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.

What I get out of Student Writing

I have been using journaling in math class since 1996 – which was a really important year in my teaching career for lots of reasons, but it was definitely because I was introduced to the idea of math journals.  Since then I’ve done many different iterations for what my expectations are.  Even this year I did something new where I allowed students to write about errors they made on assessments in order to attempt to compare their assessment problems to what they did on homework in the hope of reflecting on the work pre-assessment for future problem sets.

However, a lot of students still use their journal almost like a problem-solving conversation with me, especially after we have already gone over a problem and they still don’t understand a method.  Here is one I ran across just the other day in my lower-level geometry class and thought it just perfectly expressed some of the goals I am hoping to accomplish with journaling.

I’ll call this student Cindy and we had just introduced the theorems about parallel lines through a geogebra lab and this had been the first problem they looked at that took the concepts out of the context of the lines and threw it into a triangle.  For many students this might be an easy transfer of skills (including the algebra, other theorems, etc.) but for the kids I have – not necessarily.  Here is what Cindy wrote:file_001-1

The first thing that Cindy does in her journaling is make her own thinking explicit (which I love).  She is stepping me through her thinking and the questions that arose for her.  This is actually a major step for many students who are confused – are they able to even know what they are confused about?

She writes: “I know the problem probably deals with the parallel line theories that we dealt with.” and then lists the types of angles we studied and then with a big “OR” says “maybe it has something to do with the sum of the angles of parallelograms and triangles?”  Little does she know that what she is doing is practicing synthesizing different pieces of prior knowledge – is it overwhelming to her? – possibly, but she went there and that’s so great!  I wanted her to know that I was excited that that she even thought about the sum of the angles so I gave her some feedback about those ideas.

She wrote down what she knew about the sums of the angles which we had also studied.

She writes her first equation to think about: “5x-5/=180” using one of the angles in the top triangle.  I would’ve loved to know where that was coming from.  What made her write that?  She then notes that “but it wouldn’t work because if x is the measure of the angle than the equation should be set to 180”

There is so much that this tells me about her confusion.  She is not understanding what the expression 5x-5 is supposed to be representing in the diagram I think, or she isn’t connecting what x is “not representing” (the angle) and the whole expression is representing too.  She also is confused about between the sum of the whole triangle’s angles and just that one angle.

She then looks at the two expressions she is given, 5x-5 and 4x+10 and I think makes a guess that they are corresponding angles – she doesn’t give any reason why they are corresponding.  She just asks the question.  But the cool thing is she says “Let’s try it.”  I love that.  Why not – I am always encouraging them to go with their ideas and the fact that she tries it is wonderful.  The funny thing is she does end up getting the same value for the two angles so she asks: “Does this mean that this is correct?” and then “What do I do for “6y-4?” and still has not connected many of the ideas line the fact that these angles are a linear pair and that’s where the 180 comes into play, or even why the angles were corresponding in the first place.  So many questions that she still has, although I am encouraged by her thinking and risk-taking.

This journal entry allowed me to have a great follow-up conversation with Cindy and she was able to talk to me about these misconceptions.  I’m not sure I would’ve had this opportunity to clarify these with her if she had not written this journal entry and then she would not have done so well on the problem set the following week.  I just love it!  Let me know if you use journals and if you feel the same clarifying or communicative way about them too.

See my website for lots of sample entries and also other blogposts and resources about journaling if you are interested.

 

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!

 

Revisiting Journals: Getting Kids to Look Back

I have been using metacognitive journaling in my PBL classroom since 1995.  I first learned about it the Summer Klingenstein Institute when I was a third year teacher and just fell in love with it.  At that time, the colleagues at my school thought I was crazy trying to make kids write in my classes – it was just “something else for them to do” and didn’t really help them learn but I did more reading on it and there was clearly more and  more research as time went on that showed that writing-to-learn programs especially those that prompt for metacognitive skills really do help in learning mathematics (see my metacognitive journaling link under the Research tab for more info and sample journal entries).

Every once in a while a student will write a journal entry that I think is so thoughtful that I will write about it like this one a few years ago that just impressed me with his insight into his learning process of a particular problem. But other times kids write about their understanding of their learning overall like one I’ll write about today and I am also blown away.

Here’s a student I’ll call Meaghan reflecting on a problem that she found challenging for her.  Really, it doesn’t matter which problem it was or what topic it was, just the fact that she had a hard time with it at first, right?  The most important part was that after she wrote about how to do it correctly, she then took the time to write this: (in case you can’t read her handwriting, I will rewrite it below).

FullSizeRender

Part of Meaghan’s Journal Entry

“This problem was a challenge for me.  When I saw the question, it didn’t look that difficult but once I was trying to solve by [sic my] brain wasn’t thinking on the right track, and it was trying to use prior knowledge that was irrelevant in this case.  I wasn’t making connections to the properties of triangles that I had recently learned.”

Why is this realization so important for Meaghan?  Polya’s Fourth Principle of Problem Solving is “Look Back” – why is this fourth principle so important?  In my mind, this is where all the learning happens.  The three other principles are very clear

  1. Understand the problem
  2. Device a plan
  3. Carry out the plan

These three are all very basic – if they work, right?  But most of the time they don’t work for kids.  It’s the fourth step that we know is the most important – it’s where the critical thinking and analysis takes place.  If this part isn’t taken seriously and the right steps within it are not taken nothing happens, no moving forward, no growth.

So what did Meaghan do?  She realized that she had not made a connection between the triangle properties that we had just learned and how it applied to this problem.  She had not use the correct prior knowledge.  She  just created more openings to other knowledge that she knows- and I know what you’re thinking.  Does this mean that next time she will use the correct prior knowledge in another problem?  From my experience with kids, no, it does not.  But honestly, what I have seen is that the more they realize that there are more possibilities and also that the option of just saying “I don’t get it” or “I can’t do this” is unlikely, the more they will keep trying.

So what did Meaghan do? By just being asked to write a reflection about one problem (every two weeks) she has reinforced her own potential in problem solving on HER OWN.  That she may, in the future, weed out the irrelevant prior knowledge and possibly see the connections to the relevant prior knowledge, with more practice.  I think it’s made her feel just a little bit more confident – and they said it was just “something else for them to do.”

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 http://tutoringtoexcellence.blogspot.com/2014/08/helping-students-see-themselves-as.html

Does journaling in PBL promote resilience?

So I just read a great blogpost by Kevin Washburn of Clerestory Learning entitled “Teaching Resilience: Reflection” and it immediately made me think of the Metacognitive Journaling that I have students do in my classes.  I never really thought of it the way that Washburn was describing the reflection and the conseuences of reflection, but it’s pretty clear that if his theory is right, that a by-product of journaling could easily be resilience.

Initially, Washburn talks about the process of reflection – right out of Dewey in a way – but he narrows it down to the steps in the process (but does mention the word metacognition – thinking about thinking).  He defines reflection as “the ability to monitor one’s own thinking” which is what I tell my students the goal of writing in the journal is.  Hopefully, by the end of the year, they will have realized the way they look at problems and how they’ve had those “lightbulb” or “ah-ha” moments enough after writing about them, that when they come across a new problem, the process of being aware of their own problem solving is much more natural.

Washburn’s three steps are as follows:

1. Asking yourself “what am I thinking now?”
2.  What can I tell myself to redirect my thinking?
3.  What can I do differently?

Most students in the beginning of the year, can easily do the first step – it begins very simply as them just redoing their work (usually the correct way), which can, unfortunately, just be them rewriting their notes from class.  However, this has to be a place to start for them.  This is where teacher feedback is key.  I spend most of my time writing comments like ,”I want to hear what *you* did initially” or “is this what your first thoughts were?”  It’s really hard for students to believe that you want a record of what they did wrong.

But somewhere during the year, kids grow in their understanding of WHY their initial idea didn’t work.  This seems to be the most important part of the reflection.  That gained insight gives them not only deeper understanding, but a sense of ownership and responsibility for their own learning that can’t be had with just seeing how many points they got off from the problem on an assessment.

I’ve written about this a few times (see other blog entries) and have seen kids grow in their understanding during the 18 years that I’ve been using journals in my classroom.  However, what Washburn helped me see is how this skill of recognizing how their initial erroneous thinking has actually made them a stronger, more confident thinker.  This is an amazing gift.  As Washburn says,

In life and in the classroom, the one doing the thinking is doing the learning. When thinking ceases and self-defeating messages crescendo, we can guide students to healthier states of mind and, in the process, equip them to make such cognitive turns on their own.”

So  great!

Using Journal Writing in PBL

Over the years, especially in PBL with mathematics, I have found that students greatly appreciate the authorship and ownership that comes with keeping a journal in my classroom.  In fact when I asked my students earlier this year, “When do you feel most confident in this class?” and here are some of the feedback responses they gave me:

“When I am about to hand in a journal.”
“When I am writing a journal entry because there are various concepts that initially don’t understand, and after discussions I make big discoveries and therefore it makes writing about it easier for me.”

There is something that I have come to appreciate about the way students grow to be able to show how they understand a concept.  Recently, I read a student’s journal entry and thought it was so amazing that I asked him if I could blog about it.  I thought that it really showed how he moved through his understanding of the concept and how he struggled with it to the end. In fact, he presented this problem in class thinking that he had gotten it right and wonderfully, kept going and learned something in the process.

The student – let’s call him Pete- was dealing with a problem that was towards the end of a thread that dealt with the concept of distance – distance between two points, distance between a point and a line, distance between two lines, etc.  This question was asking students to think about two different types of distance.  Here’s the problem:

“Plot all of the points that are 3 units away from the x axis and write an algebraic expression for those points.  Then plot all the points that are 3 units away from the point (5,4) and write an algebraic expression for those points.”

Up until now all we had discussed was writing expressions for Pythagorean distance between two points and writing equations for equations of lines.  We had also talked about the fact that the closest distance from a point to a line is the perpendicular distance.  So Pete was easily able to answer the first part of the question seeing that the set of all points that were 3 units away from the x axis were both the line y=3 and the line y=-3.  He drew a diagram discussing his concept of distance from a point to a line and how he visually (in his mind and physically on paper and at the board that day in class – connected them together).

However, in his journal he wrote about how the second part of the question seemed just as easy to him at first.  “I assumed I needed to do a straight line. I then saw ‘three units’, so I put a point on (5,1), and drew the line y=1.  If (5,1) was 3 away, I thought, shouldn’t all the points on the line be 3 away?” Here’s what his first diagram looked like:

Pete’s First attempt

Pete had tried to use his understanding of distance being “three units” away from a point in the same way that being “three units” away from a line in the previous problem.  However, when he was at the board, another student told him they had thought of it another way and shared with Pete something Pete realized very soon…. “Only 1 point on each of the lines was actually 3.  The rest of the points were actually all further than 3 units from the point.”  Here’s his diagram of his realization of that:

Second attempt

So now Pete is discussing how he is using his knowledge of Pythagorean distance and seeing that only the vertical and horizontal points are actually the required 3 units away.  Huh, how does he move forward now?  At the moment when he was in class, it took another classmate to say that ALL the points were supposed to be 3 units away and that he (that classmate) thought it would look more like a circle.  Pete was still determined to correct his own work (which I just love) then attempted this drawing:

 

Third attempt

Pete writes:

“This, I thought, would cause all points on the line to be 3 units away from point (5,4).  However, I was again wrong.  The blue line on the diagram shows a point on one of my lines that was more than 3 units from (5,4). The red line shows a point on one of the lines that is less than three units from (5,4) [it would’ve been great if he went into more detail, but at this point I’m so psyched that he’s going into this much detail!]  The green lines are points that are 3 units away from point (5,4).  I have effectively created a range of lengths from (5,4) opposed to what the question was asking for which was 3 units from (5,4).”

This is some of the most insightful journal writing I’ve ever seen from a high school student.  Pete is moving through his understanding of what it means for a point to be 3 units away from another point (even when another student has imposed their understanding on him) and is trying to show me how he came to understand his classmate’s argument that it is a circle.  Realizing that there are points that are more than 3 units away (farther out than this diamond shape), 3 away (at the vertices) and closer than 3 (along the sides of the diamond)….well, that there was a range and not all constantly 3, showed him that his ideas was not correct.

Pete then draws this diagram:

Getting the concept

After understanding this, Pete writes:

“It made perfect sense!…Any point from the centerpoint of a circle to any point on the circle was the same length (the radius).  I immediately drew the connection.  3 was the radius and (5,4) was the center.  the distance between the middle point and any point on the circle was 3!”

Although Pete didn’t write about the discussion that ensued about the algebraic expression, I still felt like the goals of metacognitive journaling were reached with this entry.  Pete has articulated why he chose this problem to me, he started with his misunderstanding, how another student or the class discussion/experience had helped move him through his understanding and he could clearly write about how he now has a good understanding of the mathematical concept.  I was so proud of Pete.  Yes, it’s true, not every student gets to this point of writing mathematically by January of the school year, but the growth that occurs in each individual student is what is important – not necessarily the level of maturity in the writing.  However, I have learned not to prejudge or dismiss any student who starts off at a lower level because I believe they are all capable of growth.

If you are interested in looking at my grading rubric for journals or asking me questions about how I use them in the course, don’t hesitate to get in touch. (If you are looking for the grading rubric, make sure you scroll to the third page of that pdf).