Teaching Statistics: Textbook Considerations

I have the pleasure of teaching an undergraduate basic statistics class this fall for the third consecutive year. It's not a class I had any specific preparation to teach, but I've tried to make up for that by becoming familiar with some of the statistics education literature, bolstering my content knowledge (although I doubt it will ever be as wide or deep as I'd like), getting access to good resources, and being mindful of the needs of my students.

First, it would help to know a little bit about the course. Most strikingly, the class only meets once a week on a Thursday from 4:30 to 7. If you're used to teaching 180-day school years, you really have to wrap your head quickly around the idea that you're only going to see these students 15 times before finals. Also, despite the class being taught in the School of Education, it's not required of any education students. Instead, the class consists mostly of students from two majors: Sociology and Speech, Language, and Hearing Sciences. Honestly, most of them admit to avoiding math classes, but they usually need the stats class to apply for graduate school. As for the content of the course, here is how it is described in the university catalog:

Introduces descriptive statistics including graphic presentation of data, measures of central tendency and variability, correlation and prediction, and basic inferential statistics, including the t-test.

And that's it. As someone who works almost daily with the Common Core State Standards, building a course around such a sparse description would be quite a challenge, especially for a first-time instructor. When I talked to Derek Briggs about teaching the course, he advised that I use his preferred text, Statistics by Freedman, Pisani, and Purves. I'd recently used Agresti and Finlay's Statistical Methods for the Social Sciences for my qualitative methods courses, and while that book suited me pretty well, I was open to something different so I ordered the Freedman text for my class.

In hindsight, the Freedman text was fine, and the Agresti text would have been fine, too. Both were decently well-written and had plenty of problems to assign, but that's the thing — I was looking for a text that offered considerably more than explanations followed by problem sets. I really wanted something that supported students working together in groups during class, making sense of the material as we went along.

One book that had gotten my attention was Workshop Statistics: Discovery with Data by Rossman and Chance. I recognized Beth Chance's name immediately from some of the stats education literature I'd read, and felt good that this text would offer what I was looking for. I used the text last year and was not disappointed, and will be using it again this year. Below is a summary of some of the reasons I like Workshop Statistics.

Context Continuity

In the front matter of the book, Workshop Statistics contains a list of activities by application — in other words, they've categorized all the problems by context and indexed exactly where those contexts get used. The list of related problems appears again with each problem in the text (inset in picture above), so it's easy for me or my students to refer back or forward to where that context appears. I believe in teaching mathematics rooted in context when possible, so I found this an especially helpful way of finding problems that might be relevant or interesting to the students in my class.

Preliminaries

Every topic (lesson) in the text opens with some preliminary questions. Some involve data collection, which is great, but at the very least it gives students an opportunity to consider a question and how we might answer it. If Dan Meyer has made anything clear, it's that we shouldn't teach math as finding answers to questions that nobody has bothered to ask.

In Brief

The end-of-topic summary certainly isn't unique to this text, but the "You should be able to" statements are very handy for writing objectives for standards-based grading. (I hope to write about my SBG approach in a future post.)

Online Supports and Simulations

Besides both online instructor and student resources, the text uses a number of custom applets that often really help illustrate some of the concepts in the course. Some are Java, but a number have been converted to JavaScript for use on more platforms. I've avoided having students use software beyond a spreadsheet, and some of these applets have saved us from having to purchase SPSS (expensive!) or trying to use R (steep learning curve!).

Activities

The in-class activities use some interesting contexts and support groups working together. If anything they can be a bit over-scaffolded, but that relieves me from having to lecture much and I can spend most of my time going group-to-group in the classroom and dealing with questions more intimately.

Overall

There are a number of smaller things that I'm fine with, although they aren't deal-makers or deal-breakers. The pacing of the text is good — if we cover about two topics a week, we finish the text and pretty much everything one would expect in a basic statistics course. The order of the topics is sensible, too. Typically, it makes sense to put descriptive statistics before inferential statistics, and to work from one-variable stats to two-variable stats. This book is no different. Some texts put linear regression earlier, and where probability should land in a book seems to be negotiable. The placement of those topics in this book is fine for this course and the progression from topic to topic was very manageable.

Other than my first day activity, I haven't written much about teaching stats, but look for me to change that this semester.

NCTM Denver 2013: Johnson & Thomas's Statistical Reasoning in the Middle School

Annual Meeting - Thursday, April 18, 1:00 pm

+Raymond Johnson (Me!) - University of Colorado Boulder and Freudenthal Institute US
+Susan Thomas - University of Colorado Boulder

My colleague Susan Thomas and I took an interest in middle school statistics during our first year in our PhD program. When we read the statistics standards in the Common Core State Standards (CCSSM) for middle school, we both thought they were a major shift from prior standards, and we wondered how teachers would perceive them. For our qualitative research project, we found two willing middle school teachers who offered us access to their classrooms and time for interviews. The data we collected suggested that these teachers, both well-educated and in a high-achieving school, had some difficulty interpreting some of the standards, were unclear about some of the bigger ideas those standards might lead to, and did not have curriculum to support the level of thinking and reasoning called for in the CCSSM.

The push for more statistics is reflected in each iteration of major standards documents (NCTM 1989, 2000), the GAISE report, and the CCSSM. Some of this comes from the relative youth of statistics as a field. I've heard teachers say, "Why do we teach boxplots? We didn't learn them when I was in school," not realizing that John Tukey didn't invent them until the 1970s -- in many cases, the teachers are older than the plots. How often as math teachers do we get to teach anything younger than we might be?

More obviously, the push for statistics is part of the explosion of data in our world. Google's Eric Schmidt recently claimed that "There were 5 exabytes of information created between the dawn of civilization through 2003, but that much information is now created every 2 days, and the pace is increasing" (Kilpatrick, 2010). Even when the topic isn't "big data," increasingly we collect more and more information about our own activities. Making sense of data requires statistical thinking and reasoning.

To understand statistical reasoning, it's helpful to have a way of thinking about how statistics is different from mathematics. In 1962, John Tukey wrote, "Statistics is a science in my opinion, and it is no more a branch of mathematics than are physics, chemistry, and economics; for if its methods fail the test of experience -- not the test of logic -- they are discarded" (pp. 6-7). A more recent and helpful perspective of statistics comes from Michael Shaughnessy, who said "The twin sister of the 'certainty' in mathematics is the 'uncertainty' in statistics. We must prepare our students to deal with both types of quantitative reasoning as they grow in the mathematical sciences" (2010).

So what is statistical reasoning? According to delMas (2004), statistical thinking is knowing when and how to apply statistical procedures, while statistical reasoning explains why results were produced or why a conclusion is justifed. That means that things like stating implications, justifying conclusions, and making inferences are all part of statistical reasoning.

For this workshop, the strategy Susan and I used was to: (a) identify in each grade level of the middle school CCSSM a theme related to statistical reasoning, (b) find something in the research literature that might help teachers better understand what kind of reasoning to elicit and look for in their classes, and (c) evaluate some statistical tasks for their ability to elicit that reasoning. Susan got snowed out on the wrong side of Vail Pass, but thanks to our preparation together I was able to soldier on without her.

Grade 6: Variability and Distribution

A look across 6th grade standards

When we looked at the 6th grade standards, we saw a common theme of reasoning with variability and distribution. Yes, there's focus on measures of center and procedural things like knowing how to calculate the IQR and MAD, but all that is supported by reasoning with variability and distribution.

The work of Bakker and Gravemeijer (2004) stuck out to me in how they approached student reasoning with variability. "An underlying problem is that middle-grade students generally do not see 'five feet' as a value of the variable 'height,' but as a personal characteristic of, say, Katie" (Bakker & Gravemeijer, 2004, pp. 147-148). The suggestion was for students, instead of always assembling data points into distributions, had opportunities to deal with distributions that obscured individual points. By focusing more on distributions and variability, students will be more prepared later to reason with standard deviations, margins of error, and other topics in high school and beyond.

The tasks we chose to inspect were:

• Mean, Median, Mode, and Range
• College Athletes
• How Long Are Our Shoes? (ASA link)

Often the best part of leading a workshop is simply giving teachers time to talk and share ideas. If you look at the tasks and want to share ideas, please do so in the comments.

Grade 7: Sampling and Inference

A look across 7th grade standards

Across the 7th grade standards we see a strong theme for reasoning with sampling and inference. The focus on informality here is key, as a standard like 7.SP.B.4 is not trying to suggest hypothesis testing and t-tests, but instead a basic understanding for how variability in samples indicates uncertainty about population parameters, either as a single sample or in a comparison of samples.

Two items from the research stood out to me here. First, Rubin, Bruce, and Tenney (1991) claimed that "Over-reliance on sample representativeness is likely to lead to the notion that a sample tells us everything about a population; over-reliance on sample variability implies that a sample tells us

nothing" (p. 315). Getting more specific about the notion of sampling variability, Saldanha and Thompson (2003) found that higher-performing students "developed a multi-tiered scheme of

conceptual operations centered around the images of repeatedly sampling from a population, recording a statistic, and tracking the accumulation of statistics as they distribute themselves along a range of possibilities" (p. 261). Together with reasoning of variability developed in 6th grade, these standards address the crux of inferential statistics: sampling distributions.

The tasks we inspected were:

• What's Your Favorite Subject?
• Counting Trees
• Candy Bars

There was a lot of good discussion about the Counting Trees task and speculation around student strategies. There's a clear intersection here with students' reasoning with ratio and proportion, and one teacher suggested instead of using the diagram in the task, using aerial or satellite photography with real trees. The key idea, however, is giving students opportunities to sample in different ways and reason why different approaches yeild different, yet similar estimates for the population.

Grade 8: Covariation

A look across 8th grade standards

The clear unifying theme across 8th grade CCSSM standards is covariation of two variables. There is some real opportunity to have this support 8th grade algebra standards related to graphing lines and linear equations, and I'm anxious to see curriculum that really ties the two together.

From the research, I noticed two key things. First, Konold (2002) judged that "It seems unwise, for example, to specify ... that by middle school, students will learn how to 'make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots' (NCTM 2000, p. 248)" (p. 5). Konold isn't saying that 8th graders can't understand scatterplots, but rather that middle school students understand covariability in a variety of ways, and will develop their own alternative representations before reasoning with scatterplots. In similar work, Moritz (2005) found that students with incomplete understanding will often focus on the variabilty of one variable but not the other, or the variability of both without the association between the two. Some examples of alternative representations:

An informal and creative bivariate table

Paried and sorted case value plots

Scatterplot slices

All of the above are non-scatterplot examples that appropriately show reasoning with covariation. The tasks we inspected that deal with covariation included:

• Hand Span and Height
• US Airports
• Scatter Diagram

We ran short on time for much of a discussion on these tasks, but participants were looking at how scatterplots were used and thinking about how else students might reason with covariation. Of course, none of the tasks used in the session were meant to be perfect, and none teach themselves. Instead, it's important to find the most promising tasks we can, understand them as part of a greater curriculum, and attend to the student thinking and reasoning they might elicit.

Lastly, here are the resources we recommended at the end of the workshop. By no means is this an exhaustive list. It's just a place to start for teachers looking to supplement or revise their curriculum.

• Bridging the Gap Between Common Core State Standards and Teaching Statistics (ASA link)
• Mathematics Assessment Project/MARS
• Illustrative Mathematics
• AIMS Project

(Link to our presentation, source files, and handouts.)

References

Bakker, A., & Gravemeijer, K. (2004). Learning to reason about distribution. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 147–168). New York, NY: Kluwer.

delMas, R. C. (2004). A comparison of mathematical and statistical reasoning. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 79–95). New York, NY: Kluwer. doi:10.1007/1-4020-2278-6_4

Kilpatrick, M. (2010, August 4). Google CEO Schmidt: “People aren’t ready for the technology revolution”. Readwrite. Retrieved from http://readwrite.com/2010/08/04/google_ceo_schmidt_people_arent_ready_for_the_tech

Konold, C. (2002). Alternatives to scatterplots. Proceedings of the Sixth International Conference on Teaching Statistics (pp. 1–6). Cape Town, South Africa: International Association for Statistical Education. Retrieved from http://www.stat.auckland.ac.nz/~iase/publications/1/7f5_kono.pdf

Moritz, J. (2005). Reasoning about covariation. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 227–255). New York, NY: Kluwer. doi:10.1007/1-4020-2278-6_10

Rubin, A., Bruce, B., & Tenney, Y. (1991). Learning about sampling: Trouble at the core of statistics. In D. Vere-Jones (Ed.), Proceedings of the Third International Conference on Teaching Statistics (pp. 314–319). Voorberg, The Netherlands: International Statistics Institute.

Saldanha, L., & Thompson, P. (2003). Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, 51, 257–270.

Shaughnessy, J. M. (2010). Statistics for all -- the flip side of quantitative reasoning. NCTM Summing Up. Retrieved from http://www.nctm.org/about/content.aspx?id=26327

Tukey, J. W. (1962). The future of data analysis. The Annals of Mathematical Statistics, 33(1), 1–67.

A First Day Statistics Activity

I have the honor of again teaching our undergraduate statistics course in the School of Education, better known here as EDUC 4716 Basic Statistical Methods. Perhaps the most interesting thing about the course is that it's not required for any education programs, minors, or certificates. Instead, the course attracts students largely from the Department of Speech, Language, and Hearing Sciences (who don't need it to graduate, but do need it to apply to grad school or, more recently, to get certified) and sociology majors. So how does this course end up in the School of Ed? Probably due to the legacy we have in quantitative methods, thanks to people like Robert Linn, Gene Glass, Lorrie Shepard, and now faculty like Derek Briggs and Greg Camilli. Somehow all of their hard work and success filters down and gives a relative stats-hack like me a chance to teach undergrads.

Many of my students are upperclassmen and have spent much of their college experience avoiding math courses. In fact, on last year's FCQ (Faculty Course Questionnaire) my students' average rating for the item "Personal interest in this subject prior to enrollment" was a 1.8 out of 6 -- a response the university tells me is at the 0th percentile across campus. I like to think of this as a great opportunity in a "nowhere to go but up" kind of way, a chance for me to change the way students think of mathematics and see themselves as mathematical beings. Then again, it's hard to make big changes in only 15 class meetings of 2.5 hours each. If I'm going to make a difference, class has to get off to a solid start.

My opening activity this year started with the preparation of four simple index cards with different distribution shapes:

Four common distributions, clockwise from top left: normal, left skewed, right skewed, and normal.

I have the benefit of a small class of 14 students. So I cut my graphs into a total of 14 pieces:

14 pieces for 14 students. Note on the bottom I've provided the hints A, B, C, and D.

When class started, I mixed up the graph pieces and handed one to each student. Then I told the class to find the other people in class who had the graph pieces that aligned with theirs. Once they had a completed graph, form a group at one of the tables and discuss which of the following they thought their group's graph might describe:

• People born each month of the year
• Student GPAs at this university
• Student heights at this university
• Starting salaries of new graduates from this university

It took my class less than 3-4 minutes to find their groups and then I gave them another 3-4 minutes to discuss what their graph shape might describe. As a class, I had each group share their ideas and then we discussed them. Not everybody agreed initially about which shape matched which description, which led into important comments about how we might think about unbiased sampling of students and imagining different scales and labels along the horizontal axes.

So in less than 15 minutes I combined group-making, statistics, and active, student-centered problem solving into one activity. This activity also gets students thinking about distribution shapes, which I sometimes worry we ignore in the rush to calculate centers and spreads. If you're wondering how to adapt this for your classroom, I offer these suggestions:

• If you have a few more students, cut more slices.
• If you have twice as many students, consider making two of each distribution shape and scaling the x-axis to match one of 8 potential descriptions. (i.e., a normal distribution scaled for heights in inches could be distinguished from one scaled for SAT scores.)
• If you want to use this for Algebra 1, you can make graphs that describe things like, "Toni walked to the bus stop at 2 mph, rode the bus at 30 mph to the bike shop, then rode a bike back home at 12 mph." Such an activity begins CPM's Algebra Connections and was the inspiration for my activity.
• If you want to use this for Algebra 2 or higher, you can use graphs of functions that students will become familiar with (parabolas, cubics, hyperbolas, etc.). I don't think it's worth fretting over vocabulary at this point -- just give students an opportunity to think about how the functions behave and what phenomena they could possibly model.

Bonnie H. Litwiller, 1937-2012

I got word tonight that my undergraduate advisor, Bonnie Litwiller, passed away a couple days ago at the age of 74.

As a freshman at UNI, I had a temporary advisor until my program became more certain. After declaring as a math education major, Bonnie Litwiller was assigned as my advisor. I knew nothing about her. I remember asking Ed Rathmell, whom I had gotten to know while applying for a scholarship, what to expect from Litwiller as an advisor. I remember his response: "If you listen to her and do what she asks, she's great. She'll have your back when you need something. But don't cross her."

That's an uneasy way to know someone before you even get a chance to meet them. It felt like a description more fitting of mafia boss than a professor. But Rathmell's advice, as usual, was solid. Litwiller proved to be tough, and she made it clear to us that being a good math teacher was hard work. She set a good example: she and her research partner, David Duncan, would set aside a day a week where they'd lock themselves away in the library and write. As UNI isn't a top-level research university, the research activities of professors aren't always visible to the students. But Litwiller's dedication to research was clear, and there was no topic too small or journal too obscure. If she thought she had knowledge that would improve the teaching and learning of mathematics somewhere -- anywhere! -- she would write and submit for publication. She continued to write and publish even after her retirement from UNI in 2000, eventually passing the almost unfathomable mark of 1000 scholarly publications.

I took two classes with Litwiller, Teaching Middle School Mathematics and Teaching High School Mathematics. The classes were tough due to Litwiller being both picky about the quality of our work and her lack of clarity in describing what she wanted us to do. Some of us thought she was just being careless with her assignments, but I always wondered if this wasn't somehow purposeful. Either way, it was clear that she didn't want to do a lot of hand-holding. Some of us, ever so quietly yet respectfully, referred to her as the bulldog. A trusty companion that might just bite if you got out of line. If you didn't have the initiative and sense of responsibility to do quality work, I think she wanted a way for you to sort yourself out of the program. It happened, too; every semester some classmate would go missing and we'd try to find out what happened. Inevitably, someone would say, "They couldn't cut it. Litwiller dropped them from the program." You hear a lot today about colleges adopting GPA or test score requirements to improve the quality of their education majors. We didn't have those -- we had Litwiller. And just like letting a GPA decide who can be a teacher, I'm sure her judgement wasn't perfect and mistakes were made. (An acquaintance of mine, who shall remain nameless but now holds a PhD in math education, told me about a narrow escape from Litwiller's axe after a dispute over access to a local school.) But I think Litwiller had a sense for the quality that people expected from a UNI-prepared teacher, and a sense for giving us some survival skills that would get us through our first few years of teaching. There must have been far more successes than failures, too -- by the time I graduated in 1999, someone had estimated that a quarter of all the math teachers in the State of Iowa had been taught by Bonnie Litwiller.

My appreciation for Litwiller and her work has grown through my years first as a teacher and now as a graduate student in mathematics education. It was she who first introduced me to the NCTM Standards, and her direction of the NCTM Addenda Series was and still is an enormous contribution to the field of math education. What I believe was originally intended to be a six-book series to support the Standards grew into 22 total books, each designed to take the research behind the Standards and turn it into something teachers could use. Litwiller might have been the director and not the author of the Addenda Series, but it carried her trademark: getting as much useful information into the hands of teachers as possible. She gave me two books from the series, the middle school and high school books about statistics and data analysis. It was the first time I really thought about statistics education, and it's since become the area of school mathematics I find most interesting.

The world will miss Bonnie Litwiller, but she didn't leave without making a mark, both on the field of mathematics education and on me. Teacher education is a challenging business, and it's probably best to judge it with a certain amount of hindsight. For all of her toughness, she did have my back when I needed it, just as Ed Rathmell said she would. I may not have learned all that she tried to teach me, but maybe her most important lessons -- a sense of dedication and rejection of "good enough" -- have been most helpful in getting me to where I am today.

Sorting Out the Summative: When Standards-Based Grading Meets the End of the Semester

Source: Wikipedia

Many teachers who choose to use standards-based grading eventually find themselves facing the reality of their school's grading policies and tradition: the expectation of final, summative grades that are reported as percentages and letters. So regardless how hard you try to focus on quality feedback instead of grades all semester long (for good reason), there comes a time when, for reasons probably beyond your control, you have to turn levels and descriptions of student understanding into numbers. This is SBG's "Monday Morning Problem" that doesn't always get addressed in theory. But this week is finals week for my basic statistics students, so for me the time has come to convert standards-based formative grades into a summative grade, including calculating final exam grades. Here I'll try to describe the two steps I'll take to calculate my students' grades: (a) conversion of their formative scores into a summative score and (b) scoring and inclusion of the final exam into their semester grades.

Formative to Summative
Besides giving students a lot of written and verbal feedback about where they should try to improve, I've been using the simplest of measures to record their performance on class objectives: either students (a) "get it," (b) "sort of get it," or (c) "dont' get it/haven't demonstrated it." You could think of these as "green light," "yellow light," and "red light," respectively. I've tried discerning more levels of understanding in a gradebook and it only seems to lead to confusion and indecision (both for me and students), so I'm sticking to three levels, as suggested in Her & Webb (2004). If I need more detail, I can always go back to the copies of the work students have submitted and the comments I've made.

The gradebook we have for class is pretty primitive and as far as I can tell it only accepts numbers, so I mark my three levels as either a 2, a 1, or a 0. It doesn't take much explaining to students that a 1 shouldn't be viewed as "out of two" and therefore worth 50%. I do tell them, though, that in order to receive credit for the course they should average a 1 across all objectives. In other words, you can't pass the class without an average of at least some understanding of every objective.

Around here and in many other places, 70% seems to be the low end of passing grades. (We're not messing with Ds.) So if a student with all 1s should get at least a 70%, and a student with all 2s maxes out at 100, and we choose a linear function between the two, the "conversion formula" to percentages is simply:

percentage = 30 * objective score average + 40

If you feel a little dirty at this point because you know you just reduced all the various skills, knowledge, and abilities of your students into a single number, I say join the club. If you didn't feel that way I wouldn't have expected you to be using standards-based grading to begin with.

A "No Surprises" Approach to Final Exam Grades
Designing a final exam is often tricky business. It can't possibly assess everything in the course, but we generally want it to include the major topics and themes for the class and be possible to complete in the time allowed. We also have to think about difficulty. Trust me, your students are!

Teachers want their finals to be challenging, but they don't want to have that sinking feeling as they grade the exams that maybe the test was too hard. For whatever reason, sometimes students perform poorly and averaging the final exam grade into their other grades will look like a disaster. But ask yourself: What am I more confident in, my careful judgments of students' ability as demonstrated over an entire semester, or a fleeting, one-time judgement of students' ability on a single assessment during the most stressful time of the year? If you're using standards-based grading, I already know how you'll answer that question. If not, consider this example: I have a student who I know can do stats. She's turned in good work. She's asked quality questions. We've had good discussions. But I also know she has seven final exams this week. I still think she'll do fine, but I'll understand if she's not at her best. And I need a grading system that reflects that understanding.

In order to free myself to still give challenging, yet reasonable, assessments, without risking any huge surprises when grades are calculated, I perform a little statistical magic that ensures that the distribution of final grades has the same center and spread of class grades before the final. I'm sure many of you try "curving" your exam scores some other way, such as letting the top score count as the total possible, or even having a pre-set distribution in mind of how many As, Bs, Cs, etc. you'll allow (which is not a good idea, generally, for reasons described by Krumboltz & Yeh, 1996). I prefer my method because it accounts for the distribution of grades, not just the top score, and the distribution is determined by the students, not arbitrarily by me. Allow me to demonstrate with a couple examples.

Suppose before the final the average percentage grade is 85 and the standard deviation of those grades is 10. Then I grade my final exams and find that the average final exam grade is 60 with a standard deviation of 18. Ouch. But don't worry -- statistics will come to our rescue.

Provided you know a little basic descriptive statistics, the conversion is simple. For each student's final exam score, find out how many standard deviations above or below the mean they scored on the final (their final exam z-score), and match that with the same number of standard deviations above or below the mean they'd fall on the pre-final grade distribution (their pre-final z-score). Consider the following students and the class and exam statistics above:

• Suppose Student A scores a 51 on the final exam. That's 0.5 standard deviations below the mean. (51 - 60 = -9, and -9/18 = -0.5.) So where is 0.5 standard deviations below the mean on the pre-final distribution? If that mean is 85 and the SD is 10, then 0.5 standard deviations below the mean is 80. So I record an 80 for that student instead of a 51.
• Suppose Student B scores a 75 on the final exam. That's about 0.83 standard deviations above the mean. (75 - 60 = 15, and 15/18 = 0.83.) So where is 0.83 standard deviations above the mean on the pre-final distribution? About 8.3% above an 85, so I record their exam grade as a 93.3.
• Suppose Student C scores a 60 on the final exam. That's the same as the mean, so zero standard deviations above or below. That conversion is super-easy: their final exam grade is the mean of the pre-final mean, an 85.

For an example of how to set up a spreadsheet to do this, see https://docs.google.com/spreadsheet/ccc?key=0Anne5Z-jCkqhdDVtemkyaGhnRWFfclJoa0dIUVQ5RVE. I recommend making a copy of it for yourself and seeing what happens as you change values.

This is not a perfect system (and comments about its imperfections are welcome in the comments), but it does take away the element of surprise if the final exam happens to be way too easy or too difficult, or if other circumstances prevent grades from working out the way you'd expect. Yes, this is a norm-referenced system instead of a criterion-referenced system, meaning that the grades students earn on the final is measured largely as how they compare to their classmates and the class average. The good news is this: both the teacher and the students have an incentive before the final to master as many objectives as possible, and that is criterion-referenced. A high pre-final average helps everyone get a high final exam average, and a small pre-final standard deviation minimizes variability in final exam scores.

References

Her, T., & Webb, D. C. (2004). Retracing a path to assessing for understanding. In T. A. Romberg (Ed.), Standards-based mathematics assessment in middle school: Rethinking classroom practice (pp. 200-220). New York, NY: Teachers College Press.

Krumboltz, J. D., & Yeh, C. J. (1996). Competitive grading sabotages good teaching. Phi Delta Kappan, 78(4), 324-326. Retrieved from http://www.jstor.org/stable/20405782

I Love Good Data Visualization. This Isn't It.

Earlier this week Newsweek ran a story titled, "Classrooms or Prison Cells?" Given some of the more recent education coverage from Newsweek I wouldn't have been very surprised if the article came down in favor of prisons.

Thankfully, the article was generally informative and unbiased, and told the story of California's 30-year rise in corrections costs amidst education budget cuts. According to the article, in 1980 California spent 10% of its budget on higher education and 3% on prisons. Now, almost 11% goes to prisons while higher education spending has dropped to 7.5%. If you thought that was a tragedy, check out the graph that accompanied the article:

(Image Source: http://www.newsweek.com/content/newsweek/2010/06/28/classrooms-or-prison-cells/_jcr_content/body/inlineimage.img.png/1277695326254.png)

Do you get the feeling that somebody in the Newsweek graphics department got this assignment at 4:45pm on a Friday afternoon? I would have loved to see the graph try to predict future spending. Given the assumption that these rates are truly linear, you can predict that by the end of this century California will be spending 35% of its budget on prisons and not a single dime on higher education.

Looking ahead to PhD: Focus and Vision

As the first of my classmates were congratulating me on my PhD acceptance, the inevitable question came: "So what do you want to study?" As I started to answer, I stumbled. I didn't have the 12-second, sure-of-myself answer I was supposed to have, despite having thought hard on the question when I wrote my application essay.

I learned of my acceptance five days ago and ever since I've been thinking about how I'm going to make the most of this opportunity. CU-Boulder's School of Education is home to some of the finest researchers in the field, and my experience in the master's program has been excellent. I've always been intellectually curious, and in the PhD program I'll learn how to apply methodologies to that curiosity in ways that are both personally satisfying and helpful to others. To clearly answer the "What do you want to study?" question, I need to develop "focus." The best demonstration of "focus" that I've seen lately is Gary Vaynerchuk's "Linchpin" video he made for Seth Godin's blog:

Linchpin: GaryVee from Seth Godin on Vimeo.

Even if you think P.Diddy's hair was an odd example, I think you get the point. As I prepared my application essay, I tried to focus on one specific thing in both curriculum and instruction (my major):

Curriculum Focus: I'm interested in the intersection of policy and practice, specifically when and how state and national standards become the curriculum that is experienced by students. I studied the math wars for my undergraduate thesis ten years ago, and my new area of interest is how our standards' increased demand for statistics (including all forms of data analysis and visualization, uncertainty, and probability) is causing change in textbooks, course sequences, standardized tests, and traditional perspectives on school mathematics.
Instruction Focus:Teachers' views of standardized assessment are affecting classroom assessment practices, and in turn poor classroom practice in assessment and grading has an inordinate influence over how teachers teach and how students learn. Additionally, negativity towards assessment deters teachers from climbing the mountains of potentially helpful data produced by assessments. I'm interested in understanding both the theoretical and practical problems teachers have with assessments and grading, and how to develop pragmatic solutions that are favorable to practicing teachers.

It's good to have focus. Unfortunately, I've never been comfortable with the pigeon-holing that happens when someone declares a specialty. I remember talking to other undergraduates after we had declared our majors, frustrated with the feeling that because we had chosen a field, people assumed we were now ignorant of everything else. I might be overreacting, and I hope to "crush it" (as GaryVee says), but not at the expense of stifling my curiosity, or losing sight of something bigger, which I'm calling "vision." Vison is big. I was afraid my vision would be too broad or vague in a specialized world, but my attitude was helped greatly by Aaron Eyler's blog post about connecting research with K-12 teachers. Eyler is personally frustrated with how little research makes its way into the hands of K-12 teachers, and the little that does is presented in a "cookie-cutter" fashion that has little of the impact intended by the researcher. Why don't teachers and administrators do a better job keeping up with research, and why don't researchers spend more time working with K-12 educators?

So what's my vision? In short, I want to help teachers be better teachers. It's that simple. I hope to interact with teachers (or future teachers) whenever possible, whether it be teaching methods classes, visiting schools, giving presentations, or whatever else that might engage me with a community of teachers. When I do research, I want to always think of teachers as my audience, not some journal editor or professor who might be refereeing my work. I want to write things that teachers will want to read, and explore ways of delivering research to teachers in helpful ways. That's my vision, and I should be confidently unapologetic about believing you can't focus without vision.

CCTM 2009: Guidelines for Assessment and Instruction in Statistics Education: Progress from K to 12

Presenter: Jerry Moreno, John Carroll University

I attended this session because I'm very interested in the growth of probability and statistics education in K-12 mathematics. I realize now that it was a neglected part of my own education, but as a teacher I find statistics interesting, relevant, and powerful. I also realize, however, that placing greater demands on students to learn probability and statistics means new curricula needs to be developed and other areas of math might be compromised.

Moreno made his feelings towards high school math sequences clear up front: integrated math sequences are the only way to go, and he's helping push the effort in Ohio to rid schools of the traditional Algebra 1-Geometry-Algebra 2 sequence. (I believe he said Ohio was trying to standardize class names such as "Math Reasoning 1, 2, and 3" in their place.) Moreno sees probability and statistics as the single largest piece of a puzzle that connects mathematics, science, and social studies, and much could be gained by increasing the use of data analysis in all three subjects.

Moreno's presentation summarized many of the efforts put into the GAISE (Guidelines for Assessment and Instruction in Statistics Education) report, an effort of the American Statistical Association to guide implementation of the NCTM Data Analysis and Probability standard. The ASA has produced a collection of student investigations called "Making Sense of Statistical Studies" for upper middle school and high school students. (They're working on sets of investigations for lower grades, building off work found in ASA's Statistics Teacher Network (STN) and other sources. Collectively the projects are known as GAP, or GAISE Activity Project.) During the session we discussed/experienced several quality investigations, dealing with varied topics such as the fairness of pennies, Mentos and Diet Coke, growing dahlias, and death certificate analysis.

There is clearly no lack of quality content available to teach probability and statistics to all levels. It remains to be seen, however, if schools are willing to break with traditions and reallocate their precious time to include more probability and statistics. I'm still wondering where the textbook publishers stand on this, particularly those who sell the traditional Algebra 1-Geometry-Algebra 2 series. Could we ever see a Algebra 1-Prob/Stats-Algebra 2 series, where geometry is reduced to supplementary materials, as many teachers have to do now for probability and statistics? Or will the massive educational inertia be too resistant to such a change?