Ma, Ch. 6 and 7.

Ma talked a lot about the importance of teachers developing content knowledge in mathematics, rather than merely pedagogical knowledge.   This content knowledge, she argued, needs to be learned on the job as well as in school.   The Chinese teachers she studied spend many long hours studying textbooks, working individually and with other teachers in order to find new ways to teach material.  And the textbooks provide detailed explanations of the subject matter for teachers, in addition to telling them what to teach to the students.   She wrote, “If the teachers’ own knowledge of the mathematics taught in elementary school is limited to procedures, how could we expect his or her classroom to have a tradition of inquiry mathematics?”

I hear a lot of talk about content knowledge in current political debates in California.  Some suggest that math teachers take more advanced mathematics courses – one study, for example, showed that the better teachers have taken more advanced mathematics classes.   I know that I took math up through differential equations and still couldn’t explain why 2/3 divided by 1/2 is 4/3 without taking a class in math education — where we spent a lot of time on the content.  So I fear that simply having teachers take more college-level math courses is not the solution.

On the other hand, many elementary teachers in California come in with little or no content knowledge.   The CSET Multiple Subjects, which I took, supposedly tests knowledge of mathematics – but the mathematics section fails to ask any questions about pedagogy.   By contrast, the sections on English and on PE ask you to analyze a students’ mistakes and develop an intervention strategy, respectively.   Once I got past the multiple subjects test, I then took a class on elementary math methods.  For most of the elementary and special ed teachers in this class, this was the only math course they had taken since the beginning of college.   So much of the class was spent on content as well as pedagogy.

I took the first in the sequence of math content courses for upgrading one’s elementary credential to middle school math and to satisfy requirements for the MA in Math Education program.   I found that we explored some really interesting problems and  got into some really interesting content.   On the other hand, we didn’t get an opportunity to experience teaching the material – we didn’t practice getting up in front of the classroom and teaching – we didn’t practice facilitating discussions.   And even though many of us were currently in placements of some kind, we didn’t take the material in the classroom out into our placement or classroom.   So I wonder how much of that really stuck – as in, is remembered after the class and is actually reflected in the classroom teaching. This is why, I think, that Ma talks more about the Chinese teachers’ curricular study during their actual teaching in the classroom than about their formal education in their normal school prior to becoming a teacher.

Reflect on Ch. 11 in Adding it Up

The one thing that really stood out for me in Chapter 11 was the recommendation that negative integers be taught in the early grades along with addition and subtraction, rather than teaching rational numbers before the negative integers.   That way students could fully develop addition and subtraction before moving on to multiplication and division.

Another recommendation that stood out to me was “teachers of grades pre K-8 should have a deep understanding of the mathematics of the school curriculum and the principles behind it.”   That seems like a pretty tall order- in, say, SF State’s elementary education program or special education program, pre-service teachers are only required to take one course in math pedagogy, and no courses in math content.   The CSET exam they pass in math content addresses things on a very basic level, and does not look at the key areas that Ma and Adding it Up identify as problem spots for elementary education.

It also intrigued me to see the recommendation that mathematics specialists be available in every elementary school.   There’s been a recent push for this in California, so it surprised me to see it all the way back in 2001.

Clearly, we have a long way to go to implement the changes suggested by Adding It Up.  It’s a testament to this document that it still continues to be relevant 12 years later, and still helps to provide a guide towards what a mathematics and mathematics education would look like that would truly help students (and teachers) to learn and teach.

I walked into my local Lakeshore learning store to look from some manipulatives for my classroom. I saw these booklets labelled “Common Core mathematics.” Inside, however, it was remedial rote worksheets, with each section listing a Common Core content standard. The practice standards were no where to be found. This disgusted me, but also inspired me to look into more about the current state of Common Core-aligned textbooks and state textbook adoptions for this project.

Textbook publishing is a highly lucrative industry in the United States; according to pro-con.org, an 8-billion dollar industry. Just three publishers control 85% of the industry- Houghton-Mifflin, Pearson, and McGraw-Hill. Prior to common core, the standards of California and Texas were the primary influence on what K-12 textbook authors and publishers would cover and the approach they would take since those are the largest markets. In California’s case, this meant a set of standards based around rote application of procedures. Textbooks are rarely re-written from scratch in order to satisfy changing curricular demands – instead, sidebars with the new standards are added to pre-existing texts. So textbook publishers paid lip-service to the 1989 NCTM Standards, to the 1999 California Standards … and the fear among many educators is that they will do the same with the Common Core Standards.

The Common Core developers are not blind to this issue. They recently released a guide for alignment of high school curricula. They observed in the introduction: “Traditionally, judging alignment has been approached as a crosswalking exercise. But crosswalking can result in large percentages of “aligned content” while obscuring the fact that the materials in question align not at all to the letter or the spirit of the standards being implemented. “ The guide proceeds to cover three key elements of the standards: focus, coherence, and rigor. In other words, there should be fewer topics, they should interconnect and make mathematical sense, and conceptual understanding should be pursued as well as procedural fluency. Another key element addressed by the guide is mathematical modeling; in the criteria, they specified that “Materials include an ample number of contextual problems that develop the mathematics of the course, afford opportunities for practice, and engage students in problem solving” Finally, they address the practice standards, stating that:

“Over the course of any given year of instruction, each mathematical practice standard is meaningfully present in the form of activities or problems that stimulate students to develop the habits of mind described in the practice standards. These practices are well-grounded in the content standards.

The practice standards are not just processes with ephemeral products (such as conversations). They also specify a set of products students are supposed to learn how to produce. Thus, students are asked to produce answers and solutions but also, in a course-appropriate way, arguments, explanations, diagrams, mathematical models, etc. Materials are accompanied by an analysis, aimed at evaluators, of how the authors have approached each practice standard in relation to content within each applicable course and provide suggestions for delivering content in ways that help students meet the practice standards in course-appropriate ways. Materials tailor the connections to the content of the grade and to course-level-appropriate student thinking. Materials also include teacher-directed materials that explain the role of the practice standards in the classroom and in students’ mathematical development. “

A final key point from these guidelines is about the difference between problems and exercises:

“The underlying design of the materials distinguishes between problems and exercises. Whatever specific terms are used for these two types, in essence the difference is that in solving problems, students learn new mathematics, whereas in working exercises, students apply what they have already learned to build mastery. Problems are problems because students haven’t yet learned how to solve them; students are learning from solving them.

Materials use problems to teach mathematics. Lessons have a few well designed problems that progressively build and extend understanding. Practice exercises that build fluency are easy to recognize for their purpose. Other exercises require longer chains of reasoning.”

I remember teaching Algebra 2 over the summer, and the textbook we used (Holt Algebra 2) presented new material in the form of examples. Then the problems in the back of the book presented a series of exercises, with the example that was similar clearly identified for each exercise. After about 30 or so of these, they would start to get to more interesting problems – and by the time my students finally got to interesting problems they were so exhausted from the exercises that they’d refuse to do the problems. Problems also weren’t properly scaffolded – either the problem would be assigned for homework and give no guidance on how to approach it, or it would give a step by step method for solving and there was no opportunity to explore and create with mathematics.

So, how well to available materials align with the Common Core? In 2011, the Indiana State Department of Education commissioned the Dana Center to review textbooks to see how well they aligned with the mathematical practices. The results are rather dismal; out of 12 Algebra 1 textbooks reviewed, for example, only 4 textbooks received a rating of moderate. (The three ratings were minimal, limited, and moderate – so moderate is the highest rating). I brought two of those textbooks (CPM and Discovering Algebra) to a seminar class of experienced mathematics educators for them to look over, and of the two, the class felt that only CPM and Discovering Geometry really followed the mathematical practices. The results for Geometry from the Dana Center study were even more discouraging, of 11 textbooks reviewed, only 2 received a rating of moderate. And for Algebra 2, only 2 out of 9 received the highest rating of moderate.

So, these guidelines from the Common Core organization hadn’t been written at that time, and these weren’t textbooks that promised Common Core alignment. Now, there are clear, unambiguous guidelines about what a Common Core aligned textbook should look like, and textbooks are starting to come out claiming such alignment. I know I am personally quite skeptical, though; will the 10th edition of the textbook suddenly be labelled Common Core and have lots of Common Core jargon cut and paste-d in, or will the textbooks really be redesigned from the ground up?

Ann Henson compiled a list of ten questions to ask vendors. The one that particularly resonated with me were: “Q8. Explain how all students, regardless of skill and ability level, can be successful using your product?” – the Common Core team is very careful to explain that students should be taught grade-level material in a way that avoids rewinding to remedial content – that, for example, place value can be taught in the context of division in a higher grade rather than rewinding all the way back to teaching lower grade material. Textbooks, though, will need to be designed with the principle of universal access – including special supports in the teachers’ edition for teaching skills like math content area reading. 

There seems to be a cautious optimism regarding textbook alignment and Common Core. California’s going to start the new textbook adoption process in 2014 for K-8. Smarter Balanced assessments will be rolled out in California in 2015.

References

http://www.doe.in.gov/achievement/curriculum/mathematics-textbook-reviews-2011

Hensen, A. http://blogs.edweek.org/edweek/on_innovation/2012/11/top_10_questions_to_ask_common_core_vendors.html

Read and reflect on Liping Ma, chapter 3, Fractions.

Chapter 3 tackles kind of the classic problem of math education – how dos one teach division by a fraction in a conceptual way, rather than say (as the apocrypal tale goes) having the students chant “ours is not to reason why, just invert and multiply.”

Reading chapter 3 and the discussion of how Chinese teachers understood that dividing a number is the same as multiplying by its reciprocal … reminds me of an article I recently read about the connections between various types of inverse operations and how the superscripted -1 tends to be used to signify those kinds of operations. The 5^(-1) is actually part of an inverse operation, being (5)(5)^(-1) = 5^(1 + -1) = 5^0 = 1  or 5 * (1/5) = 1.   So division being the inverse operation of multiplication, you can multiply by a reciprocal in order to divide.   So  5 / 3 = 5 * 1/3 = (5*1) /3 = 5/3.

Another alternative that the Chinese teachers offered was using decimals.   Say  5 / 10 divided by 1/4.   In decimals that would be 0.5 divided by 0.25.   Looking at this, we can easily see that 0.25 * 2 = 0.5 (two 0.25s added together give us 0.5), so the answer must be 2.

Ma also asked the teachers to come up with a story to explain division by 1/2.   The United States teachers tended to give a problem like having 3/4ths of a whole pie, and then splitting it among two people.   But that’s 3/4 divided by 2 … NOT 3/4 divided by 1/2.

A better example might be something like – I’ve got 5 pizzas, and each person is going to get a half slice of pizza.   How many people could we feed.   So, we’ve got 10 half-slices in 5 pizzas, so we could feed 10 people.

Ma also points to a fallacy that a lot of people have about mathematics – that it needs to connect with real-world examples.  The US teachers had plenty of real-world examples, but failing to understand the mathematical concept, they were not able to pick the right examples and both they and their students failed to understand the concept.

• Read and reflect on chapter 4, Adding it up

I’m going to pick a different strand to address this time: Productive disposition. Disposition has to do with one’s beliefs and attitudes towards mathematics. What’s interesting about Adding It Up is that they consider this to be a part of *proficiency*, not merely some separate domain of mathematics education.

One of the things that I have been struggling with lately when working with my students is their tendency to look to me to find out if an answer is correct. Thus, mathematics becomes a matter of teachers’ fiat rather than a logical system of ideas that fit together.

Adding It Up makes it clear that productive disposition involves “the tendency to see sense in mathematics” – to “believe that mathematics is understandable, not arbitrary” (p. 131). When you are able to see this, it helps to build strategic, conceptual, procedure, and adaptive capabilities. And as one builds those capabilities, it improves their disposition.

For me, this was a major issue in college mathematics. It often didn’t make sense to me, and I would hit these dead ends and start to feel like it was rather arbitrary and disconnected. I ended up stopping after the techniques of proof class, and never really picked up upper division mathematics again. Working my way through the math education masters has given me new confidence in my abilities as a mathematician, and given me a new appreciate and joy in mathematics that I had kind of lost in college. I’ve really started to build confidence in my abilities, and am starting to see how all the different areas of mathematics connect. Now the question is how to develop that in my students – to help them see that math makes sense, and that they are capable of making those connections themselves, with a little bit of help from me.

My student teaching ended last Monday, and I’m taking some time to reflect on the experience. My formal reflections are in my portfolio, so this is more of an informal reflection.

The first thing that I wasn’t prepared for is how sad that I am to leave, particularly two weeks before the end of the school semester at the school I was working at. I’m particularly sad about leaving this one boy that I really connected with, especially knowing that he doesn’t have a father. And knowing that I’ll probably never see him again, unless I happened to end up in another school that he’s at and he remembers me.

I also learned how challenging lesson planning really is. I’d work for weeks on a lesson that seemed so engaging and relevant, only to discover students lacked the background knowledge and I should have been building it up over those weeks.

I learned that if you’re going to do mathematics education in a way that gets students to think, you need to start that from day one, not simply try to bring that in half-way through the second semester.

I learned that most of the curriculum isn’t relevant to middle schoolers – and that in the era of standards we have to be really creative to connect and relate it to students’ present and everyday situations. I learned that standing up at the front of the room and jotting down notes about the differences between, say, the samurai and the knights of the ancient world on an overhead projector does not capture the attention of a class of 7th graders. I learned that students often rebel when you ask them to justify their answers in a math class when they’ve never before been asked to do so. And that they’ll get upset if you refuse to tell them whether they are “right or wrong.”

And I learned that I still have a lot to learn myself as a teacher – my content knowledge, my pedagogical knowledge, my ability to keep the attention of a group of middle schoolers who would rather be wandering the school or playing on the yard than doing a math lesson.

As I embark upon my search for a teaching job, I now have this new experience to draw upon, and new insights into what needs to be done not only in my own classroom but in the entire system of education. I’m grateful for having had this experience at the middle school, and am looking forward to having my own classroom.

It started one day with responding to a 20 year old gay guy who posted to craigslist looking for a place to crash. I, living with three people in a bunkbed in an illegal in-law apartment, had no such space to offer. I emailed him some referrals to some local community resources. We stayed in touch over the years, and eventually he decided to enroll in community college. I met with him and helped him to choose a major, to pick out his classes, and to apply for financial aid. I still remember the incredulous but happy smile on his face when his financial aid award letter came and they were handing him $2,000 just to go to school. In the 5 years of his adult life, no one had ever just handed him money to further his goals – so he was shocked that there was money available to go to school and you just had to know how to fill out the right forms in order to receive it.

I since have worked with a number of other students that were enrolling in community college, and found a few things – I’m particularly interested in working with GLBTIQQA students and those from the leather community – I especially like helping those going back to school after a long period of time or those who make the decision to go to college as an adult – and I’m also interested in the transfer process and how to make that less complicated. I enrolled in a graduate course through UC Irvine about helping students with community college transfer, and recently completed that class.

So, I’m looking for one or a few students who are in community college and looking to transfer to a 4-year institution or to enroll in a bachelor’s completion program to work with in order to practice these skills from the class. Being self-identified as or part of GLBTIQQA, leather, or poly communities is helpful as that’s my particular expertise but not required. Just starting out in community college would be fine, too, even if they’re nowhere near the point of thinking about transfer.

•  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.

COMET is a newsletter produced by the California Mathematics Project.   I’ve been on the list for the past year, and it gives regular updates on what’s going on with the State Board of Education, the Commission on Teacher Credentialing, the Common Core, and various other projects and initiatives within and outside California.  The headline story for the current issue is about how the SBE has restored individual district’s ability to make decisions about placing students in Algebra 1 in 8th grade.   One of the students in my class related about how parents push to have their students in high school geometry in 8th grade (perhaps to be in multivariable calculus by their senior year?)   Another article that particularly intrigued me, given that I serve on the Advocacy Committee of the California Association of Mathematics Teacher Educators (CAMTE), is the article about teacher preparation and possibly requiring a mathematics pedagogy class to add mathematics to an existing multiple subject or single subject credential.

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.