I needed to find an option that did not have the excessive scaffolding and the extensive quantity of questions that I saw in most textbooks.  Many of the  worksheets I found online has the same characteristics at the text books, so I began to think about how I would design problem sets myself.

Designing the problem sets

When I sat down to think about how I would design my problem sets I wanted to keep some design criteria in mind these criteria were:

• There needed to be variety in the difficulties of the questions, these needed to be clear to students
• The questions needed to target depth rather than breadth of understanding.  Harder questions was not about work at a higher year level, but about the same ideas with great requirements for flexible thinking.
• Being extended should not be about doing more work, it is about doing more difficult work.  If it is 10 questions correct that is required to pass then it should not mean that doing another 20 that look the same should get you a better grade, it should still be 10, but at a higher level of difficulty
• There needed to be enough questions to show understanding, but not so many that it just becomes endless repetitions of the same question type.
• Students would need to have choice about whether they started with the easier or harder question types

One of the question sets I came up with are shown below.  Underneath that image I will explain the design of them.

• The questions are arranged into level 1, 2 and 3. Level 1 aligns with a C level, our minimum passing grade, level 2 aligns with a B and level 3 with an A. 
• Level 1 questions typically only involve the simple application of one skill at a time (collecting like terms, expanding, factorising).  Level 2 and 3 questions require students to make use of multiple skills within the topic in the case of the questions above they needed to decide which of the three skills were needed and in what order. The level 3 pushes a little deeper, requiring them to think about other topics such as fraction operations
• They don't need to do every question, if they can do 6 it is clear to me what level of understanding they have of the work.
• They can choose to start anywhere they like.  If they feel they can make a start on the level 3 questions then they can start there, they do not first need to show they can do the level 1 and 2 questions as the level 3 questions contain all of the same skills as the level 1 and 2.  They can also choose to split their 6 questions over different levels (e.g. they can do 2 level 2 and 4 level 3 questions)
• They have more than just 6 questions to choose from in each of the levels, they can choose the ones that they feel will give them the best chance of success.
• If they complete more than 6 questions they are required to select the just the 6 they are happiest with for assessment.

Impact of the problem sets

Greater levels of metacognition

​One of the most interesting observations I have seen from students engaging with these problem sets is that they seemed to become much more metacognitive.  This is evident in where they choose to start with the question sets. Some have talked about believing they can do the level 2 questions, but really want to work on a few level 1 questions first to make sure they have it.  Others have attempted the level 2 or three questions, given it a go for a while, haven't made progress and have moved back to try the level 1 questions.  There have also been students who have looked at the level 1 and 2 questions and have made the determination that they know how to do them and have spent their time only working on the level 3 questions.  This process of having three different levels of question to choose from has made them much more aware of themselves as learners and of what they need to do to move their learning forward.

All students have had the time they have needed to work on their questions of choice

In looking at the problem sets with my class now, I feel that they all feel as if they have enough time to work on the questions they have chosen and feel comfortable attacking. If I take the level 1 questions for example, I know that some could be through those questions very quickly and there are some that will take much longer.  If I look at my classroom about 5 years ago I would say that the time I gave them to do the questions was aimed at the the middle, the ones who had it finished early and got bored and the ones who were struggling never had enough time to finish them.  With students working on different difficulties of questions they all seem now to have the time they need to finish the questions.

They are attempting much more challenging work

The comment I get a lot when students are working on these problem sets is "why do you have to make it so hard".  This is normally from students who are working on the level 3 questions.  My response in this situation is always the same "Doing the level 3 questions is your choice not mine, so you are making it hard on yourself" to which they normally reply something along the lines of "yeah well those other ones are too easy".  What is clear through this is that they they are no longer just happy to do the easy ones, the could do that and finish really quickly, but they don't.  Students seem to be really working on questions that they feel are just beyond their current level of understanding and they are striving to understand them.

I can now see the slow, deep mathematical thinkers

Having done the Jo Boaler courses one point that is emphasised a lot is not to make maths about speed.  It talks at length about how mathematicians do maths, how they are deep slow thinkers.  Reflecting on previous practice I realised that for those deep slow mathematical thinkers, there was a time that my classes did not offer anything to those students, I didn't even know they were there.  I didn't see them because their diagnostic data backed up what I saw in class, but that was because I was asking them to work in class in similar ways to which the test was administered.  What I am seeing now is that there are quite a number of students who do not perform well under the time and pressure of diagnostic testing, but have flourished with these problem sets as they have the time to sit with the problem and think about it rather than being pushed through endless problems.  They are showing much greater levels of mathematical thinking than some others who score much more highly on those high stakes tests.

The quality of their work is much better

I have seen a noticable improvement in the quality of the work I recieve since using these problem sets. Much of it I think can be attributed to giving them problem types that allow them to demonstrate their understanding and then the time to work on them.

What I find challenging about these problem sets

Despite the massive, dramatic change these problem sets have made to my class, I still struggle with some aspects of them. Most of these are still the tug of war I have inside myself about getting the balance of mathematics right in my classroom
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• I don't like how I have named the different levels, level 1, 2 and 3, are named, but I was also not happy with any others that I found such as "exemplary, accomplished, proficient" or "running, jogging, walking" or any other of those descriptive ways of describing the levels.  I felt those label the student and their understanding rather than describing the difficulty of the tasks.  I felt the level 1, 2 and 3 were the best way to label the work and not the student, but I am sure there is a better way.
  
  
• I don't necessarily like that the levels are tied to grades.  I want them to attempt the more difficult questions not because they are hoping to get a better grade, but because they find them more interesting and intriquing. I want them to attempt them because they want a challenge.  Ultimately when I look at their work I need to assign a grade to it, and the more complex the problem the  do, the better grade they will get, but I don't want them to just try those questions for the  grade alone. Ultimately I went with this so that it is really clear to the students about my grading policies, I aim for there to be transparency in the way I operate in a classroom.
  
  
• I still wonder if I am giving them enough questions.  When I look across an entire unit of work on algebra that includes expanding brackets, factorising, collecting like terms and solving linear equations, students are maybe doing 20 to 30 of these practice problems total, no per part. There is of course some other work we do such as problem solving and investigations, however in relation to the problem sets, 20 to 30 questions over a 10 week period is it.
  
  
• I often wonder if I level the questions correctly, are the jumps between level 1 and 2, and between level 2 and 3 too big. I often look at the questions in level 3 and think is this too much for a year 8 student, have I taken it too far.  However when I look at students engage with them, they are able to do them, despite how horrendous some of them look.