Category: Inclusion

How do you use empathy to teach math?

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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.

PBL and second language learners

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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.

Yours, Mine and Ours

Yesterday we had a speaker in our faculty meeting who came to talk to us about decision-making process in our school.  He spoke about the way some colleges, universities, independent schools are very different from businesses, the military, and other governing bodies that have to make decisions because we are made up of “loosely-coupled systems.” These are relationships that are not well-defined and don’t necessarily have a “chain of command” or know where the top or bottom may be.  They also don’t necessarily have a “go-to” person where, when a problem arises, the solution resides in that location.  The speaker said that this actually allows for more creativity and generally more interesting solution methods.

About mid-way through his presentation he said something that just resonated with me fully as he was talking about the way these systems come to a decision cooperatively.

“The difference between mine and ours is the difference between the absence and presence of process.”

Wow, I thought, he’s talking about PBL.  Right here in faculty meeting.  I wonder if anyone else can see this.  He’s talking about the difference between ownership of knowledge in PBL and the passive acceptance of the material in a direct instruction classroom.

Part of my own research had to do with how girls felt empowered by the ownership that occurred through the process of sharing ideas, becoming a community of learners and allowing themselves to see others’ vulnerability in the risk-taking that occurred in the problem solving.  The presence of the process in the learning for these students was a huge part of their enjoyment, empowerment and increase in their own agency in learning.

I think it was Tim Rowland who wrote about pronoun use in mathematics class (I think Pimm originally called it the Mathematics Register). The idea of using the inclusive “our” instead of “your” might seem like a good idea, but instead students sometimes think that “our” implies the people who wrote the textbook, or the “our” who are the people who are allowed to use mathematics – not “your” the actual kids in the room.  If the kids use “our” then they are including themselves.  If the teacher is talking, the teacher should talk about the mathematics like the are including the students with “your” or including the students and the teacher with “our”, but making sure to use “our” by making a hand gesture around the classroom.  These might seem like silly actions, but could really make a difference in the process.

Anyway,  I really liked that quote and made me feel like somehow making the process present was validated in a huge way!

Considering Inclusion in PBL

It’s always refreshing when someone can put into words so eloquently what you have been thinking inside your head and believing for so long.  That’s what Darryl Yong did in his recent blogpost entitled Explanatory Power of the Hierarchy of Student Needs.  I feel like while I was reading that blogpost I was reading everything that I had been thinking for so long but had been unable to articulate (probably because of being a full time secondary teacher, living in a dorm with 16 teenage boys, being a mother of two teenagers of my own and all the other things I’m doing, I guess I just didn’t have the time, but no excuses).  Darryl had already been my “inclusive math idol” from a previous post he wrote about radical inclusivity in the math classroom, but this one really spoke to a specific framework for inclusion in the classroom and how in math it is necessary.

 

In my dissertation research, I took this idea from the perspective of adolescent girls (which, as I think towards further research could perhaps be generalized to many marginalized groups in mathematics education) and how they may feel excluded in the math classroom.  These girls were in a PBL classroom that was being taught with a relational pedagogy which focuses on the many types of relationships in the classroom (relationship between ideas, people, concepts, etc.)  – I did not look at it from the perspective of Maslow’s Hierarchy of Student Needs and this is really a great tool.

Interestingly,  I came up with many of the same results. My RPBL framework includes the following (full article in press):

  1. Connected Curriculum– a curriculum with scaffolded problems that are decompartmentalized such that students can appreciate the connected nature of mathematics
  2. Ownership of Knowledge – encouragement of individual and group ownership by use of journals, student presentation, teacher wait time, revoicing and other discourse moves
  3. Justification not Prescription– focus on the “why” in solutions, foster inquiry with interesting questions, value curiosity, assess creativity
  4. Shared Authority – dissolution of authoritarian hierarchy with deliberate discourse moves to improve equity, send message of valuing risk-taking and all students’ ideas

These four main tenets were what came out of the girls’ stories.  Sure many classrooms have one or two of these ideas.  Many teachers try to do these in student-centered or inquiry-based classrooms.  But it was the combination of all four that made them feel safe enough and valued enough to actually enjoy learning mathematics and that their voice was heard. These four are just a mere outline and there is so much more to go into detail about like the types of assessment (like Darryl was talking about in his post and have lots of blogposts about) the ways in which you have students work and speak to each other – how do you get them to share that authority when they want to work on a problem together or when one kid thinks they are always right?

The most important thing to remember in PBL is that if we do not consider inclusion in PBL then honestly, there is little benefit in it over a traditional classroom, in my view. The roles of inequity in our society can easily be perpetuated in the PBL classroom and without deliberate thought given to discussion and encouragement given to student voice and agency, students without the practice will not know what to do.  If we do consider inclusion in the PBL classroom, it opens up a wondrous world of mathematical learning with the freedom of creativity that many students have not experienced before and could truly change the way they view themselves and math in general.