Journals: Paper vs Digital: The Pros and Cons

I was totally honored the other day when I saw some tweets from TMC16 from @0mod3 and @Borschtwithanna


And yes it’s true, I’ve been writing and practicing the use of metacognitive journaling for very long time – probably since 1996 ever since I read Joan Countryman’s book about mathematical journaling and heard about it in many workshops that summer.  I wrote a rubric (make sure you scroll to the 3rd page) while I was at the Klingenstein Summer Institute for New Teachers (that’s how long ago it was) and since then I’ve been refining that rubric based on feedback from students and teachers. A few years ago, I finally refined a document called How to Keep a Journal for Math Class to a degree that I really like it now.  However, please know that lots of math teachers do journaling differently and without the metacognitive twist. I do believe that metacognitive writing is essential to the PBL classroom (read more here)

So this morning, I was asked this question on twitter


Which is something that many people often ask so I thought I’d respond with a more in-depth answer.

Here are the pros, I’ve found over the years of having students journal digitally:

Speed/complexity: Students are used to typing, using spell-check, inserting pictures, graphics and naturally including documents, links and thinking in the complex way that digital media allows them to.  It allows their journal to be more rich in content and sometimes connect problems to each other if their journal is say on a google doc that can connect to other html docs.  If they create, for example, iBooks or Explain Everything videos, there is even a lot more richness that can be embedded in the file as well – their creativity is endless.

Grading/Feedback: I found grading in Notability or on Google docs or some other digital platform really nice that allowed you to add comments with a click or audio extremely easy and quick.  I did not receive feedback from the students very often about how the feedback helped them though.  If you use an LMS like Canvas that integrates a rubric or integrates connection to Google it’s even nicer because you can have those grades go right from your assignment book to your gradebook.

I love having kids use digital platforms for writing/creating in mathematics when it is for a project or big problem that I want them to include many pieces of evidence, graphs, geogebra files and put it together nicely in a presentation or portfolio.  Not necessarily for their biweekly journals. Some guys who make use of digital journals in interesting ways are @GibsonEdu and @FrasiermathPBL at the Khabele School in Austin TX.

Here are the cons, in my mind of using digital journals: (which might be the “pros” of paper journals) – which is the side I have come down on.

the “real” writing factor: there is some research about the actual physical process of writing and the time it takes for kids to process their thoughts.  I do believe that when i want kids to be metacognitive about their learning and also want them to be thoughtful and take the time think about their initial error, think about what happened in class discussion to clear up their misunderstanding and also then what new understanding they came to.  That’s a lot of thinking. So I want them to take the time to write all that down.  Sometimes typing (like what I’m doing right now!) is a fast process and I’m not sure I do my best writing this way.

practice in hand-writing problem solving: this is re-enacting doing homework and sitting for assessments (in my class at least) and I want them to do this more regularly.  If in your class kids take assessments digitally or do homework nightly digitally then maybe they should do their journal digitally as well. This also give me practice in reading their handwriting, getting to hear their voice through their handwriting and seeing what it looks like on a regular basis.  In a time crunch on an assessment it honestly helps me know what they are thinking.

Conversational Feedback: I feel that when I hand write my feedback to them I can draw a smilely face or arrows or circle something that I want to emphasize more easily than when it is on something digitally (this is also true in a digital ink program – so that is something to consider, like Notability for example). I give feedback (see some journal examples on my blog) that is very specific about their writing and want the to improve not only in the math aspect of their writing but in how they are looking at their learning.  I want them to respond and I want to respond in the hope that we are starting a mathematical conversation about the problem.  I have received more questions about the feedback in the paper journals (like “what did you mean by this?”) than on the electronic feedback – not sure why.

Portability: I find that small composition graph paper notebook is extremely portable and easy for me to carry home to grade.  The students bring them to their assessments and there is nothing else in the notebook (no homework at all and no access to the internet) so I am not worried about academic honestly.

There are probably more but this is it in a nutshell – please add your comments below or tweet me to let me know your thoughts!


What does “making students metacognitive” mean? – answering “why should someone learn?” in Math

So I recently tweeted a nice article that I read that discussed “12 Questions to Help Students See Themselves as Thinkers” in the classroom (not specifically the math classroom


and appropriately, Anna Blinstein tweeted in response:


So I thought I needed to respond in a post that spoke to this question. First of all, I should state the caveat that even when I am in a more “standard” classroom (i.e. not a PBL classroom) – which happened to me last year – I try as much as possible to keep my pedagogy consistent with my values of PBL which include

1) valuing student voice
2) connecting the curriculum
3) dissolving the authoritative hierarchy of the classroom
4) creating ownership of the material for students

I find that helping students to be metacognitive helps with all of this. An important aside her is also Muller’s definition of 21st century learning* which is much more than that 20th century learning and education that often comes with direct instruction in the mathematics classroom (not always).I think it’s important to note that the more fluid concept of knowledge that is ubiquitous with technology today and is no longer static in textbooks or delivered by teachers.  Students can go find out how to do anything (procedurally) nowadays, but it is the understanding of it that is more important and the true mathematical learning and sense making.

Anyway, I think I would write way too much if I responded to every one of the questions, but how would I use these questions in my direct instruction class that I taught last year?  What I tried to do was introduce a topic with some problems (and then we would do some practice with problems from the textbook so I could keep up with where my colleague was in the material).  Well, this course was Algebra II, which often referred to prior knowledge that always reminded students of something they had studied before.  I let them use computers to look things up on the internet and use the technology at hand, GeoGebra, Graphing Calculators, each other to ask questions about the functions we were studying.  They could look up topics like domain, range, asymptotes (why would there be an asymptote on a rational function)…but then the bigger questions like “what am I curious about?” had more to do with how did those asymptotes occur, what made vertical vs. horizontal asymptotes and then I would have them do journal entries about them (see my blogposts on metacognitive journaling – journaling and resilience, using journal writing, page on metacognitive journaling).

The more “big picture” questions like “Why learn?” and “What does one *do* with knowledge?” I find easier to deal with because the students ask those.  I think that all teachers find their own ways to deal with them, but I enjoy doing is asking students about a tough question they are dealing with in their life – I use the example of whether or not I should continue working when I had my two kids.  Was keeping my job worth it financially over the cost of daycare? and of course I had to weight my emotional state when I wasn’t working – this is why I enjoy learning and what I do with my knowledge.  When kids see that there’s more to do with functions than just points on a grid, it becomes so much clearer for them – but you know that!

What I really like about Dr. Muller’s list is that he lays out some nice deliberate ways in which we as math teachers can get students to think more clearly and reflectively about mathematics as a purposeful process as opposed to a just procedures that they can learn by just watching a Kahn Academy video.


*”Learning – here defined as the overall effect of incrementally acquiring, synthesizing, and applying information – changes beliefs. Awareness leads to thoughts, thoughts lead to emotions, and emotions lead to behavior. Learning, therefore, results in both personal and social change through self-knowledge and healthy interdependence.” Muller

Six of one, half a dozen of the other…I think not

(Sorry, this is a long one! and caveat: I am not claiming that Wikipedia is the be all and end of definitions!)

So according to Wikipedia, PBL means two things (well, three if you count Premier Basketball League, but that’s neither here nor there). If you look up PBL on Wikipedia, the first hit is, in fact, Problem-Based Learning. Why, you may ask? I believe that this is because Problem-Based Learning has been around in various forms longer than Project-Based Learning, but the term itself was coined in the 1960s by Howard Barrows at McMaster University in Ontario, Canada ( You can read more about Barrows’ specific definition of Problem-Based Learning at this site and in my tab at the top of my website that says “Problem-Based Learning.” However, what I feel is one of the biggest parts of Barrows’ definition is the fact that “the problem is a vehicle for the development of problem solving skills” – that is it, that it is the problem – hopefully well developed and set in a context that is interesting, challenging and meaningful to students – through which the students will development and learn the problem solving skills.

Wikipedia names Project-Based Learning ( Learning) almost the same thing, however, they connect it more to Greeno’s theory of situated learning – “learning by doing” and “teaching by engaging students in investigation.” However, all of these theorists ideas range from about 1991-2006, so it would seem that a PBL by any other name…is not really the same?

One of the most important distinctions in Project-Based Learning (which I will write as PjBL, because you know, it came 2nd, for the record) is the authenticity of the task that is motivated by a larger “driving question” – students learn by creating a project and investigating what they need to do in order to organize or structure their presentation for the project.

So what does this mean in mathematics? A few weeks ago, there was quite a discussion going down on Twitter about what constituted Problem-Based Learning.




Dan Meyer seemed to be criticizing Problem-Based Learning saying that it was discussed by others as “too much curriculum and too little time for PrBL” and he says that he saw “a lot of fluff in PrBL” – to which I would say, there’s much more to Problem-Based Learning than just doing problems, Mr. Meyer. Nat Banting also asked for a clarification of what the difference was between a project and a problem in math education.

In response to this discussion and Mr. Banting’s question, I posted an image of a table I created listing differences that I saw between project- and problem-based learning in mathematics education and had hoped for some feedback.


Then I thought that maybe my description on the image of the differences wasn’t clear enough, so I thought I’d try one more time to make the distinction between the authenticity clear.

If we revisit the idea that Problem-Based Learning has at its core, problems as the vehicle for learning and constructing knowledge, I think this is at the heart of the difference of the learning/teaching experience. When I look at purposefully scaffolded and written curriculum for problem-based learning, yes there are outcomes that need to be met, there are topics that are discussed that are set by the teacher (or curriculum writer). Mathematics has within it many very interesting abstract concepts that are worthwhile to teach even though the “real-life” applications don’t have much context for students (will a student appreciate the logic and problem solving skills that are learned in factoring a polynomial, for example, or do they just accept and take it as a skill they need, or use technology and not use that part of their brain – who knows?) The point I’m trying to make is that in problem-based learning the problems would lead students to figuring out (through discussion and with prior knowledge and experience) the skills needed to create a process of factoring and perhaps “unfactoring” an algebraic expression and hopefully what that expression might represent. This is what’s known as “preauthentication” – when the curriculum writers or teachers try to come up with some kind of experience for the students that simulates the mathematicians authentic experience or “ah-ha” moment of understanding or realization. Then there are other problems that allow the students to dig deeper and apply those ideas to other areas of mathematics, and yes, real-life problems too.

In Project-Based Learning, “emergent authenticity” allows the mathematics to emerge from the ideas of the project (or driving question at hand) which is up to the students to then find out what they need to know. This is where Project-Based Learning can fall short in the area of secondary mathematics – in a world of standards. Where does the learning take place and how does it happen? Once students understand a concept, having gone through rather traditional instruction and some type of formative assessment, the project can then be given. Yes, Mr. Meyer, I believe that there is a hard balance to make there between traditional instruction and when to do the projects – quite a dilemma of time (although schools do it with very nice interdisciplinary time schedules).

However, I do believe that in problem-based learning the rigor, content, problem-solving and all the other “4C’s” skills that project-based learning also promote end up happening in the discussion and presentation of problem ideas and solutions. So I would have to argue with those who say “it doesn’t matter what you call it” and “they’re all the same thing” because the learning process in mathematics is so very different in these two methods. Hopefully, I can shed some light on the differences between PBL and what I hope will be called PjBL soon!

Teaching Students to Become Better “Dancers”

So the other day I read a tweet by Justin Lanier that really sparked my interest.

 We all know the scenario in classroom discourse where a student asks a question – a really great question – and you know the answer, but you hedge and you say something like, “That’s a great question! I wonder what would happen if…”  So you reflect it back to the students so that they have something to think about for a little while longer, or maybe even ask a question like “Why would it be that way?” or “Why did you think or it like that?”  to try to get the student to think a bit more.  But what Justin, and the person who coined the phrase “authentic unhelpfulness” Jasmine Walker (@jaz_math), I believe were talking about was hedging because you really don’t know the answer – sincere interest in the uniqueness of the question – not because you’re so excited that student has helped you move the conversation forward, but because of your own excitement about the possibilities of the problem solving or the extension of the mathematics.

I think what got me so excited about this idea was how it connected to something that I was discussing earlier this summer with a group of teachers in my scaffolding in PBL workshop in late June.  In a PBL curriculum, the need to make sure that students have the right balance of scaffolded problems and their own agency is part of what Jo Boaler called the “Dance of Agency” in a paper she wrote in 2005 (see reference).  My understanding of this balance goes something like this:

(c) Schettino 2013

So initially, the student is confused (or frustrated) that the teacher refuses to answer the question although you are giving lots of support, advice and encouragement to follow their instincts.  The student has no choice but to accept the agency for his or her learning at that point because the teacher is not moving forward with any information.  But at that point usually what happens is that a student doesn’t feel like she has the authority (mathematical or otherwise) to be the agent of her own learning, so she deflects the authority to some other place.  She looks around in the classroom and uses her resources to invoke some other form of authority in problem solving.  What are her choices?

She’s got the discipline of mathematics – all of her prior knowledge from past experiences, she’s got textbooks, the Internet, her peers who know some math, other problems that the class has just done perhaps that she might be able to connect to the question at hand with previous methods that she might or might know how they work or when they were relevant – that discipline has had ways in which it has worked for her in the past and lots of resources that can help even if it may not be immediately obvious.

But she’s also got her own human agency which is most often expressed in the form of asking questions, seeing connections, drawing conclusions, thinking of new ideas, finding similarities and differences between experiences and thinking about what is relevant and what is not.  These pieces of the puzzle are not only important but a truly necessary function of the “dance of agency” and imperative to problem solving.

Interweaving both of these types of agency (and teaching kids to do this) have become more important than ever.  Yes, being able to use mathematical procedures is still important, but more important is the skill for students to be able to apply their own human agency to problem and know how and when to use which mathematical procedure, right?  This “dance” is so much more important to have every day in the classroom and if what initiates it is that deflection of authority then by all means deflect away – but the more we can “dance” with them, with “authentic unhelpfulness” and sincere deflection because we need to practice our own human agency, the more we are creating a true community of practice.

Boaler, J. (2005). Studying and Capturing the complexity of practice – the Case of the ‘Dance of Agency’

So How Do We Shift Gears?

OK, OK, I get the idea – not everything on the Internet is true and, for sure, not everything on the Internet is meaningful or helpful.  Since April of this year I have started following a bunch of people on Twitter (before that I really didn’t even know what it was or care) and thought that there were so many people out there that I wanted to learn from.  I would read other people’s blogs and try my best to think about what I had to learn from others. Mind you, I know I am definitely not the god of teaching, that’s for sure, but many of the things that are written out there – should I guess – with the hope of being “inspirational” or meaningful to others, I find less than helpful.

One site that I have really enjoyed reading which often has some great links and blogposts is Mindshift.  But they just tweeted this blog entry that cited an article about creating a business that fosters creativity.  OK, I see the connection to education, but honestly, it is a very different machine.  Kids and adolescents have a very different mindset than adults who are out there making money.  Not to mention the consequences of risk-taking in the classroom vs. risk-taking in the office have the potential for being very different.  (Assessment for grades has a different meaning possibly for a 13-year-old mind than brainstorming on the job, vs. assessment for a salary raise, etc for an adult who we hope can handle the pressure a little more.)

Then the blogger writes two short paragraphs at the end about how schools are just “incurious and risk averse” places.  Basically stating that schools don’t ever allow students to practice risk-taking or mistake making at all:

“Too few schools are incubators of curious and creative learners given their cultures of standardization, fear, and tradition. No doubt, external pressures exist that drive that culture. But if there ever was a time to shift gears, this is it. “

No doubt…sadly, our blogger, Will Richardson doesn’t really give us any advice on what to do about it….except, to do something about it. (Admittedly, he may have written something someplace else that I missed.)  And I don’t want to single out Mr. Richardson – I find tweets and blogs like this all day long – “Exploration, inquiry & problem solving are powerful learning mechanisms…” or “asking good questions and promoting discourse is an integral part of teaching and learning”…. Hmmm, well let me think about ways in which we can talk to teachers  in terms of mistake making and risk-taking:

  • Blogpost on making mistakes and classroom activity tied to Kathryn Schultz’ TED talk On Being Wrong
  • Discussion about article “Wrong is not always bad” with teachers
  • Modeling risk-taking in Problem-Solving in my course at ASG conference in June
  • Discussion of Relational Pedagogy to foster Risk-taking
  • Using a PBL curriculum to foster mistake-making and communication

I found that many teachers that I work with and who contact me are entirely dedicated to changing the culture of the mathematics classroom in the U.S. and making it (as Mr. Richardson writes) an “incubator of curious and creative learners.”  We need to make changes to our curriculum, our classroom relationships, our classroom culture and the authoritarian hierarchy that traditionally is prevalent in our mathematics classroom.  Students need to be able to feel safe enough, from judgment, alienation and failure to make those mistakes while learning.  We, as teachers, need to begin the discussion with each other about how to move forward with these initiatives and make sure that student voice is heard in the mathematics classroom as they question each other and us, the teachers, with true questions – ones we may not be able to answer.  These are the important aspects of creating curious learners who make mistakes and learn from them.  But we, as the adults in the room have a responsibility to let them feel safe in doing that.

I think teachers are aware of the fact that it’s time to “shift gears” – to make the classroom more conducive to students working together and taking chances.  There are so many subtleties to making this shift, however.  Students who need to shift, parents who are not used to that, assessment changes to be made – the list goes on and on.  I am doing what I can to help people with this conversation.  The pedagogy of relation (I believe) is at the heart of all of this – keeping in mind that in order for people to be vulnerable and make mistakes, we need to consider the interhuman aspect of learning.  In a classroom where this connection has for too long been typically so acceptably removed, it will take a lot of work to make this big “gear shift” but I’m up to it – bring it on!