•  Read and write a reflection on chapter 3,  Adding it up

Chapter 3 seemed to mostly be covering the content knowledge around number.   I was surprised that it didn’t talk at all about the pedagogy, but I’m assuming that will be elaborated on in the later chapters – and that they wanted to make sure that readers of the book had the basic understanding of the content before delving into how to teach it.

The two things that intrigued me were — the whole numbers lack an additive inverse, and the integers lack a multiplicative inverse – necessitating the invention of positive numbers and rational numbers, respectively.   i had never seen it spelled out quite so systematically before.   It makes me want to go back and read some of the higher math books that I have on integers and rational numbers and real numbers and work some of the problems in those books.

Read Chapter 1, Knowing and Teaching Mathematics and be prepared to discuss.

Some quotes I particularly liked from this reading: “Subtraction, with or without regrouping, is a very early topic anyway.  Is a deep understanding of mathematics necessary in order to teach it?  Does such a simple topic even involve a deep understanding of mathematics?  Would a teacher’s subject matter knowledge make any difference in his or her teaching, and eventually contribute to students’ learning?   There is only one answer for all these questions:  Yes.”   (p. 43-44)

“‘We can’t subtract a bigger number from a smaller one’ is a false mathematical statement.”   (p. 45)

“The proportion of Chinese teachers who held such procedurally directed ideas, however, was substantially smaller than that of the U.S. teachers (14% vs. 83%).”  (p. 49). borrowing  decomposing

“What differed was that most Chinese teachers said that they would have a  class discussion following the use of manipulatives.   IN these discussions students may report, display, explain, and argue for their own solutions”  (p. 60).

Bruner: “Grasping the structure of a subejct is understanding it in a way that permits many other things to be related to it meaningfully.  To learn structure, in short, is to learn how things are related” (Bruner, p.7, in Ma, p. 64).

Select a section in Lessons Learned from Research that you are interested in, and then read introduction to the section that you’ve selected, and look over the chapter titles:  select 3 chapters that interest you…..

Chapter 12: Interference of Instrumental Instruction in Subsequent Relational Learning

Chapter 14: Developing Concepts of Sampling for Statistical Literacy
Chapter 15: When a Student Perpetually Struggles  (was assigned to someone else)

Chapter 2: Adding it Up ( write a short reflection: was there anything that

surprised you? In your experience, do you agree?)

I was surprised by the 1996 NAEP survey that showed that 46% of fourth grade teachers had little or no knowledge of the NCTM standards.   It makes me wonder – for all the talk about Common Core in NCTM and CMC, is this knowledge about Common Core filtering down to the average elementary school teacher?   What kind of data, I wonder, exists in terms of how prepared elementary school teachers are to teach with the Common Core standards?

• Read Introduction: Knowing and Teaching Mathematics, LiPing Ma

The thing that really stood out for me from the Introduction to LiPing Ma’s book was the distinction she makes between “teacher’s mathematics subject matter knowledge for teaching” rather than knowledge of advanced mathematical topics.   I often worry that we get these two things confused, assuming that taking, say, abstract algebra or linear algebra will make someone a better algebra teacher and taking, say, real and complex analysis will make them a better calculus teacher.   I mean, it might, if those courses are really taught in a problem-solving fashion and where the teacher works to connect the knowledge learned to what you are going to be doing as a teacher.   Otherwise, though, I often think that the knowledge from the advanced mathematics courses fails to transfer into the practice of teaching the K-12 material, and the time would better be spent learning the material being taught in K-12 well.

• Read Introduction: Lessons Learned from Research

(hasn’t arrived yet)

• Given your research interest, write down a possible (or several possible)
research question. Think about the question as something for which you will
need to be able to collect data on which to base a possible answer.

I’m currently coming at things from more of a theoretical angle … as in, what would it take to rethink the disciplinary apparatus of special education from the point of view of queer studies and disability studies?  How might I draw on the various literature that I’ve been surveying in order to more fully elaborate the theoretical framing?   How does Foucault fit into this picture?  Judith Butler?   Lacan and Lee Edelman?   I might also critique some of the literature and research in terms of special education and mathematics from this perspective that I’m looking at.

If I were to focus more on doing actual fieldwork, I could see possibly doing a case study of a teacher who’s using a problem-solving approach to teaching students who have been labelled as having disabilities.  Or doing a study of a district that’s implementing reforms in how they deal with special education and mathematics.   Or sit in on some IEPs of students who are being evaluated for mathematics disabilities and critically analyze those IEP meetings.   Or even take a critical lens to my own teaching and analyze my own interactions with the students during my student teaching.

Strands of Mathematical Proficiency

1.  Write a short personal reflection on some aspect of the executive summary and chapter one of Adding it Up.   In the executive summary, I particularly found the section on the strands of mathematical proficiency interesting, as it gives a clear and concrete definition of what it means to know mathematics.   In doing so, it expands the definition beyond procedural fluency to include four additional strands of proficiency.   As a math leader, I will have to help other teachers to see these additional aspects of what it means to have learned mathematics.   I will have to also come to terms with the fact that i mostly learned procedural fluency in school, and will have to acquire conceptual understanding, strategic competence, and adaptive reasoning myself in order to be able to teach it to my students and to help educate my fellow math teachers about it.

Furthermore, preschool children start with the ability to “formulate, represent, and solve simple mathematical problems and to reason and explain their mathematical activities.   They are positively disposed to do so and to understanding mathematics when they first encounter it.   For the preschool child, the strands of mathematical proficiency are especially closely knit”  (p.6)  As we discussed in class, as early as kindergarten we start to teach kids that it’s only about procedural fluency and the rest of their mathematical skills start to atrophy.

In Chaper 1, I particularly liked the section about research methodology.   First, the study authors pointed out how both instructional decisions and research must be guided by values.   In my my paper on research methodology, I propose that an author having a value-based theoretical commitment is an important part of the validity of a research project… meaning that, I contend, any research claiming to be value neutral is fundamentally flawed in terms of attempting to critically analyze and transform society.   I also like Adding It Up’s focus on how “the conditions of practice make the success of any procedure contingent.”   I was talking to the educator the other day about my research work, and he observed that a lot of times researchers try to import a solution they saw work in one classroom directly into another classroom, only to have it fail miserably once the researcher leaves the picture.

Third, the authors observed, it can be hard to control irrelevant variables.   I see this as kind of a red herring — of course we can’t control the variables in education, and that’s okay – education practice is messy, and research can inform the work we do, but it’s always going to be a guide to be shaped in the hands of an experienced teacher.  Finally, the authors contend, “information obtained from research therefore is particularly useful when it goes beyond the sought-after effects.”   (p. 26).   This is something that my approach to research is particularly useful at — digging deeply into what’s going on in a situation using a value-based theoretical modality in order to give researchers and practitioners new tools and frameworks to interpret reality.

2.  Explore these websites: Mathematically Sane and COMET.

Mathematically Sane is a blog and resource site about reform approaches to mathematics education.     Although the site hasn’t been updated in a while, I could see it as being potentially useful for educating teachers and parents about the true nature of reform mathematics.   It has some guides for administrators and families that were developed by NCTM, for example.   There’s also background information on the common core and some philosophy documents on teaching mathematics.

3.  Think about the issues in mathematics education; choose one or two issues

that you’d like to explore in more depth. Write them down….

I’m interested in a number of issues in mathematics education:

• technology/MOOCs/calculators/computer algebra systems/khan academy for one
• gender and race in math, and how it relates to both geek identity and the notion of being good at math
• teacher preparation and what it will take in order to prepare teachers for the Common Core
• multicultural mathematics and ethnomathematics
• should we assign homework, and what place should it have in the curriculum?

4. Generate a few topics that you might be interested in exploring further in your

Field Study. Write them down……

I’ve already done a Master’s thesis, so I won’t be specifically enrolling in a field study for mathematics education.   However, I’m working on a concept for a culminating project that will be a theoretical piece about taking a queer approach to mathematics disability.   Right now i’ve written it up as a 10-page conference paper and have submitted in abstract form to the DC Queer Studies Symposium.   I’d like to continue to work on this project throughout the summer, and coming into Fall have about 20-30 pages — and really elaborate on both the queer theory aspects, the disability studies aspects, and also connect it to issues faced by English language learners in mathematics as well.

I would also like to find a way to distill the key information from the project into something that I could present at the mini-conference that my math leadership class is putting together for May, and possibly even turn into an Asilomar workshop.

(This was a class assignment tonight, and I’ve decided to post it on my blog and share it).

What have you done?

In high school, I worked for a project called the Technology Institute where as high school students, we taught K-12 teachers how to integrate the Internet into their curriculum.   I taught a semester-long course on queering and transforming your teaching and learning, offered through a local nonhierarchial school called Corvid College.   And I’ve presented three conference papers on various educational topics.  Oh, and we learned and practiced facilitating professional development sessions last semester in the cases class.

What would you like to do?

I’d like to learn how to work one-on-one with a teacher to help them learn the skills that we developed on mathematical discourse in the cases course.   I’d also like to have some real-world experience facilitating case-based professional development.    I would like further practice writing, submitting, and presenting conference papers and designing conference sessions.    I would eventually like to write my own e-book about my previous thesis research.    And eventually I would like to go on to earn a Ph.D in Education and get a job doing research and teaching future teachers.

What do you need?

I need my classmates to help hold space for this personal development work.   But at the same time i need their honest critiques of my work and the feedback that i need to improve and grow.

I’m excited, but nervous.   Sometimes I feel like an imposter – what could I possibly have to tell someone else about education?   I have to remind myself that I’ve been preparing for many years to take on exactly this kind of role, and that I’m ready to, as Dr. Foster would say, “go forth into the world!”

it’s okay for someone else to be wrong

I was waiting in San Jose for a Santa Cruz Metro Highway 17 bus to Santa Cruz.   I ask someone how much the standard fare is so that my friend can pay for his fare.   I note that it’s \$2.50 for the disabled fare, and the guy asks about how he can get the disabled fare.   It turns out that i know something he doesn’t, namely that his Medicare card will get him the discount.  He sees me with a two-dollar bill and asks if he can trade me two \$1s for it, and I agree since I’m about to put it in a farebox anyway.  (Incidentally, he opts to pay the full \$5 fare instead of the \$2.50 disabled fare so he can keep the \$2 bill…)

So then I mention how that when I first started riding the highway 17 bus it only went to Scotts Valley and not to the Metro Center.   He disagrees, and starts telling me about how the bus always went to Santa Cruz and used to be run by Amtrak.   Mind you, I rode the bus many a time and it ended at Scotts Valley and you had to transfer to the 35 to get to downtown Santa Cruz.   I also served for a year on the Santa Cruz Metro Advisory Committee.    So I’m pretty sure I’m right about this.  He strongly disagrees with me, and continues to tell his version of the tale.

Then I had an epiphany.  It’s okay for him to be wrong.   I won’t die if he leaves thinking the Highway 17 never terminated at Scotts Valley.   The world won’t end (well, it might, but not from this belief).   I want to be right, but not only that, but I want to one-up him and win the discussion and have him admit that I’m right.  The “male role belief system” (to borrow a term from Humanalive) tells me that I have to prevail when I’m challenged.   Instead, I chose not to argue with him.   I made a mental note to double-check my facts, reaffirmed the existence of my own experiences that I felt he was negating, and changed the subject.   And ended up having a reasonably nice hour long conversation with him.

So I propose something a bit radical here:  It’s okay for other people to be wrong.   Through a buddhist lens – how do we practice with other people being wrong?

I’m going to start noticing my own emotions and what’s going on with me when I think someone else is wrong.   When someone told me my own experiences didnt happen, I was hurt and I felt invalidated.   I felt unsafe.    Rather than arguing back and trying to regain authority over him, I can intimate with myself and be present in those feelings.   And changing the subject’s a good idea too, to keep myself from continuing to get hurt.   I can live with him being wrong.   I won’t die.

reflections on Robin Ward’s “I’ve been programming since I was 7″

I was reading Robin Ward’s post “I’ve been programming since I was seven” and thinking about my own experiences in an undergraduate computer science program.   I had been working with computers from a young age, since my father had been a software engineer as far back as I could remember.   I was surprised, then, to find out how challenging the introductory computer science sequence was.   It was almost taken for granted that most students wouldn’t pass the introductory programming course.  Those that made it past that introductory class faced a second class on data structures that was about three times as challenging.   When you added to the requirement of three semesters of calculus, only a very small number of students made it past the introductory sequence.

One might conjecture that it is intelligence that one would need to get through that program.   But in reality, it’s about privilege.   As a kid, I didn’t have to work other than occasional chores and school homework.   I could spend as much time as I wanted fiddling with computers (aside from a caveat that I had to walk around the block for every half hour I spent on the computer).  I had access to a parent and any number of friends of his that could answer questions and challenge me to explore new things.

What would a computer science program look like that didn’t privilege those that have been programming from a young age and wash everyone else out of the program?   One idea that’t I’ve been pondering for a long time would be the creation of a bridge course  between the intro course and the data structures course.  Something that laid the foundations for the data structures course for people who are still trying to understand what they had just learned.   There also needs to be more support for those encountering the material for the first time.

I also think that more attention to what Lee Shulman calls “pedagogical content knowledge” in computer science would be helpful.   Pedagogical content knowledge is the knowledge of how to best teach the specific content of a discipline, as opposed to pedagogical knowledge (general techniques of teaching) or content knowledge (the knowledge of the discipline itself).   We know a lot about how people learn the so-called core subjects such as mathematics, science, social studies, and language because entire schools of education are devoted to researching that teaching and learning process.   But computer science pedagogy is a very minimally studied area, and those that write and publish on it are almost entirely experts in computer science rather than the discipline of education.

Finally, people that work in computer science need to give up the assumption that it’s something that can only be learned by people that are “intelligent” enough to grasp it.   Computer scientists, as much as they’d like to believe so, aren’t necessarily “smarter” than those who study history, music, literature, philosophy, or politics.   We would never flunk half of the students out of an introductory politics class and believe that only a “prodigy” (to use Ward’s term) can learn politics.   Why is this okay for computer science?

MA in Math Education: Cases Class

The third class in the Math Education program at SF State is a class on analyzing cases of mathematics teaching.  The class has a particular focus on mathematical discourse.    A secondary goal of the class was to teach us how to facilitate our own professional development sessions for teachers. Each week in the class, a group of students facilitated discussion of a case, which consisted of a description of what transpired during a lesson or a video showing an excerpt from a lesson.

The biggest take-away from this class was the practical experience facilitating the cases.   At the end of each class session, the teacher and the class would critique the presentation.   Getting that feedback on what I did was particularly helpful in fine-tuning my own approach.   There were even little practical tips that I wouldn’t have thought of, such as: Always have a handout.   People think that if you take the time to write your main points down on a handout, they must be important — and will take you more seriously even if they never look at the handout ever again.

A secondary thing that I learned from the class was how to structure and plan a problem-solving based lesson.  The problem-solving class presented the why behind problem solving and helped us to have problem-solving experiences ourselves, but the big question in my head had been how to actually practically implement group problem solving and classroom discussions.   Watching the videos and discussing them with other students gave me a lot of opportunities to think about how I might do this in my own classroom.   When i start my student teaching in the Spring, I will have a lot of opportunities to test and fine-tune my own approach and figure out my own style as a teacher.

The final thing that I took away from the class was strengthening of my content knowledge.  This was kind of a side effect from the course as the focus was really on discourse.   But as we went through and I read the cases and watched students grapple with the different topics, it helped to strengthen my own understanding of the variety of topics that i’ll be having to teach at the middle school level in my student teaching next semester.

MA in Math Education: Assessment

The second course in the MA in Math Education program that i moonlighted in focused on assessment.   At a first glance, one might think this means grading or testing.   But it turned out to encompass a lot more than that.   And assessment isn’t always about a student working solo on a problem.   Sometimes we want to assess how a student performs working with others – after all, how often in life do you encounter a problem that must be solved entirely on your own?   For example, I broke my favorite file box today and attempted to fix it on my own.   After about 20 mins of that, I realized that I would need to consult someone else in order to get ideas.

The course followed naturally from the problem solving course in that the question is, if you’re going to give complex, thoughtful, interesting problems for students to solve – how do you know if they’ve learned what they’re supposed to learn?   How do you grade the work they do in groups and as a whole class in solving problems?   After class time is spent solving problems, do you then turn around and give a test on rote procedures as a summative assessment?   Is it a reasonable problem solving environment to have a closed-book test and a student working silently on a problem that they’ve never seen before?   But is it really a reasonable assessment to give students a problem just like one they’ve already done but with different numbers?

My biggest take-away from the assessment course was the notion of a portfolio – the idea that you have a student compile a sample of their work and then write some sort of reflection on each piece of work in the portfolio or on the portfolio as whole.  I mean, we did what were called portfolios in high school English – but it mostly just consisted of putting all our essays in a folder and calling it a portfolio.   We never really did the reflective aspect of it – how does my work fit together?   why did i choose this piece?  What did I learn from doing this assignment?

Ideally, I’d like to see a shift away from setting grades based on tests and homework completion to grades based on a portfolio.   Understanding your learning trajectory, knowing what you knowing, and knowing what you don’t know is, in my mind, more important than a teacher deciding whether or not you know how to FOIL two binomials in a closed-book, closed-note, no talking test.   Not to say we must always do away with exams, but that they would be assessed in context through a portfolio.

Another thing I want to address from the class is the question of rubrics.   I must say, I’m not a big fan of rubrics.   In middle school, one of my teachers loved to use rubrics to grade essays.   She, however, thought that a 1-5 rubric in two categories would naturally equal a grade, so the only way to get an A on an essay was to get a 4-5 or 5-5 score on the essay.   So rather than the rubric being a tool to help a student identify the teachers’ expectations and figure out how to improve, the student simply saw it as a fancy way of grading.   I think all too often we fall into the trap of thinking rubrics are for grading, rather than for assessment.

MA in Math Education: Problem Solving

Over the past year and a half, I’ve been “moonlighting” in the Masters program in Math Education at San Francisco State University.   The core courses for the program are a sequence of four courses that are taken in order with the same cohort of students – starting with a course in problem solving, assessment, analyzing cases / discourse, and ending with a course in math leadership.

The problem solving course was an experiential course designed to give current math teachers experience in both solving problems themselves and pointers about how this kind of approach might be used in their classes.   On most class days, we would review problem-solving strategies and then use them to tackle interesting and challenging problems – sometimes so challenging that it would take a dozen math teachers working the entire three hour class period to even make a dent in them.  We also had to hand in a problem journal in which we worked through problems – complex problems involving many hours of work, not the kind of five to ten minute exercises found in most math books.   A major component of the class involved learning what a problem was, and why it’s a misnomer to claim that the 80+ exercises at the end of your average textbook section constitute “problems.”  (A problem is something that you don’t already know how to do, whereas a typical textbook consists of problems that directly apply the technique shown in one of the example problems.   One textbook I once taught from would tell you which example to use to solve each problem, just in case you hadn’t really understood the examples and would rather follow one by rote…   A problem, by contrast, is an interesting problem that students don’t already know how to do.

The class also served as an introduction to doing graduate level research in mathematics education.   We were required to prepare a literature synthesis on a chosen topic in mathematics education and to connect that topic to problem solving.   (Mine can be found here: http://www.jamessheldon.com/math-ed/mathematics-reform-and-learning-disabilities/ )

One of the biggest AHA! moments for me in this class was when a student confessed that they had asked someone for help on one of the problems.   The teacher’s response was something along the lines of, “That’s a wonderful problem-solving strategy!   What did you learn from it” rather than reprimanding the student for cheating.   I mean, what professional mathematician would hesitate to walk down the hall to speak with a colleague when stuck on a difficult portion of a mathematical research question they were investigating?   What math teacher would dream of writing a lesson plan on a topic they had never taught before without consulting a colleague, a textbook, the Internet, or a teacher’s mailing list?  What engineer would tackle a complicated engineering problem without reference books?  But we tell our students it is cheating to ask someone for help!   And we even think it’s cheating when we ask someone for help when we’re told to solve a math problem for a graduate class.

Another big take-away from this course was the importance of having many different sources to draw on for problems and the importance of working with colleagues to plan courses.   You can’t necessarily rely on the district or state-adopted textbook to have these kind of problems that will really engage students in them.   And giving an ill-chosen problem at random is sometimes worse.   For example, I took a problem that we did about classifying three dimensional shapes from my credential math methods class and threw it in randomly into a 7th grade honors math I was substituting for.  The students were bored, because they had already mastered the classification of three dimensional shapes.   In essence, it wasn’t a problem for them because it was something they already knew how to do.  And it didn’t link rationally into the content that they were currently studying, and there was no curricular alignment.

The final key take-away from the class is the importance of mathematical community.   The kind of changes that mathematics reform asks us to make in our classes aren’t easy.   They go against everything we were taught in our 17 years of mathematics instruction in school.   What this experience showed me more than anything was the power of creating a group of teachers that meet on a regular basis to improve their practice.

intelligence as a compliment, stupid as an insult: rethinking normal, rethinking attraction, rethinking society

I’m reading Tim Chevalier’s post on “alternatives to intelligent” and am thinking about my own experiences in geek communities, in the dating world, and working in special education.

With my geeky friends, it’s not uncommon to refer to non-geeks as being stupid, particularly when they are obsessed with things such as fashion, sports, or other sufficiently non-geeky things.   I know for me I was ostracized as a kid for not being into sports and then as a gay adult for not being into fashion, and so the primary way to win approval was to head in an academic direction and work on developing my mind.

I built my own identity around thinking of the average person in my high school as “stupid” and thought things would be different in college.   Imagine my shock to find out that all those people in high school were still there in college — they just had a higher GPA and more controlling parents pushing them to attend college.  So then i turned to computer science, but was surprised to find out how many people were just in it for the money rather than being intrinsically interested in the subject.   (It was somewhat satisfying to see that they got lower grades than the people interested in the subject, though).

The question, though, that I have for geeks is … can you give up your need to have been right, the self-righteousness that comes with putting down those who were putting you down all those years for being geeky?   And what are the consequences that come with attempting to judge others’ intelligence.   Many people process information differently or have different perceptions than you do.   And there’s a long history in our society of those people being marginalized and oppressed for being that way.

In my special education classes, we were talking about person-first language and how to talk about the students that we work with.   So, for example, you might have someone who’s pejoratively called a retard.   So then you might say, well, let’s call him or her an intellectually disabled person or a learning-disabled person.   But then that’s assuming that that’s a defining characteristic of them as a person.   So we might say, a person with an intellectual disability or a person with a learning disability, the so-called person first language.   But we still are comparing them to a standard and finding them wanting.   I worked with a group of classmates once to design a conceptual model for disability, and my classmates came up with the idea of a tiramisu, with each layer describing an aspect of normal functioning (visual,  intellectual, hearing, executive functioning) and that some people have the whole tiramisu and some only have part of each layer.   But that means that we’ve defined the normal person, and then judged and found almost everyone as wanting.

So I suggested in class a radically different approach.   What if we were to instead describe their positive qualities and call them a thoughtful and creative person, instead of a person with a disability.   What would it mean to really honestly look at someone’s strengths and not always focus on comparing them to a standard?    And what would it mean to not insist that a person’s worth is defined by their relationship to capitalist productivity?   Might someone still deserve a decent standard of living even if they aren’t able to work in a traditional environment?   Can we take the principles of UDL (Universal Design for Learning) and instead create Universal Design for Living and Working, so that all our spaces are accessible for everyone, regardless of physical or mental differences from the “norm”?

By the same token, though, we need to heed Tim’s call to give up calling people intelligent as well as stupid.    Looking at things from the point of view of deconstruction, we’ve got a binary: smart people / stupid people – where the smart are defined by the existence of the stupid.   We need a list of positive characteristics we can use to describe everyone that aren’t structurally connected to terms that put people down.   As we say in education, this list is left as an exercise to the reader   What are some terms that exist outside of these binary dichotomies?   Do such terms even exist?   Do we have to give up characterizing people all together?

I know I’m going to be rethinking my online dating profiles too.   Instead of looking for smart and intelligent people, I’m going to have to take a serious look at what I’m actually attracted to.   What kind of characteristics define someone that I’m interested in?   And how do those relate to ableism?   I mean, listing well-read puts someone who has difficulty reading in the average person’s way at a disadvantage.   My friends who identify as Aspies might not fit the traditional mold that one thinks of collaborative.   So I’m going to have to seriously rethink what I construct as attractive.   (See the Dating from the Margins series for more on taking a critical look at our attractions.)   Looking at ourselves in an honest fashion.   The introduction to my site talks about how we choose to be ignorant because we can’t bear the implications of knowledge on our own selves.

So, when replying to this post, please keep in mind the two questions that I pose in the Welcome page: What does this information do to one’s own sense of self? What does the knowledge ask me to reconsider about myself and the subject studied?