Examples of Journal Entries:
Handout of Sample Journals Entries to be Discussed
Some Blogposts of mine about Journaling over the years:
Using Journal Writing in PBL (2014)
Journals: Paper vs. Digital: Pros and Cons (2016)
What I get out of Student Writing (2017)
Revisiting Journals: Getting Kids to Look Back (2016)
Does Journaling in PBL Promote Resilience? (2014)
Assessment of Journals:
Journal Writing Rubric
Keeping a Journal for Math Class
I have received some great inquiries since I wrote my last blogpost and I’ve spoken to many teachers who have ideas and questions about teaching online with PBL math. I thought I would share some of those ideas so that everyone can benefit from these great thoughts.
Collaboration
One of the aspects of the PBL classroom that can be missing or disappear altogether is the relational aspect of discussion or collaboration. It is difficult to have students interact, especially if your school has decided to not have synchronous classes (students in different time zones, allowing families to have their own schedules, etc.). Collaborating asynchronously is difficult, but can happen. Here are some tools where this is possible:
Voicethread:
See my last post – I think Voicethread is the best app for asynchronous collaboration. It does take time to set up, but once it is, students can post the problem they have a question on and students can asynchronously post a video, audio, text or drawing question in response to each others’ comments. The thread keeps the order of the comments and the teachers can put their two cents in when necessary as well.
Flipgrid:
Some teachers that I am working with are using Flipgrid to allow students to continue doing problem partial-solution presentations and then having other student post video responses. The responses are either in the form of feedback on the video or in the form of a question or clarification. Of course, this creates a great deal of work if you are going to watch all of these videos, so you must find a way to randomly watch videos every day so the students know that you will be looking at them.
Canvas Conferences:
You might want to check whatever LMS you have, but I now that Canvas has a wonderful feature called “conferences” where you can set up a video of you chatting (it even has a whiteboard where you can write) and students can type questions. This can be recorded and posted on canvas for students that cannot be there synchronously. I guess is the same as using Zoom, but it is nice because there is communciation through the LMS automatically for the students the time and they don’t have to log onto another app.
Explain Everything Whiteboard:
If you already have an account with Explain Everything, that allows you to make videos on your iPad (and I think on your laptop too), Explain Everything also has a web whiteboard where you can create an online lesson and invite people to watch as you talk. I have not used this before but iti’s kind of an interesting add-on.
Twiddla:
This Free Online Public whiteboard is pretty cool, but it does have pop-up adds. You can make it private in the room settings and you don’t have to sign-up for an account. Of course their goal is for you to sign up for an account, so I’m sure there’s some thing in there about getting you to join, but if you just need something in a pinch, I think this is pretty great for synchronous collaboration. You can also leave anything that was written up and have kids go in later and see what was there. So it somewhat works for asynchronous as well.
Zoom Breakout Rooms:
Boy, I wish I had bought stock in Zoom about six months ago. Zoom has become the most-used app, I believe for online and distance learning in the past three weeks. I hope you are all aware of the break-out rooms that you can do in Zoom, you can triage students by putting them in these break-out rooms in groups to work on topics or separate problems as well. Of course, this is also assuming you are using synchrous time for these classes, but it a really great way to have kids collaborating in smaller groups while you can jump in between the groups – sort of like in class.
“Lessons”
I know it seems weird to think about giving direct instruction during a PBL class, but I think at this time in a crisis, it is important to think about the stress, anxiety, isolation, and other emotions that these students are feeling. All of those feelings are exacerbated in a PBL classroom so whatever we can do to help ease any of those feelings without totally moving to a direct instruction or lecture based class I think is good for the students.
One thing I have done when I teach online, especially at the beginning of the year, is try to make small videos with an app like Educreations (which giving away free Pro Accounts right now) and Explain Everything, which I use a lot (and are also giving free accounts to closed schools). These are wonderful apps and I think easy to use. Here is a video below that is a quick tutorial on how to use Explain Everything on an iPad if it would help anyone.
When I make a video to help a student(s) move forward with a problem, I try to keep it the way I would motivate their thinking in class. If they are not seeing something that is holding them back from what is needed in the problem like visualizing moving from a cone to a sector as in this video:
Another thing that I think is important is to remind students of prior knowledge that they may not see and cannot benefit from the discussion and ability of other students in the class who may have that prior knowledge more accessible to them. This is a great advantage in the face-to-face classroom that many PBL teachers rely on, is that there is usually at least one student who can say “Oh, yeah, I remember doing that before” and can give a short recall of what that prior knowledge is. Online, it is more difficult for students to access that prior knowledge that might be needed. So another type of video I make for students is a recap of prior knowledge that they might need for problems that are coming up:
Now you might be saying, “I don’t have time to make all these videos, Carmel” and that may be true. You can feel free to search on YouTube, but what I’ve found is that many of the videos on YouTube are not what i want. They are either very much based on mneumonic devices or not the type of understanding I would want my students to have. So feel free to use any of mine or email me (cschettinophd@gmail.com and I will try to make a video for you if I have time and post it on my YouTube channel.
Assessment
So it is clear that giving tests with distance learning if very difficult. Parents are not really going to sit and proctor written tests or quizzes especially if they are working from home. The idea of test integrity is also quite difficult with different age groups that you are all working with. Many students are extremely trustworthy and some schools have honor codes which kids would probably adhere to even when at home. However, the stress of being online now with a new way of learning, would be enough to make any student crack under the pressure. So here are some of the ideas that you can use to be more formative in your assessment and then perhaps a bit summative as well.
LMS Quizzes:
Many of the LMSs that schools use already have built in systems for making quick quizzes (Canvas, Schoology, Moodle, just to name a few). If one of your colleagues makes a quiz, it is very easy to share them among members of the same LMS. These quick quizzes can be open notebook and you can have 3 or 4 a week with only 3-4 questions, just to see how the students are keeping up with material. They could count towards a grade or not, but this is a wonderful way to check in. They can also be used as exit or entrance tickets.
Socrative App:
When I was still in the classroom I used this app many times in order to have a quick quiz that was open notebook and gives students real-time feedback about whether they answered a question correctly or not. You can take the time to make your quizzes or you can use the many quizzes that are already in their data base at the Socrative Quiz Shop and use those.
Summary of Topic Arcs:
I believe that writing is the best way for students to show their understanding of a topic, but writing online is very difficult. So it does make sense for students to make videos of themselves. I think a good summative assessment is for the teacher to assign to 2-3 students one of the topic arcs that has just been discussed in the recent 2-3 weeks. Students should work together synchronous or asynchronous (make a google doc perhaps) to :
The way these are graded should be up to the teacher, but what I would suggest is a collaborative grading system:
I do believe that a combination of these types of assessments will help students stress less about their grades before the end of the school year and will also help you see how much they have retained in terms of material. There is also daily contributions and completion of work that should be recorded, but I don’t think I need to tell teachers that!
I wish you all luck with this new venture and would love to hear any other ideas and your experiences with teaching PBL online.
About three years ago, I was invited to the great challenge of finding a way to teach Problem-Based Math online with a great school named Avenues: The World School, whose online school is called Avenues Online (AON). Little did I know then that COVID-19 would come in 2020 and the idea of online learning would be not only popular but necessary in this time of crisis.
Awhile ago, I wrote a blogpost about teaching with a relational pedagogy with technology, as it is something that has been on my mind for a long time. When we started the pilot of AON, one of my first challenges was to find a way for students to interact that somehow mimicked the classroom process and community of discourse and interaction that a Relational PBL classroom has. In the past, I had experimented with the wonderful app called Voicethread that allowed students to have threaded discussions about problems with video, audio, text and drawing. I had students try to have conversations before we came to class about some problems they were grappling with. Here is an example of one of those voicethreads:
Of course, this was always followed by a class the next day, so we could follow up on students’s questions and comments. After I did this small experiment, I did a survey of the students in my classes:
I think what I learned from this experiment was that using an app or interface that allowed for some type of interaction where students could hear, see and interact with each other helped them feel more heard, not judged, and more confident in their learning and discussion. They also did not feel like it was media overload, which was not surprising for this generation for sure.
This experiment helped inform the work I did with AON of course. There was no work that was F2F at all since this school is totally online. The coursework for other disciplines were split into World and STEAM, which are totally project-based (to learn more click the AON link above). But we kept math separate, in what we called the Math Inquiry Program, in order to make sure that the materials were consistently covered in a problem-based way.
The next question is, how do we get students in an online environment to have those relational interactions that were important to the students that used Voicethread? Clearly, it was seeing, hearing, interacting and being able to read each other’s work.
My ideas came to fruition with the amazing help of some exceptionally talented developers in our R&D team. We are lucky to have an interface now that allows for students to see, hear, write and watch each other write while discussing problems. The interface looks like this:
We have it to be very effective for interaction between students and teachers and to allow for collaborative problem solving. Is it perfect? Not quite, there is much that needs to be done, including teacher training and improvement in bandwidth needed for the video, audio and writing capabilities.
Here is a 5-minute video of a small cohort of 7th graders discussing a Mathalicious problem that you may be familiar with. It shows some of the ability of the interface and the ways in which students and teachers can interact in real-time.
The most important piece of teaching online with PBL is attempting to keep the relationships together – student-student, student-teacher and student-material. Keeping students involved in metcognitive activities – writing, making videos about their ideas, interacting with each other – this is what will pay off in the long run. It’s not something we are going to create in the next month or so, but if we keep in mind the relational aspects of learning, we may come out of this crisis with the strong connections we always had with our students.
“A self-compassionate attitude could help us feel comforted when we witness the fallibility of other humans.”
Newman, 9/4/19 Greater Good Magazine
This is the conclusion of a research study that was done by researchers at the University of Waterloo, when they asked 100 recruited students to record a video about themselves to be rated on a scale from 1-9 on how “great” they were (see study for more details).
Unbeknowst to them, there were no objective observers, the study actually began when students were shown peers’ scores sometimes mediocre and sometimes outstanding. The question really was “how do you react when you have a sense of failure in the face of others’ success?” Of course, the findings were interesting. It depended on the person. If they were the type of person that had a great sense of how life is wrought with failure and learning from our mistakes, they were more likely to have an adaptable, compassionate, and sympathetic reaction to others’ failure. However, if the student had the habit of being hard on themselves for failing, that is often what they reflected onto others and had a hard time being sympathetic.
As I read this research study, I could not help but put all of this into the context of the PBL classroom. I immediately thought “this is why whole-class discussion with sharing authority is so important.” In order for the failure of the presenter to be shared by the class and experienced as the shared experienced of failure, students should hear ideas, discuss the pros and cons and loopholes and catch the mistakes together. This could not have happened in isolation or in pairs (caveat: not to say that whole class discussion is the only way it can happen or is the only PBL should happen). The shared experience of confusion, being sympathetic to the confusion, the risk-taking of embarassment or being wrong, needs to happen for everyone to say “I know what that feels like.”
Of course, this needs to be done a classroom that fosters the idea that this is a good thing – we need to make mistakes and learn from them – in a safe, non-hierarchical environment where status and positioning are being closely reflected upon by the teacher. What this study shows is that the other mindset (e.g. fixed, as we’ve come to call it) won’t necessarily embrace the shared failure and could make the shared failure counterproductive. This is a hugely important part of the dynamics of a PBL classroom.
I can’t help but recall a student that I interviewed for my dissertation research and how she captured her feeling about shared failure, in that by contributing ideas, the failures or mistakes become a shared success:
You could kind of add in your own perspective, and kind of give you this sense like, “Ooooh, I helped with this problem” and then another person comes in and they helped with the problem, and by the end, no one knows who solved the problem. Like, everyone contributed their ideas to this problem and you can look at this problem on the board and you can maybe only see one person’s handwriting, but behind their handwriting is everyone’s ideas. So yeah, it’s a sense of “our problem.” It’s not just Karen’s problem, it’s not just whoever’s problem, it’s “our problem.”
Kaley, Schettino Dissertation 2013
Framework for PBL Classroom
Student Self-Assessment
Student Analysis of Contribution Behaviors
Rubric for Class Contribution
Rubric for Feedback before Grades
Journal Writing Rubric
Keeping a Journal for Math Class
Avenues World Elements Website
Rubric for Grading Mathematics Work for Avenues World Elements
Links to Blogposts about Assessment:
Why Teachers Don’t Give Feedback instead of Grades, and Why We Should (2015)
Someday I’ll get this assessment thing right… (Part 2 of giving feedback before grades) (2015)
Buyer Beware…when using rubrics for critical thinking skills (2013)
Powerpoint Presentation for this Session – Assessment
Handout of Sample Journals Entries to be Discussed
Some Blogposts of mine about Journaling over the years:
Using Journal Writing in PBL (2014)
Journals: Paper vs. Digital: Pros and Cons (2016)
What I get out of Student Writing (2017)
Revisiting Journals: Getting Kids to Look Back (2016)
Does Journaling in PBL Promote Resilience? (2014)
Assessment of Journals:
Journal Writing Rubric
Keeping a Journal for Math Class
Link to page on my website about Metacognitive Journaling
PDF of Powerpoint of Presentation of this Session – Journals
I’ve always considered myself a rational person. I mean, I majored in math and then went on to get a masters in pure math. I really always loved the logic and thought that went into proof and how I could think rationally about so many things in life. Even when life threw me curve balls, I could find a way to rationalize most things and think them through in some way.
In 1996, my mother died of colon cancer at age 56 – probability of a female getting colon cancer at that age around that time was about .018%.
In 2015, my father succumbed to a number of cancers (but mainly breast cancer that metasticized in his brain) at the age of 83. He was quite a fighter – probability of a male getting breast cancer now is about .01%.
In November 2018, my son, at age 21 has been diagnosed with Acute Myeloid Leukemia – probability of this happening is 0.2%. (all from cancerstatisticscenter.cancer.org )
All of the doctors say that none of these are related to each other at all – independent events, right? So the probability of this happening in my life is 3.6 x 10^-5. Pretty crazy, huh? Next to impossible, probably isn’t supposed to happen to anyone.
After trying to deal with this rationally, I decided to deal with it irrationally. I wouldn’t talk to anyone about it. Clearly no one could help – what was the point of discussing it with anyone? Loss, grief and sadness were always a part of my life and this was nothing new. I just had to suck it up and get over it. Do my best and move on. Get through it, I always did.
My husband refused though. We deal with things differently. He wanted to tell the world. He writes emails giving people updates about how my son is doing – people I haven’t even met and I felt it was a huge invasion of our privacy. But something happened that totally surprised me. These people, some I knew and some I didn’t know, started sending us things – cards, notes, emails, gifts cards to Panera, cookies, offers to make us dinner, and many other things I never expected. I couldn’t fathom why they would do this. I would say to them, “I don’t know how to repay you.” And their response would always be, “You don’t have to, just pay it forward.”
180 degree turn to the world today – students in MAGA hats harrassing indigenous peoples at a protest rally, supreme court upholding the ban on transgender peoples in the armed forces, our bullying, self-absorbed president keeping federal workers from having their paychecks, Republican elected officials losing their integrity and concept of their job to look out for their constiuents – what is happening? Where is the kindness and care in the world today? I was losing it quite regularly.
In all of this confusion in my brain, I felt the need to let people know that I had to cancel some public speaking and courses that I had committed to this spring. I was nervous about tweeting about something so personal, but I know some folks who follow might have been trying to sign up for my course or looking forward to my NCTM talk.
That’s when it hit me. I received so much love, support and kindness from the virtual PLC of math teachers that I did not expect. I could not believe it. People I knew, people I knew virtually, people I’ve never met – all reaching out and telling me that they cared and wished us the best in this struggle. It was an interesting comfort at a time when my personal struggle was being exacerbated by the complex chaos of the world today.
I do believe that there is no balance between chaos and calmness. We just learn to take both as it comes. The peace that was given to me from the #MTBoS and #iteachmath community definitely allowed me the moment of calm that I needed and I will always remember that. You all are surely paying it forward from someone who did something for you and I hope that I can do the same someday. I do think that teaching has something to do with paying it forward as we do every day, honestly. People say they teach because they love it. But in some way, the relationality that it takes to be a good teacher, be open with people and understand our students is the way in which we pay it forward every day.
Thank you so much for giving me this moment of calm at this time in my life.
In a traditional Geometry text, there is a chapter on circles – usually around 8 or 9 – right? Where they introduce a definition, the equation in a big blue box and students learn how to write equations, find the area, circumference and do some interesting problems negative space if they are lucky. The question then is how are circles connected to the rest of the geometry curriculum? There is so much more that students could be able to see that goes unseen when circles are compartmentalized like this. Circles are often introduced in middle school mathematics classes and by the time they get to high school geometry students have seen area and circumference and perhaps even understand a little bit of the irrationality of pi. What then can a PBL curriculum do to help students understand circle concepts more deeply and in a more connected way?
One of the first types of problems that students are asked to do in the curriculum that I have adapted goes like this:
Find two lattice points that are 5 units away from each other that are not horizontal nor vertical.
This question is clearly asking for two integer-values points on the coordinate plane that are 5 units from each other but students can’t just count them since the distance is diagonal. This is often a difficult idea for students at first who are not comfortable with the idea of Pythagorean distance (see previous post about Pythagorean distance). However, since the beginning of the Math 2 book has so much work with the Pythagorean Theorem, it may be that students are thinking of the distance as the hypotenuse of a right triangle and can think of the 3-4-5 that has shown up in many other problems. It does take some time for students to be able to think of two points like (1,2) and (4,6) as 5 units away from each other.
Students are asked other questions about the configuration of points that meet certain criteria and how it would be represented algebraically. For example:
Describe the set of points that are all 3 units away from the x-axis. How would you describe this configuration in an equation?
This is trying to get students used to visualizing ALL points that satisyfing a given distance condition and when other conic sections are discussed in later courses, they are prepared to be thinking of configurations as a set of points. So finally when they are asked to
Write an equation using the distance formula that says that P=(x,y) is 5 units from (0,0). Plot several such points. What is the configuration of all such points called? How many are lattice points? [See here for a wonderful journal entry by a student on this question].
This question is the first time that circles are actually introduced and the word “circle” isn’t even in the problem. The discussion that occurs can be very deep and interesting (as is what happened when the student who wrote that journal entry presented this problem) or it could be very straightforward and benign. I have been impressed over the year with how well some students understand that a circle must contain points that are all the same distance from the point that is the center. It is introduced with the distance formula because so much of their work to the point has been based on the distance formula.
Students have a great time discussing the number of lattice points that lie on the circle and where they are. If they can harken back to the earlier problem about points that are 5 units apart, they recall the 3-4-5 right triangle easily and get the lattice points in the first quadrant.
Most students can then use their knowledge of transformations and the symmetry that the circle has and find the coordinates of the of the other lattice points. This is also a wonderful conversation about reflection over the axes or origin.
If the student presenting this problem leaves the equation in the distance formula form without simplifying, that’s even better and sets up the next question:
Explain how you could use the Pythagorean Theorem to obtain the same result.
At this point, it is important to connect the distance of five units, the center of the origin and the idea of the radius being a hypotenuse of a right triange with the coordinates the students just found. This will be so important later on when students work with the unit circle in trigonometry for thefirst time.
Ask students to start by drawing right triangles where they think they might be in the circle. Interestingly (and maybe obviously) many of them try to draw it like this (see below) where the right angle is at the origin. This is a great time for a conversation about where the points are that are “5 units away from the origin,” where that distance is, and which point are they saying is on the circle with the right triangle. If some student can connect the idea that the radius is supposed to be the hypotenuse and let another student co
me up to the board and make an attempt at the drawing, it is much better than the teacher drawing it for them. As the discussion moves forward and a student can draw the correct right triangle, I have always tried to get other students to draw other triangles. The first that usually happens is that students generally draw the right triangles that have the lattice points as the point on the circle. It takes some time for students to think about the idea that (x, y) can be any point such that x²+y²=25. This takes some time and discussion.
With a group that is ready, I have also asked students to find the y-coordinates that corresponds to the point on the circle that has the x coordinate that is equal to 1 or 2 and see what they come up with. This is a great time to see if they truly understand what the equation is telling them.
There are many more problems that come after this – some ask for what circles have in common from their equations, some ask for lengths of chords, some are area and circumference problems. I will write another post on how inscribed angles and arcs are introduced but the idea that circles are all connected to the distance formula and the Pythagorean Theorem is a deep one that runs through the whole curriculum and is important for students to see the connections between the right triangles and the circle itself.
This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures — from problem solvers and theorists, musicians and artists, explorers, alchemists and wrestlers, to “makers of patterns.” Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking.
One of the things that is interesting about teaching with PBL is how students often describe enjoying this type of math class more than others they have had in the past. It’s hard for students to paint a picture of what it is that produced their enjoyment. The interesting thing is that it is often not the mathematics they enjoy, but the class itself – the interactions and relationships between the people in the class, and should they be solving some interesting problems that pertain to mathematics, that’s pretty great, too.
What one girl, Isabelle, described enjoying about my class once, was the way in which she saw mathematics as no longer black and white – with only the teacher’s information as what counts. In a research interview, I asked her to describe for me what that was like:
Isabelle: Like it’s, if you have a question you can just ask it and then that can lead into, like, some conversation or [the teacher] can ask a question and then kind of leaves it out there for us, the kids, to answer it, so…
Ms. S: OK, and why do, why do you like that better?
Isabelle: Um, because it’s not so uptight and [laughs], like it’s not like focused, “memorize all of this stuff…”
Ms. S: Hmm
Isabelle: It’s more relaxed, and that helps me learn better I think.
Isabelle’s more traditional view of the mathematics classroom with its “uptight” and rigid nature reminds her of memorizing facts and formulas and she stated that she responds better to a classroom that, in her eyes, is more “relaxed” and interactive allowing her views and responses to matter. This is extremely consistent with Frances Maher and Mary Kay Thompson’s (2001) view of the feminist classroom’s responsibility to “deliberately position students as academic authorities” in order to allow them the input for the feeling that their responses matter, but so that that they do not “dismiss their own emerging sense of themselves.” Also, Isabelle’s feelings are consistent with what Fox Keller (1985) once called “dynamic objectivity” which she defined in terms of how we might be inclined to think about the idea of integrating student input with factual mathematical knowledge.
Dynamic objectivity is a form of knowledge that grants to the world around us its independent integrity but does so in a way that remains cognizant of, indeed relies on, our connectivity with that world. In this, dynamic objectivity is not unlike empathy, a form of knowledge of other persons that draws explicitly on the commonality of feelings and experience in order to enrich ones’ understanding of another in his or her own right (Fox Keller, p.117).
We can view this more flexible way of viewing knowledge as necessary for including students like Isabelle who find the more rigid mathematics classroom not conducive to learning. She would rather remain connected to the material and the persons in the classroom with her in order to facilitate learning for herself. Many students truly enjoy the fact that students are the contributors to the knowledge and part of the authority presence in the classroom. Because of the openness to the dynamic objectivity of the knowledge, the students are able to accept that their input is valuable. When I asked Isabelle why she thought the students felt so compelled to participate in a PBL classroom, she had this to say:
Ms. S: Yeah, there’s almost a guarantee that people will… I wonder why? I wonder what guarantees that everyone will have something to say.
Isabelle: Well [both laugh] it’s probably just because geometry has like twenty… like a lot of different ways to do certain problems so there’s a lot of variations in the way that people do them, so…
Ms. S: Hmm.
Isabelle: That might be it, or it might just be that people feel comfortable in the situation they’re in to participate and it’s not like, “OK nobody ask questions so we can leave now.”
Ms. S: [laughs] Yeah. Ok. So there’s a certain amount of like motivation to want to talk about it?
Isabelle: Yeah.
Ms. S: because it’s like interesting to hear what other people did? [pause] Um, yeah, I can’t figure that out.
Isabelle: I think everybody like shares the same curiosity level and like when somebody… like I know in our physics class he never tells us the answer to questions and it drives everybody crazy…
Ms. S: Huh…
Isabelle: And then we all start talking about it to try and figure out if like we can find out the answer ourselves so and the same thing happens in my math class so…
Ms. S: Yeah?
Isabelle: I think it’s just the motivation to find the right answer and like, because I know everybody in my class wants to understand.
Isabelle had described a mathematics classsroom culture with a tacit understanding of the dynamic objectivity of the part that students play in the formation of knowledge. When presented with a problem where the solution is unknown and the teacher presumes a certain lower level of authority than the students, the students take on a higher level of responsibility and curiosity in finding solutions and methods for those solutions. Being open to a view of dynamic objectivity and the empathy that it needs, allows many students to have their comfort in this type of learning environment and fosters more equity in learning for students who have previously been disenfranchised in mathematics and science classes.
Fox Keller, Evelyn, (1985). Gender and Science. New Have: Yale University Press.
Maher, F. A., & Thompson Tetreault, M. K. (2001). The Feminist Classroom. New York: Rowman & Littlefield Publishers, Inc.
When I was in elementary school, I was lucky enough to have a teacher named Mrs. Bayles who believed that what it meant to be “cool” was enjoy solving really interesting problems. I remember one time she gave everyone in class a piece of pie and asked us all “What’s the best way to start eating this piece of pie?” and everyone else in the classroom immediately took their fork and stabbed it right in that pointy corner, where, they argued, they would get the most of the juicy center of whatever type of pie they 
had. I was sitting with my group of friends (mostly girls) who were self-defined math “geeks” (although I think back in 1976, that’s not what they were called). We all kept thinking about that and eventually came to the conclusion that we wanted to start with the crust because thought saving the middle for last was a great idea.
Mrs. Bayles thought that was so awesome and asked the four us to come up to the front of the room, draw a diagram on the board and give evidence as to why eating pie from the back of the piece of pie was somehow better than eating it from the tip. We thought we were the Albert Einsteins of pie-eating. We just loved it. Even though the other kids in the class thought it was kind of weird, since we could justify our choice with a good argument, we stood together and most importantly, Mrs. Bayles respected our evidence and let us have our authority in our say.
One of the things in recent years that has become a passion of mine in the mathematics classroom, and more precisely, the mathematics probem-based learning classroom, is the idea of status and positioning of students in the discourse and learning that occurs. This has become such an important issue that I invited one of our keynote speakers this year at the PBL Math Teaching Summit to speak on this very subject. After many, many years of hearing teachers’ concerns about how to handle a student who tends to dominate a conversation, or who doesn’t speak enough, or what happens when kids get off on the wrong the track when discussing a problem – it is about time that this socio-emotional topic (which includes race, gender, privilege, equity, and all things relational that made me start studying this pedagogy in the first place years ago) be moved to forefront of the mathematics classroom once and for all. (Aside – huge thank you to Teresa Dunleavy who gave an awesome talk on this BTW!)
I have shared this specific story with so many people at this point in time, but I find it so important that I want to repeat it here for Sam Shah’s “How does your class move the needle on what your kids think about …. who can do math?” prompt for this “Virtual Conference on Mathematical Flavors.” I feel it is something I’ve worked on for almost twenty or so years and I still don’t have it down to a science, I just know that I can’t let it go anymore.
In my classroom, I allow students to use dynamic geometry apps or technology as much as they want to justify their answers or as evidence for their thinking, as long as it doesn’t go awry (and of course, as long as it is correct and they can describe their thinking). Two years ago, I had a student (for whom I will use a pseudonym here because I’ve used his real name in the story, but not on the Internet), I’ll call Ernie. Ernie was one of those kids who could do no wrong – very popular in his current class, very successful academically, which made him very outspoken in his ideas, very good-looking in our white, hetero-normative, social class acceptable way and to top it off – (what Dr. Dunleavy says is usually one of the highest privileges in white schooling) – an exceptional athlete. Mix all of these privileges together and what automatically came with him into this class? Mathematical status.
Mathematical status doesn’t mean that he was not a good math student, that’s for sure. Ernie worked very hard and had excellent intuition, as well as good retention from his past math courses (–hmm, I feel like I’m writing comments from the fall term here..) These were neither here nor there however to the rest of the class. When students bring their own thoughts and impressions of a student into the class with them, its the class itself that priveleges that other student (in this case Ernie) the high mathematical status that he had. There might have been other students in the class who should have had higher status but because they were not as outspoken, had different relationships with others, were messy or not as articulate about their ideas, asked “stupid” questions (you know that’s not what I mean) or whatever the behavior that was exhibited – the other students in the class would assign a low mathematical status to other students by the things they say, brushing aside questions or simply by just not listening.
So one day we were discussing a question about the congruence of two triangles that were in the different orientation, plotted with coordinates. Students were supposed to come up with some triangle congruence criteria (I believe it was supposed to be SSS) for why the two triangles were congruent – this was at the very beginning of the concept of Triangle Congruence. Ernie had plotted the triangles on GeoGebra and simply said, “These two triangles are not congruent because all of the correponding sides 
are not equal” stepped back, matter-of-factly with pride in a job well done.
There was thoughtful silence in the room as the class looked at this diagram up on the board projected from his laptop. There was no arguing with the fact that the sides of his triangles were not all the same. However, there was still confusion I could see in some of the students’ faces. Some kids asked him to find the lengths of the sides. “That’s what I got,” “Oh I see what I did wrong,” and “Thanks for clarifying” were some of the comments that Ernie received. Under her breath, I heard one girl just whipsering to herself, “That doesn’t make sense” and I tried to follow up on the comment, but she would have none of it. We spent maybe 5 more minutes of me trying to get anyone else to make a comment. It got to the point where I even got out my solutions because even I was doubting myself (the power of Ernie’s status) because I had sworn that those two triangles were supposed to come out congruent. I knew some of those kids knew it too. Why weren’t they all saying something? It was as simple as a misplaced point. Not a huge problem, why couldn’t anyone call him on it? I decided to do a little experiment. “OK, well let’s move on then, but I really think there’s a way to show that these two triangles are congruent.” Ernie was intrigued. He couldn’t be wrong so tried to start finding his error, but couldn’t. I said, “no, no, I want everyone to go home tonight and try to see if we can find a way to show that these triangles are congruent.”
Jump to the next day in class and kids are sharing their solutions from the previous nights struggle problems. Before we start discussing them, I say, “Did anyone think about the problem that Ernie presented yesterday?” Radio silence….I wasn’t sure that anyone would actually do it, so I had come prepared with a geogebra diagram of my own. I projected it on the whiteboard and asked if they noticed anything. Still no one said anything (outloud so everyone could hear, but I could tell that some students were at least talking to
each other). Suddenly, Ernie says, “Oh my gosh, I plotted the wrong point! It was supposed to be (6,1).” There was this huge metaphorical sigh of relief from the whole class at this moment that could be felt by everyone. I just coudn’t understand it. Although no one was willing to speak up that they knew Ernie had been wrong, they were all relieved that that he realized his own error.
I expressed my concern with this dynamic in my classroom. Simply asking them why didn’t anyone help Ernie with the problem yesterday in class? or what kept anyone from speaking up when they thought the triangles were congruent? wasn’t getting us anywhere. So what I did was tried to let them know how much I wanted to hear their ideas – similar to what Mrs. Bayles did with the pie. If students can see and hear evidence that the teacher values all voices equally, not just those that the students have given high status, can truly make a difference in how they start placing their status beliefs.
What I saw change in the class slowly, wasn’t the status that the kids all gave Ernie. In fact, if anything he got even more from finding his own error – but what happened was that girl who had spoken under her breath, spoke a little more loudly. Students who presumed that Ernie was correct, asked an interesting question that Ernie had to justify. These other students were growing in the status that the others were giving them. I believe that it is very hard for us as teachers to control what the students come into the classroom believing about each other, but we can have an impact on what they believe is valuable and meaningful about what they do in the classroom.
As I write curriculum, I am constantly scouring the Internet for ideas and ways to improve my own work, as we all do. I was just on the NCTM resources page the other day at their “Reasoning and Sense Making Task Library” and found this description of a task called “As the Crow Flies”:
“The distance formula is often presented as a “rule” for students to memorize. This task is designed to help students develop an understanding of the meaning of the formula.”
OK, wait – shouldn’t this just say, we shouldn’t be presenting the distance formula as a rule for students to memorize? Instead we should be teaching it for understanding from the conceptual level and allowing students to realize the connection between the Pythagorean idea of distance and how it allows a student to find the distance between two points? Why should we have a specific task designed to create the understanding after learning the formula when the formula is actually secondary?
There is a series of questions in the problems I have written/edited that allow students to come to this realization on their own.
First, a few basic Pythagorean Theorem problems to practice the format, remind themselves of simplifying radicals, Pythagorean Triples, etc. Second, some coordinate plane review such as:
This is really an interesting discussion question for many reasons. First many students have trouble understanding where the right angle is supposed to be. If they incorrectly read that the angle that should be right is ABC, then they are picturing a different right angle (and also doing a harder problem that we’ll get to soon!) but if they are reading ACB, it’s still an interesting question because there is more than one answer.
Students can sometimes visualize where the right angle can be (even both of the points) but may not be able to get the coordinates. This discussion is important however because in order to come up with the distance formula later in general (with the x’s, y’s and subscripts – whoa, way confusing!) they need to realize what’s so special about that vertex’s coordinates. So if there is a student who is confused I usually ask the student presenting this problem, “Can you describe the way you found the coordinates for C?” Their answer usually goes something like this: “You just take the x of the one it’s below and y of the one it’s next to.” and other kids are either totally on board, or totally confused. So then they need to make it a little more mathemtical so every else is on board. Other kids often chime in with words like horizontal and vertical, x-coordinate and y-coordinate. This is a really fun, useful and fruitful mathematical discussion in my experience.
We can then move to a problem like this:
2. Find the length of the hypotenuse of a right triangle ABC, where A = (1,2) and B = (5, 7). Give your answer is simplest radical form.
This is generally a problem that is given to students individually to grapple with for homework or in class in groups at the board. After doing the one discussed above, they at least are prepared to find the vertex of the right triangle and see where it should be.
It’s honestly rare that a student can’t even draw the diagram – especially if they can make the connection with the previous problem. (Connection is one of the four pillars of the PBL Classroom). One of the things that is often difficult for students is the idea of subtraction of the coordinates. The can easily count the units to get the sizes of the legs in order to do the Pythagorean Theorem, but in order to generalize, for a later purpose…sorry, don’t want to steal the thunder…subtraction would be an interesting alternate solution method if someone comes up with it – and they usually do.
At this point if someone does come up with it, I usually do ask why can you subtract the coordinates like that to get the lengths of the sides and (you guessed it) there was an earlier problem that had student finding distance on a number line, so, many kids make that connection.
So finally we get to this, maybe a couple of problems later:
Again, students are asked to use their prior knowledge and contemplate a way that they might be able to describe of finding a way to express the distance between two points in a plane. This is after discussing notation, discussing how to visualize that distance, discussing subscripts, and discussing the purposes (in other problems) of why we might actually need to find the distances between two points. Because the Pythagorean Theorem squares the lengths of the sides (BC and AC) I’ve never had a kid get all upset about the fact that we don’t put the absolute value signs around the difference for the sides – we’re gonna square it anyway, so who cares if it’s negative? Kids usually say, “if it’s negative, let’s just subtract the other way and it’s be positive.” We just get right to the point that all we are finding is the hypotenuse of a right triangle which has been the Pythagorean Theorem all along.
I generally have students write a journal entry about this amazing revelation for them at the beginning of the year and voila! It’s right there for them, in their journal for the whole year – no memorization needed. They understand the concept, know how to use it and actually love the idea because now they can just see a right triangle every time they need a distance. It’s how so many of my students say that have “never learned the distance formla” – they just use the Pythagorean theorem to find distance. I love it.