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.


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).