Blog 6

1. First I’d ask the president a question - If security weren’t an issue, would you send your girls to a local public school? The reasons you enrolled your girls in a private school, are the things that need to be ‘fixed’ in public education.

Specifically, in order

Provide early childhood education and healthcare to ALL children. Make headstart and transitional kindergarten mandatory for all students.

Develop a national curriculum that includes math, English, science, arts and social sciences. Curriculum must be separate from political and special interest agendas. Number of math standards must be reduced in favor of mastery of basics - number sense and operations.  

Make all testing meaningful to students as well as teachers, schools and districts. Develop levels of degrees/diplomas - completion without proficiency, completion with proficiency, completion with advanced proficiency. 

Students with chronic attendance, discipline and failure issues need to be removed from mainstream public education. Provide an environment in which these students’ issues can be addressed without detracting from general population.

Fund all students equitably - private donations, community funding to individual schools should decrease state and national funding per student, with the difference redistributed to schools without private funding opportunities.  

2. Meaningful about Ravitch - 

Reading Ravitch was frustrating. While admitting her position has changed is laudable, we are in this position because short sighted advisors and policy makers like Ravitch ‘drank the kool-aid’ and adopted the free market, choice and testing policies. Class discussions were also frustrating in that individual points were made without discussion regarding that point. Each person made a statement, and then the next person made a statement with very little discussion - agreement or disagreement.  To me, it seemed that Ravitch was making the point that we should question everything - verify opinions, demand proof of research for policies, yet all we did was regurgitate a statement and apply it to experience. I would have liked an opportunity to really discuss issues - both sides of Alvarado’s policies (not everything was negative), ways to redirect and focus billionaire boys club and not just condemn their interference.

What can I do as a teacher? Take care of my 190 students a year. Teach for understanding, instill an inclination to learn (not just an ability to take a test), question administrative policies and still find a way to work within them.

What can I do as a citizen? Educate myself in policies, politics and politicians. Vote.

4. Research teaching associations

NCTM - National Council of Teachers of Mathematics

From the website: ‘The National Council of Teachers of Mathematics is a public voice of mathematics education supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development and research.’  

There are lots of research articles, professional development opportunities, lessons and resources. There is also an advocacy page with NCTM’s legislative platform.

AWM - Association for Women in Mathematics

From the website: ‘The purpose of the Association for Women in Mathematics is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity and the equal treatment of women and girls in the mathematical sciences.’

Association sponsored events include an essay competition, middle school and high school mathematics days, student charters, lectures, awards and charters.

5. 

Local Experiential resources: 

Bay Area Math Project - through Lawrence Hall of Science

IISME - Summer Fellowship Program with Lockheed Martin

Skyways math field trip - partnership through NASA and Hiller aviation

Math Midway - at Lawrence Hall of Science

Summer Teacher Institutes - Exploratorium

Books:

Mathematical Sorcery by Calvin C Clawson

A Tour of the Calculus by David Berlinski

Mathematicians are Peaple Too: Stories from the Lives of Great Mathematicians by Luetta Reimer

Problem Solving 101: A Simple Book for Smart People by Ken Watanabe

Poblem Solving through Problems by  Loren C Larson

Journals

Journal for Research in Mathematics Education

International Journal of Science and Mathematics Education

Conferences/Events

CMC-N Asilomar Mathematics Conference

National Conference on Singapore Math Strategies

Experiential anywhere:

I am still researching programs, but I want a higher degree in either MAthematics or Mathematics Education.

Museum of Mathematics (NYC)

Mathematikum (Giessen, Germany)

The Garden of Archimedes (Florence, Italy)

Puzzlemania Maths Exhibit (Glascow Science Center - Glascow, Scotland

Interdisciplinary Teaching and Learning July 17

Blog Assignment July 17

Interdisciplinary Teaching and Learning

One of the benefits of teaching as a second career is I have many programming and  engineering applications I have use in my classroom. This has been invaluable answering the incessant whine “When are we ever going to use this?”. I believe students are more vested and engaged when they can see the impact and importance of what they are learning. A topic covered in multiple disciplines carries more weight with my students. Math seems theoretical until they have a realistic application to which they can apply what they learn.   
 

Functions are covered in both Algebra 1 and Algebra 2. I have used cryptography as a real world application of functions (Algebra 1) and inverse functions (Algebra 2). Cryptography is the science of secure communication - codes, ciphers, security encryption. Encryption demonstrates the concept that a function assigns one output value for each input. I use a linear equation to mimic a private key encryption method. Deciphering requires students to find the inverse of the function. (I use my own experience and information for the activities which are on my work computer, but I found this essay that has a lot of good information http://www.ridex.co.uk/cryptology/#_Toc439908860 )

Another interdisciplinary topic I’ve used is binary code/assembly language to deepen understanding of number sense and place value as well as evaluating equations. We have an amazing physics and teacher at CHS who has built a robotics program - both classroom and extra-curricular. Students program their robots to accomplish different tasks. Binary code and assembly language can show how what they program communicates with their robots. 

I responded to the Minding the Gap blog entries of:
Jenna Joseph
JennaJosephMAT.blogspot.com

Mary Jane D'Arcy
mjdarcy.blogspot.com

Wilson Pang
misterwilsonpang.blogspot.com

Content Knowledge MAIT 402

Mind the Gap / Session 2

1.   In recounting her journey through many educational reforms, Diane Ravitch makes a number of provocative statements. Choose two, quote them, and personally respond.

“Market reforms have a certain appeal to some of those who are accustomed to “seeing like a state”. There is something comforting about the belief that the invisible hand of the market, as Adam Smith called it, will bring improvements through some unknown force. In education, this belief in market forces lets us ordinary mortals off the hook, especially those who have not figured out how to improve low-performing schools or to break through the lassitude of unmotivated teens.”

I don’t see how choice - charters and vouchers would have ever been a viable option for education. Comparing education to a free market should reveal serious contradictions not a plan of action.

Some of the principles that must exist in order for a free economic market to function are lacking or impossible in public education. Individual rights as well as fair and equal treatment must be assured for a market to naturally reorder. Understood in the concept of individual rights is the tenet that individuals will act in their own best interest. So, applied to education, all parties, administrators, teachers, students, parents and communities, have a predictable role to play. In public education, many are unable or unwilling to act in their own interest. Students or parents, due to language, economic or other barriers can’t enroll, apply or move to the best schools. Communities are financially unable to support local schools at a level equitable to more affluent districts. 

If, however, a free education market did exist, and was able to reorder. The best public and charter schools would survive, leaving the weakest, poorest schools to close. This leaves the students needing the most with the fewest options contradicting the free market principle of fair and equal treatment. 

*********

“I was also concerned that accountability, now a shibboleth that everyone applauds, had become mechanistic and even antithetical to good education. Testing, I realized with dismay, had become a central preoccupation in the schools and was not just a measure but an end in itself. I came to believe that accountability, as written into federal law, was not raising standards but dumbing down the schools as states and districts strived to meet unrealistic targets.”

First, this may be an unpopular opinion, but I do not like Diane Ravitch’s writing style. She has a spiraling, nonlinear organization with statements and statistics, and at times makes questionable vocabulary choices. Shibboleth? Really?

So the quote may not be that provocative. Most can see that standardized testing has morphed from a measure of accountability to the only method and measure. In my personal experience, the actions my school and department have taken to improve test scores do not improve overall education. We have adjusted pacing guides to the STAR test schedule. We eliminate enrichment concepts and activities that won’t appear on the tests. We use STAR test formatting (exclusively multiple choice) for common department assessments teaching students to take the test and not to think critically. We have completely eliminated preparatory classes because we are penalized for students in classes below the level of Algebra 1. 

The new common core standards seem to focus on standards for math practice that value critical thinking - problem solving, modeling, abstract as well as quantitative reasoning, constructing arguments, using appropriate tools and attending to precision. But the actual standards for particular concepts have changed little. I’m not sure how the additional standards for practice will change the nature or focus of the exams.

2.  How would you characterize a well-educated person? What should any well-educated person know in today’s world?

“A well-educated person has a well-furnished mind, shaped by reading and thinking about history, science, literature, the arts and politics. The well-educated person has learned how to explain ideas and listen respectfully to others.”

To Ravitch’s definition, I would add that a well-education person has an ability to reason, form plans and solutions to practical arithmetic, algebraic, geometric and spatial problems. Neglected in a list of concrete attributes a person should know would be the undefinable benefits of an education. Students form ethics and morals and standards of behavior throughout their school experience. Adolescent brains continue to develop until their early twenties. Synapses are connected during this development that an adult will use for reasoning, memorizing critical thinking.

3.  Thinking about the class discussion on the book, what stands out for you? What would you have liked to say that you did not say?

I can’t remember much discussion on how standards disintegrated into standardized testing.  I would have liked to have shared my anecdotal evidence supporting the disconnect between tests and standards. We have departmental common assessments written to mimic the STAR tests. They are multiple choice, worded similarly to released STAR questions, formatted to look like STAR tests. Last year I gave the departmental exams as well as my own exams for several tests. My exams changed just the numbers in the problems and eliminated the multiple choice options. My students averaged 25% higher on the STAR like tests. I contend that the STAR tests aren’t even accurately assessing mastery of concepts.

4.  Choose one gap you listed from your subject area and identify 3 resources that can help you fill that gap. List these and discuss what you learned from one.

My guiding question coincides with my gap in subject area (pedagogical) knowledge, so I may have had a head start. One of the new common core standards for mathematical practice is to make sense of problems and persevere in solving them. 

I have found that students are reluctant to solve word problems. They also have issues with problems that may not match exactly  problems they have already solved. If they do not know every step to solving a problem, they will often not even start. I want to know how to teach this concept/practice better.

Resources:

Web site:  www.mathforum.org

article:

Kapur, Manu. (2010). Productive failure in mathematical problem solving. Instructional Science: An International Journal of the Learning Sciences, v38 n6 p523-550 Nov 2010

book: 

Polya, G (1945) How to Solve It A New Method of Mathematical Method. (1945) Princeton: Princeton University Press 

www.mathforum.org is a website dedicating to problem solving. It is maintained by Drexel University. It includes methods of problem solving, teaching strategies, problems of the week and solutions, rubrics for grading problems.

5.  How were two of my annotated resources useful to topic/question?

Kapur, Manu. (2010). Productive failure in mathematical problem solving. Instructional Science: An International Journal of the Learning Sciences, v38 n6 p523-550 Nov 2010

Helpful in that it described a productive failure method of instruction that resulted in scores higher than the direct instruction/practice method. It may not be the best source though, just due to the small number of participants/students (75).

MathLeague.org competition. (2012, March) DVC, Pleasant Hill, CA Field Experience

Helpful in that it identified a gap between the best seniors at CHS and 8th graders at other competing schools. I want to find a way to prepare my math club kids to be competitive, in local competitions as well as in college and later. I also got practice problems and solutions used in the competition to help prepare for next year. 

     

Class discussion of Ball’s ‘The Subject Matter Preparation of Teachers’

One of the issues discussed in class was the ability to field students’ questions. Concern was raised about maintaining students trust if you can’t answer their questions. I don’t believe you need to or should be able to answer every question. Many of the best discussions and investigations in my classes have started with a question I could not readily answer. Students do not need a teacher to be omniscient. In fact, I’ve found students trust me more when they know I won’t tell them something I’m not entirely sure is accurate. They appreciate my saying “I really don’t know; let’s see if we can figure it out.”

One of the major issues I face in my math classes is getting students to let go of algorithms and tricks they have learned in previous classes. Ball states that “critical knowledge about mathematics also includes relationships within and outside the field - understanding the relationship among mathematical ideas and topics and knowing about the relationship mathematics and other fields.” Many teachers have a good understanding about the standards and concepts in their particular math subjects and teach the tricks necessary to solve problems in those areas. However, due to a weakness in overall or higher level math, they do not see where their subject fits within mathematics as a field. Many do not see that by teaching students one algorithm, they limit conceptual understanding and hinder learning extensions of their subject taught in later years. One example is teaching students to multiply binomials using FOIL without ensuring understanding of the concept. Students will then have difficulties factoring and multiplying polynomials with more than two terms. Another example is teaching and allowing students to change every fraction to its decimal equivalent. Students do not understand fractions conceptually. They are then unable to manipulate fractions which will affect their Geometry in proportions, Algebra 2 in rational functions, PreCalc and Calculus in limits and differentiation.  

Outside experience and subject matter knowledge

Unlike many in the class, I have little recollection of my high school math classes. I remember my Geometry teacher was the basketball coach, and that I sat behind the varsity first baseman in Elementary Functions. I was a fan. (Of the first baseman. Not the Elementary Functions.) What I do remember is the feeling that girls weren’t of any value in the upper level math classes. I was one of very few girls in Elementary Functions and PreCalculus. All the math teachers at my high school were male. I remember a relaxed atmosphere in classes, and the easy rapport the teacher had with the male students. I also remember the teacher not knowing my name at open house in May, after the entire year in his class. This feeling was exacerbated in college. I was in the third coed class after my private catholic school began accepting female students. Some of the professors were still obviously resisting the change. My Differential Equations professor was a Benedictine monk who refused to acknowledge any of the female students. When test score distributions were shared, our scores were not even considered. I responded by making sure I destroyed his curve on every exam, but most female students dropped the major.   So while Ball states “teachers influence students through their own engagement in ideas and processes”, their influence on me was less subject specific. I left high school and college with a drive to prove I was as good as anyone else in my field.

So almost all of my conscious acquisition of subject matter knowledge has come from actual teaching, professional development and non-school experiences. I am extremely lucky to work in a department of well educated, experienced professionals. We have daily lunch room discussions about the best ways to teach particular concepts as well as ‘spirited’ conceptual debates about mathematical concepts. Our last argument/discussion the last week of classes was about the derivation of the equation of an ellipse.

My two other outside school experiences that have contributed to my subject matter knowledge are my years as a software engineer and my lifetime of figure skating. I have drawn several lessons directly from programming - sorting exercises to order/understand rational and irrational values, cryptography to learn functions and inverses, binary/hexadecimal systems practice functions and number sense. Students are more engaged when I can get out of the textbook and answer their incessant “When are we ever going to use this?” with a concrete, real example. I can also honestly tell my students that I use math while teaching skating. (I don’t usually tell them about my skaters’ reactions - rolled eyes, groans, “school’s out” exclamations -  to my using vectors to describe how to correctly perform a turn or jump.)

Line of Inquiry

I have narrowed my line of inquiry down to two general possibilities. 

I am interested in how to get students to problem solve and actually think for themselves. Many students do not begin unless they know how to solve a problem completely. If they are not sure of the answer by looking at the problem, they do not even try. If there are more than four words in the directions, they skip the problem, and they do not even read the word problems.

My second possibility is how to prepare a math club. My math club has attended several competitions over the last 4 or 5 years. My bright, high achieving seniors have been completely demoralized by 8th grade students from Silicon Valley schools. I want to know how to bring math club, and by extension all students, up to a level that will be competitive.

I am behind in research, but the skating competition ends tomorrow, so I plan to be back on track after the weekend.

Content Knowledge Guiding Questions

1. What is the history of the math and mathematicians covered in Alg/Geom/Alg2?



Pedagogical Content Knowledge

1. How are successful math club/teams prepared for competitions?

2. What can be done to improve students' problem solving and critical thinking abilities in math?

3. Why is there gender inequality in higher level engineering/science/math courses and careers? How can this be addressed at the high school level?

4. What interventions for low performing students are successful? What methods/interventions make students independent?