Category Archives: Statistics

Blogs Related to Statistics

Trying to Increase Box Office Attendance

So I’m listening to NPR a few weeks ago and hear how they are attemping to increase new movie attendance by increasing frame rates.  They have this idea that the better the technology in the movies, the more likely we are to come.  The real issue isn’t us being bored at the movies, it’s the prices!  Stop increasing the 3-d and frame rate technology and make it more economical to attend.  I don’t see why a trip to the movies should cost me almost half a days wage.  I propose that pricing of movies should have some kind of function that determines cost.  I would think possible factors would be projected attendance and length of time in box office.  Strong movie goers would continue to watch premiers while increasing attendance to movies when they are unlikely to draw much interest. 

Other easy methods would be to transfer production costs of movies to customers.  Why should low budget films cost as much as others?  Why does a movie that has one celebrity playing multiple rolls cost as much as movies with multiple celebrities?

Quadrant Count Ratio and Pearson’s R in Education (GAISE)

Reading through the 2007 GAISE report, I’ve found some very cool teaching ideas and knowledge I didn’t know.  My first observation was that the report was bland and teaching elementary concepts, but as I progressed through the report I picked up on ideas to help incorporate in my own middle school and college classroom.  The report used some methods that I have used in my classes with the MAD to help students understand variation, but later used a Quadrant Count Ratio(QCR) to help students understand strength of a relationship.  This quadrant count directly relates to the covariance between two data sets, which is of course used in Pearson’s R.  I like the idea of using this to help students understand how Pearson’s R is either positive or negative relative to the relationship of the trend line.  I don’t see a page devoted to the QCR on wiki, but it does have some statistical usefulness (Holmes 2001).  If you have not heard of this method for relationship, it relates to what you see in many textbooks for an explation of the covariance between two random variables.  The scatter plot is divided into 4 quadrants based off of the mean for both variables.  The number of data points in quadrants that represent values that are both above the mean and both below the mean are added together and subtracted by the number of points in the 1st and 3rd quadrnants, then divided by the total observations to produce a number between -1 and 1.  The larger the number, the strong the relationship is positively.  It’s seem to be a mix between a parametric and non-parametric estimate on strength of a relationship.  It’s weakness lies in the fact that the values may lie anywhere within a quadrant and not particularly related to the linear relationship of the pairs of points.  It would be interesting to see how this relationship holds in extreme cases and any effectiveness of using the median instead of the mean to create quadrants.

Big Data

Looks like the world is changing fast!  A recent article by the New York Times emphasizes the importance of math and statistics in future jobs.  Math is the new cool thing!  http://www.nytimes.com/2012/02/12/sunday-review/big-datas-impact-in-the-world.html?_r=1&scp=1&sq=big%20data&st=cse

Ramblings

I have so many different thoughts rambling through my head constantly about a class I’m taking at Auburn University (Issues and Trends in Mathematics Educaiton) and how it will influence me as an educator.  Will all of this knowledge make me the Gingrich of math education?  Is there nothing new under the sun?  How can we learn from past mistakes?  Is the decline of American education related to reform practices or just correlated?

I’ve heard to be careful what you read or watch, as it will certainly impact your thoughts.  (Been reading John Dewey and The Decline of American Education)  I’ve read a study that says we believe everything we read or see until we have time to judge its truthfulness or how it may contradict we already hold as true.  There are so many different views of almost any topic or situation.  Philosophy and qualitative studies I wouldn’t say is my forte, but it is becoming increasingly interesting.  I’m really having to wrap my thoughts around different arguments and find out what parts are truth and what parts are noise from people just wanting to be heard.

What is truth?  How do we prove causation?  Qualitative and quantitative research, I believe, should be mixed not disjoint.  So many people in the world do not mix these together.  An example by Obama wanting to lengthen drop out ages in all states to 18 does not address the real issue at hand and is based purely on quantitative studies.  AL state senator, Shadrack Mcgill, says that increasing teachers pay will not increase the quality of teachers because teachers who teach will work for any amount of money, because they are called to teach according to the bible (Click Here for Article).  Old Shadrack seems to be basing his vote purely on qualitative studies.

How do we mix these two types of research?  This is essential if we want to show causation!

Issue Brief for Common Core State Standards for Mathematics

The Issue

In 2010, forty seven states and territories in the United States decided to unite as one voice for the use of common mathematical standards throughout the nation, the Common Core State Standards for Mathematics (CCSSM).  These standards have increased rigor and cognitive demand in many state curriculums in addition to remapping standards to different grade levels (Porter, 2010).  The combination of these new demands has provided increased stress for teachers in middle grades who come from varying backgrounds.  Experts within curriculum development and assessment at a recent conference in Arlington, VA noted a key recommendation for successful implementation of the CCSSM was to “focus attention to content changes at the middle grades (Garfunkel, 2011, p. 12). “  These expert participants claimed there were “substantial increase in the number and nature of learning goals… at middle grades. (p.12)” These increases have many current educators in need of professional development and future teachers in need of curriculum changes at the post secondary level.

Though how well individual states standards are mapped to the Common Core vary tremendously from state to state, a recent study by Porter tried to quantify these mapping using a content analysis procedure called the SEC, Surveys of Enacted Curriculum (Porter, 2011).  To understand how well the middle grades CCSSM standards are related to current state standards, it is important to understand the SEC method in slight detail.  The SEC helps determine the “extent to which cell proportions (topics by cognitive demand) are equal cell by cell across two documents (p. 104).”  These proportions will show direct relationship at one and no relationship at zero, hence stronger relationships will be larger decimal numbers between one and zero and weaker relationships smaller numbers between one and zero.  When the average proportional relationships from all states of grades kindergarten through eighth and high school are ranked from Porter’s analysis weakest to strongest of the ten proportions, grades five through eight represent the 4 weakest related standards (see chart below).

National Proportions

             Weaker                                            Stronger

Grade

6

5

8

4

7

9-12

1

3

K

2

Proportion

.19

.21

.22

.22

.23

.24

.25

.27

.31

.34

Table 1:  SEC analysis on relationship of CCSSM to average state standards from Porter ranking grades from weakest to strongest relationship (p. 106)

Although these are averages across all states, these proportions represent a shift of content knowledge and new standards for many and most states.  The Institute for Mathematics and Education’s workshop, “Gearing Up for the Common Core State Standards in Mathematics,” focused on pivotal standards for grades k-8 professional development (2011).  Their emphasis in grades six and seven were on ratios and proportional relationships, and grade eight’s highlighted domain was geometry.  The emphasis for ratios and proportional relationships comes in view of the fact that “the language of the standards is different from what teachers are used to” and “standards that calling for understanding rather than a problem solving type problem may be challenging. (p. 8)” Grade eight’s highlight in geometry is essential because of the concepts being based on transformations, “an approach that is significantly different from previous state standards. (p. 9)”

These proportions representing content shifts and new standards with new mathematical practice standards directly relate to many middle grades teachers’ concerns on successful implementation of the CCSSM.  On top of elementary education majors reporting the highest rate of mathematics anxiety of any college major, new or shifting standards can make many teachers overwhelmed (Beilock, et al, 2010; Hembree, 1990).  A study on the effects of teacher certification on student achievement indicated “that the majority of middle grades teachers who instruct in core subject areas are certified in elementary education, a handful are certified in a secondary subject area, some have a specific middle grades certification, and some are certified in special education or other fields (Young, 2009).” Similar studies suggest that teachers who are certified in a secondary subject have greater student academic growth in mathematics and science than those whose teachers are elementary certified (Hawk, Cobble, & Swanson, 1985; Mandeville & Liu, 1997).  Attention from states and schools districts should focus on sixth grade teachers’ relationships to standards not only because of the weakest relationship to prior standards (.19 for Alabama sixth grade alignment), but because of the highest levels of standards for teachers with elementary certification in most states.

History of Certification

It may be surprising to know that certification requirements of school teachers have varied largely throughout the years.  This variation has had a direct impact on the content and pedagogical knowledge of teachers.  The beginning of US History to the early 20th-century had no mandated requirements for teaching professionals in essence requiring school districts and boards to govern the hiring of competent professional educators (Mirel, 2011).  In the first half of the 20th-century colleges of education became integral parts of American universities and had many differences in how uniting specialists in subject matter and pedagogical techniques should be merged.  There was a large divide between the liberal arts schools and schools of education; “for most of the 20th century, dialogues between “ed school” faculty members and their liberal arts colleagues about how to train prospective teachers were scarce (p.7)” In the early 1900s, many state governments made “schools and colleges of education the main institutions legally permitted to train perspective teachers. (p.7)” The split of these two portions within secondary education, largely continued today, has painstakingly required elementary teachers to take many of their courses in the school of education and lack course work within the liberal arts.  Currently, approximately twenty states have tried to lower this divide by offering a middle grades certification with one to two content area focuses (Mandeville, G., & Liu, Q., 1997).

Progressive movements from as early as 1902 by John Dewey have sought to reshape curriculum standards and pedagogy.  He urged mathematics reform to learn mathematics using real world applications and not just memorize abstract mathematical formulas.  Other reformation activities continued approaches by Dewey but problems persisted in curriculum and disciplinary knowledge of teachers.  Paul Hanna, an education activist from 1934, stated “ I struggled for a long time to get some kind of structure that did not represent merely the traditional categories of economics, political science, sociology, anthropology, history, and geography, because these would scare most teachers not having had anything in these fields (Halvorsen, 2007).” Interestingly, many of these reforms failed to take hold for two large reasons, many elementary teachers did not have the liberal arts knowledge necessary to teach new curricula and many programs did not provide adequate resources for professional development to aid teachers in proper implementation (Church, R. & Sedlak, M., 1976).  If higher education institutes training teachers, organizations fostering professional development and other like minded stake holders in education do not take note of the past, we will be doomed to repeat it.

Responses to Issue

            Teacher preparedness to implement the CCSSM standards effectively currently can be addressed in two major ways, teacher preparation programs and professional development.  The Science and Mathematics Teacher Imperative (SMTI) and The Leadership Collaborative (TLC) says that not only should higher education focus on aligning higher education curriculum with K-12 curriculum, but be involved in preparing and educating teachers, both prospect and practicing, and conduct research related to the implementation of the CCSSM  (2011).  A recent survey by the Center on Education indicated that only half of higher education institutes foresee changing teacher academic or pedagogical content by 2012 while the others see this by 2013 or later (2010).  Teacher preparation to teach the CCSSM must be addressed within university settings quickly to limit teacher knowledge gaps.

            The alignment of higher education curriculum should be addressed in multiple ways.  Assessment from the Partnership for Assessment of Readiness for College and Careers (PARCC) and Smarter Balanced Assessment Consortium (Smarter Balanced) should be used in some facet for college admission and placement (SMTI, 2011).  Using these assessments can help colleges identify individuals to recruit for middle and high school education programs.  Lastly, and most importantly, the “content of introductory courses in higher education… should be revised to ensure alignment with the content of the high school curriculum, to build on what high school students have already learned and challenge to learn more. (p. 2)”  Just as there are clear expectations for students in k-12 through the CCSSM, there should also be coherent outlines of the mathematics that teachers need to effectively teach the CCSSM (SMTI, 2011).

            Opportunities for teachers to master this new, more challenging content, can be developed in largely three ways says the Conference Board of Mathematical Sciences:

  • Immersion experiences… Mathematical habits of mind, mathematical practices, and mathematical disposition,…   Such experiences may be summer institutes, year-long professional development, online mathematics experiences, or incorporated in undergraduate courses.
  • Greater emphasis on field and clinical experiences
  • Professional learning communities… teachers at all levels, mathematics (at two and four year institutions, and mathematics educators (CBMS, 2011, p.15)

These are experiences that many have been accustom to in East Alabama from undergraduate practicum experiences, Transforming East Alabama Mathematics summer institutes, Alabama Math and Science Technology Initiative, and organizations such as Alabama and East Alabama Conference for Teachers of Mathematics.  These types of organizations build a community of professionals that share ideas, share manipulatives, and build stronger teachers.  Experiences for future teachers inside the classroom help prepare for student questioning, handling classroom management, and treating misconceptions of future students before and after they arise.  Continuation of these programs requires funding and in unsure economical times they should not be disregarded.  It is imperative that these type organizations continue their development of school personnel in professional development related to the CCSSM. 

             Though there are many great things coming from these different organizations and events, many of these organizations professional development activities have room for improvement.  Largely, teachers learning standards well enough to teach understanding rather than methods should be addressed.  Many middle grades teachers have never learned fundamental implications of and reasons for using different methods for central tendency or aspects of distributions to name just two standards from the middle grades CCSSM.  Activities related to field experiences, professional learning communities, and immersion experiences largely focus on mathematical practices rather than standards, leaving an unfulfilled gap for middle grades teachers’ knowledge of standards.  These organizations’ leaders and university personnel should acknowledge the fact that they have graduated students who are unprepared to teach specific standards through misalignment of their own curriculum.  It is important that these avenues for teacher development also include rigorous review of standards that ensure teacher understanding of content at more than a surface level.  When teachers understand material themselves, they are more likely to teach fundamental understanding to students and help students make better reasoning for practical implications.

            The variation from state to state and teacher to teacher makes a one size fits all professional development strategy impractical, but is a great starting point for many districts.  Districts and universities who have been making strides in research based mathematical practices should strive to do more.  It is actually ironic to read how many articles emphasize that professional development should focus largely on mathematical practice standards within the CCSSM when so much relationship to understanding of standards is linked to student achievement.  Teachers are encouraged to differentiate instruction in k-12 education, but how often does one see differentiated instruction in higher education classes in liberal arts or in the education department?  How often has one been to a seminar, conference, or other professional development setting that implemented differentiated instruction?  Are higher education and professional development sessions excluded from good educational practice of meeting learners where they are?  Organizations offering professional development and training for educators should strive to make sessions that meet each teacher where they are in either content or pedagogical illiteracy.

Conclusion

Overall the nation is heading in a great direction by creating rich standards that are student centered on research based standards and practices.  Ensuring these standards are met with excitement and reliable teachers should continually be addressed.  There is still future study needed for the study of statistical factors that contribute to student achievement in middle grades in alignment with the CCSSM such as certification, teaching methods, content level proficiency, and others.  Professional development through local school districts, higher education initiatives, state programs, etc. for current middle grades educators should focus not only on mathematical practices but on standards as well.  Higher education should work on alignment of its own curriculum and teaching programs to the CCSSM and reach out to local schools in proper implementation techniques.  With stake holders within education in all different areas working together for the education of our future leaders, we will continue to be the strongest nation in the world.

Bibliography

Beilock, S., Gunderson, E., Ramirez, G. & Levine, S. (2010). Female teachers’ math anxiety

affects girls’ math achievement. Proceedings of the National Academy of Sciences,

USA, 107(5), pp. 1860- 1863. Retrieved from http://www.pnas.org/content/

107/5/1860.full

Conference Board of the Mathematical Sciences (2011).  Common standards and the

mathematical education of teachers:  Recommendations from the October 2010 forum on content-based professional development.  Washington, D.C.:  Conference Board of the Mathematical Sciences.

Church, R. and Sedlak, M. (1976).  Education in the United States.  Dow Schoolhouse Politics.  .

New York:  The Free Press.  14-417. p. 263-264. 

Garfunkel, S., Hirsch, C., Reys, B., Fey, J., Robinson, E., & Mark, J.  (2011).  A summary report

            from the conference “Moving Forward Together:  Curriculum & Assessment and the

            Common Core State Standards for Mathematics”.  COMAP.

Halvorsen, Anne-Lise. (2007). The Origins and Rise of Elementary Social Studies Education,

1884 to 1941. p. 317-319.  Retrived from

http://books.google.com/books?id=tz4OrzFDk98C&pbooks.google.com/books?isbn=0542920522.

Hawk, P., Cobble, C., & Swanson, M. (1985). Certification: It does matter. Journal of Teacher

Education, 36(3), p. 12-15.

Hembree, R. (1990). Th e nature, eff ects, and relief of math anxiety. Journal of Research in

Mathematics Education, 21(1), p. 33– 46.

Institute for Mathematics & Education (2011).  States’ progress and challenges in implementing

common core state standards.  Washington, DC:  Center on Education Policy.  Retrieved  http://commoncoretools.files.wordpress.com/2011/05/2011_05_07_gearing_up1.pdf.

Kober, N., Rentmer, D., & Center on Education, P. (2011)  State’s Progress and Challenges in

Implementing Common Core State Standards.  Center on Education Policy. 

Mandeville, G., & Liu, Q. (1997). The effect of teacher certification and task level on

            mathematics achievement. Teaching and Teacher Education, 13, 397-407.

Mirel, Jeffrey.  (2011). Bridging the “Widest Street in the World”:  Reflections on the History of

Teacher Education.  American Educator, 35(2), 6-12.

Porter, A., McMaken, J., Hwang, J., & Yang, R. (2011).  Common core standards:  The new US

intended curriculum.  Educational Researcher, 40, 103-116. 

Science, and Mathematics Teacher Imperative/The Leadership Collaborative (2011).  Common

 core state standards and teacher preparation:  The role of higher education a discussion

draft by the SMTI/TLC working group on common core state standards.

Young, B. (2002). Common core data: Characteristics of the 100 largest public elementary

and secondary school districts in the United States: 2000-01 (NCES 2002-351). Washington, DC: U.S. Government Printing Office.

New Endeavors

For those of you who aren’t close to me, I’ve recently been accepted into the Teacher Leader Academy at Auburn University.  I’m really excited about this, not only because it’s some extra income for my family, but they pay for my graduate studies completely.  I’m going to have the chance to not only pursue my interest in statistics but to pursue them in a meaningful way in education.  I’m going to be able to merge my loved profession of education with my knowledge of statistics.  This is going to be a great time for me to actually pursue these interests with about 41 states, including Alabama, adopting the Common Core State Standards for Mathematics.  There is some definite increase in rigor and cognitive demand for students and teachers alike.  Hopefully I will be able to share some of my statistical and pedagogical knowledge with teachers around the nation as being a part of the T.L.A.

Set Theory

I just completed a lesson with my 1st period class on the cardinality of the intersection of sets. This is not something in my course of study, but I thought it would be a great activity for a day when many of my kids were on a field trip. It was amazing how well a student in my 7th grade class did. She represented the cardinality accurately for two sets and three with no prior experience with sets besides yesterdays lesson involving intersections, unions, and subtraction. I challenged her to show me the equation for the cardinality of the intersection of 4 sets in attempt for her to find a pattern in these formulas. I love it when kids climb farther than what I anticipate.