TIMSS research plan

This last spring, I wrote a research proposal for a ph. d. position in "behavioral measurements", focusing on understanding international differences in mathematics understanding as measured by TIMSS and PISA. Recently, I've seen a renewed interest in TIMSS results online, as for example in Michael Pershan's video critique of Khan Academy.

So I'm thinking if I post part of the research proposal here, maybe people will find the "research overview" part interesting and relevant to the times. Parts of the proposal are about Sweden, but from what I understand much is highly relevant for the US as well.  Sorry for the sketchy formatting which happened when I copy-pasted from MsWord. 

Oh, and I did get that ph. d. position, it fit me like a glove and I happily accepted. Unfortunately it would have required me to relocate to a different town, and recent family developments made relocation currently impossible. Oh well - I'll always think of this position as "the one that got away."

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Specific Objectives and Aims

The overarching aim of this project is to use existing international TIMSS data to understand the factors that influence the quality of mathematics education in Swedish schools.  Ever since the first international comparisons of mathematics knowledge in middle-school students, Sweden has positioned itself at or below the average score of participating nations (Hellerstedt, 2011). Between TIMSS 1995 and TIMSS 2003, the results of Swedish 8^th grade students decreased by 41 points, which is more than any other country among the 16 that participated in both 1995 and 2003 (Skolverket, 2004), and then decreased even further by TIMSS 2007 (Skolverket 2008).  By contrast, other nations, such as our neighbors Finland and Russia, have shown consistently higher results in international comparisons.  

Such differences between nations deserve attention because they signify that mathematics education can be more effective than is the case currently in Sweden. By identifying the causes behind the relative successes of high-performing nations, Sweden might be able to emulate them and thus achieve more efficient use of school finances as well as a more mathematically literate population. However, what works in one country may not work in another cultural and economic context.  It is therefore necessary to take into consideration factors that affect mathematics education within Sweden, as well between Sweden and other nations.  While TIMSS tries to be curriculum-neutral, so that it can be applied to all nations, it can be argued that the mathematics knowledge measures by TIMSS does not constitute mathematics knowledge in its entirety, that the questions target only specific aspects of mathematics knowledge such as specific subject areas or skills.  In order to enable research of these international differences, TIMSS and PISA are accompanied by in-depth data regarding the questions in the test, as well as a wide range of detailed contextual data about variables at the student, teacher and school levels of the participating nations.

The three main objectives of this proposal are:

To identify what aspects of mathematics knowledge are targeted by the TIMSS questions and how they are related to the aspects of mathematics knowledge valued in Sweden To analyze TIMSS contextual data to determine in what relevant ways Sweden differs from nations with higher TIMSS results To analyze TIMSS contextual data to determine what factors cause differences in mathematical knowledge within Sweden

Overview of the Research Area

The achievement of the above stated objectives will be made possible by a close analysis of relevant parts of the large amount of data collected in the TIMSS mathematics reports. This data includes results for each participating nation on different types of questions in the different areas of mathematics tested in TIMSS.  Also included is contextual data such as statistics on student, teacher, school and curriculum variables in each participating nation.   Next, we shall see several research studies that to varying degrees, and with different aims, make use such contextual data.

One of the most relevant studies regarding international differences in TIMSS mathematics results is the TIMSS Videotape Classroom Study (Stigler, 1999a) which was created together with the 1995 TIMSS mathematics study (Beaton, 1996).  Stigler used video recording in order to compare instructional practices in 8^th grade mathematics lessons in Germany, Japan, and the United States.  In a large sample of in total 281 classrooms, chosen to be representative of classrooms in each country, one lesson per year was randomly chosen and filmed.  Results show that, among many other differences, Japanese classrooms include more complex problem-solving tasks and higher difficulty mathematical content than do their German and United States counterparts.  In a popular description of this study and its findings, Stigler and Hiebert claim that it is such differences in instructional practices which influence some international differences in mathematics knowledge (Stigler, 1999b).

A related study points in a somewhat different direction.  Leung (2005) analyzed the data from the larger scale TIMSS video study that was made in conjunction with the TIMSS 1999 mathematics study (Mullis, 2000).  In this larger and more recent video study, 7 countries (Australia, Czech Republic, Japan, Hong Kong, the United States, the Netherlands, and Switzerland) were included with a total of 638 videotaped lessons. When comparing the East Asian nations to other nations, Leung observed that East Asian nations provide students with learning opportunities involving complex mathematical problems often featuring high level mathematical difficulty and logical reasoning such as proofs. However, the two East Asian nations differed from each other in that Hong Kong classrooms are highly teacher-directed, while Japanese classrooms very much less so.  Leung concludes that East Asian classrooms are highly heterogeneous, and that the success of East Asian nations in international comparisons must be understood as resulting from interactions of cultural factors such as perceptions of education and high expectations in the classroom.  These video studies show that although there are differences in instructional factors between high-performing and low-performing nations, not all such differences are causal factors of mathematics knowledge.  Also high-performing nations may have some common and some different strategies to ensure high levels of mathematics knowledge. 

While the video studies have yielded much valuable data about instructional practices, other research has focused on psycho-social differences such as attitudes towards oneself in relation to mathematics.  Shen (2008) aimed to investigate the relationship between self-perception in mathematics and TIMSS results in 8^th graders in the 1995, 1999, and 2003 TIMSS studies.  Using statistical analysis of relevant TIMSS contextual data, Shen found that within each country, there is a positive correlation between mathematics results and perceived competence in mathematics, how much the student likes mathematics, and how easy the student perceived mathematics to be.  Between countries, however, the relationship is reversed such that the students in the highest performing nations are the ones who report liking mathematics less, judge it to be difficult, and have low opinions of their competencies in mathematics. Shen attributes this surprising relationship to higher academic standards in high-performing nations, and lower standards in lower-performing nations.

Findings such as those in Shen’s study can be questioned on methodological grounds, since the assumption is that the surveys used in the collection of TIMSS contextual data are valid for all participating nations. Eklöf (2007) challenges this assumption by conducting in-depth analysis of the Swedish TIMSS data on mathematics self-concept and students valuing of mathematics. While the former of these is shown to be consistent and correlated to mathematical achievement, this was not true of the latter.  Eklöf’s research, among others, illustrates the need for careful statistical investigation of the scales used in the TIMSS contextual data before using the contextual data for secondary analysis.  Eklöf and Shen show that TIMSS data must be analyzed for both between-countries and within-country differences if we wish to understand the factors that influence mathematics knowledge.

Another area of research into student variables is illustrated by Boe (2002).  Boe investigated whether student task persistence (a variable not included in the TIMSS contextual data), as measured by the percent of background questionnaire questions students completed, was related to mathematics results on the TIMSS 1995 test. The results indicated a surprisingly strong correlation (ranging from 0.72 to 0.79 for 7^th and 8^th grade students) between task persistence and mathematics results on a between-nations level of analysis.  The relationship between task persistence and results appeared much smaller at the classroom and student levels, however, and in total task persistence accounted for about 0.28 of the total variation between students participating in the TIMSS 1995.  Such findings are difficult to interpret. First, we do not know whether the strong correlations indicate any causal relationships between the variables. There could be a third  factor on a cultural level, such as ability to delay gratification (Mischel, 1989, shows a moderate correlation between delayed gratification and SAT scores), or test-taking motivation (Eklöf, 2006, finds a weak but significant correlation in the Swedish TIMSS 2003 sample), that causes both increased task persistence and higher mathematical achievements on the TIMSS tests.  Also, it is strange that the correlation is smaller on the student level than on the national level.  From Boe’s study, it is clear that TIMSS contextual data demands analysis beyond just looking for correlations, and that the relationships found require careful interpretation in terms of findings from cultural and psychological research.

Of special interest for this research proposal is the research that has been done on mathematics knowledge in Nordic countries in general, and Sweden in particular. Kjaernsli (2002) investigates similarities and differences between the Nordic countries, excluding Finland, and finds that their results on the TIMSS science and mathematics tests are similar and may be connected to cultural factors such as the reluctance to put academic pressure on young children.  Finland, by contrast, has seen a dramatic rise in mathematics results as measured by TIMSS and PISA since 1999. It is of great international interest to determine the factors behind Finland’s success, and recently much research has been made with this aim. Välijärvi (2003) aims to present a broad look at factors influencing Finland’s rise to success.  Välijärvi identifies factors such as educational equity in comprehensive shools, cultural homogeneity, and highly educated teachers.  Interestingly, Välijärvi also points out that some factors seem to be more important in Finland than in other OECD nations.  The within-country correlation found by Shen (2008) between self-perception and mathematics achievement is significantly higher in Finland than elsewhere.  Research such as Välijärvi’s further illustrates the need for both between-nation and within-nation investigations of factors influencing mathematics achievement.

TIMSS aims to establish the success of mathematics education in terms of how well students achieve the educational goals formulated by their own nations, whether at the state or local levels. It is therefore very important to investigate to what extent TIMSS questions are aligned with the Swedish curricula, both in terms of the subject matters covered (geometry, algebra, etc.) and the skills which students are meant to develop (reasoning, application of procedures, etc.). However because of the loosely formulated goals in the government-issued curriculum documents, we should be wary of using those documents to understand the implemented curricula in the Swedish schools. Instead, it makes better sense to analyze teacher responses about their intentions and expectations within their implemented curricula (Skolverket, 2004).  

Teacher responses to TIMSS questionnaires indicate that students have received relatively more instruction in arithmetic and measurement, and less in algebra and geometry, compared with students in other nations as well as compared with the proportions that each subject area has in the TIMSS examinations (Skolveket, 2004).  Also, Alger (2007) finds that Swedish teachers compared to teachers in other nations report using a larger proportion of class time on independent practice with mathematics exercises, and less time going over homework and lecturing.  Lindström (2006) considered differences between Swedish national tests (though he used an old test from 1992) and TIMSS 2003 and PISA 2003. He found that the exercises are about equal in difficulty level, but the Swedish test had much less emphasis on reasoning and on applications, and more emphasis on identifying and carrying out procedures.  One major limitation of Lindström’s study is that the Swedish national tests have changed considerably since 1992. However, Lindström’s results find support in a more recent in-depth analysis (Skolverket, 2009) of student responses to the TIMSS 2008 Advanced and the Population 3 responses to TIMSS 1995.  This analysis is based on the patterns of correct and incorrect solutions, and frequent mistakes, made by Swedish students and reveals that Swedish upper secondary school teachers since 1995 have increasingly focused on procedural knowledge rather than conceptual and reasoning based understanding of mathematics. Together, these studies indicate that there are variables at the instruction level that affect Swedish students’ results on the TIMSS assessments.

On a school level, there are other important variables identified in the responses from teachers and principals. Overall Swedish students in grades 4 and 8 receive substantially fewer instructional hours compared to the OECD average, this difference in grade 8 is approximately 25% and is even larger (closer to 40%) in grade 4. In addition, few Swedish students receive extra mathematics instruction outside of school and report much less frequent homework and less time spent on homework compared with OECD averages (Skolveket, 2004).  Thus, it is reasonable to assume that school-level factors also play a role in determining students’ mathematical knowledge.  

Research questions

In light of the background research presented above, the main research question in this research proposal is:

Research question: To what extent do school- and teacher-level variables influence Sweden’s mathematics results on international tests?

In order to investigate the main research question, it is necessary to consider several related questions:

1.       What types of mathematical knowledge is measured by TIMSS?

2.       Are the TIMSS measurements reliable and valid?

3.   What statistical analysis methods are relevant for studying TIMSS secondary data in search of potential causal relationships?

4.       To what extent are the aforementioned statistical methods valid?

Potential causal factors can be divided into several categories:

5.       What instructional factors influence mathematics achievement within Sweden?

6.       What instructional factors influence mathematics achievement differences between Sweden and other nations?

References

Alger, S. (2007). Svenska skolans lärare och undervisning i matematik och NO i ett internationellt perspektiv: Några resultat från TIMSS 2003. (BVM 32:2007). Umeå: Umeå universitet, Department of Educational Measurement. 

Beaton, A., Martin, M. O., Mullis, I., Gonzalez, E. J., Smith, T. A., & Kelley, D. L. (1996). Mathematics

Achievement in the Middle School Years: IEA's Third International Mathematics and Science

Study. Chestnut Hill, MA: Boston College.

Boe, E. E., May, H., & Boruch, R. F. (2002). Student task persistence in the Third International Mathematics and Science Study: A major source of achievement differences at the national, classroom, and student levels (Research Rep. No. 2002-TIMSS1). Philadelphia, PA: University of Pennsylvania, Graduate School of Education, Center for Research and Evaluation in Social Policy.

Eklöf, H. (2006). Motivational Beliefs in the TIMSS 2003 Context. Theory, Measurement and Relation to Test Performance. Doctoral  dissertation. Umeå: Umeå University, Department of Educational Measurement. 

Eklöf, H. (2007). Self-concept and valuing of mathematics in TIMSS 2003: Scale structure and relation to performance in a Swedish setting. Scandinavian Journal of Educational Research, 51(3), 297-313.

Hellerstedt, L., & Reistad, H. (2012, January 22). Matteutveckling för miljarder. Origo, 1. Retrieved March 22, 2012, from http://www.lararnasnyheter.se/origo/2012/01/22/matteutveckling-miljarder

Leung, F. (2005). Some characteristics of East Asian mathematics classrooms based on data from the TIMSS 1999 video study. Educational Studies in Mathematics, 60, 199-215.

Lindström, J. (2006). Med fokus på matematik och naturvetenskap: en analys av skillnader mellan internationella jämförande studier och nationella kursplaner. Stockholm: Skolverket.

Mischel, W., Shoda, Y., & Rodriguez, M. I. (1989). Delay of gratification in children. Science, 244, 933-938.

Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Gregory, K.D., Garden, R.A., O’Connor, K.M., Chrostowski, S.J. & Smith, T.A. (2000) TIMSS 1999 International Mathematics Report: Findings from IEA’s Repeat of the Third International Mathematics and Science Study at the Eighth Grade. Chestnut Hill, MA: Boston College.

Nyström, P. (2006). TIMSS fixpunkter. En analys av vad elever med olika resultat  i TIMSS 2003 vet och kan göra. (BVM 20:2006). Umeå: Umeå universitet, Department of Educational Measurement. 

Nyström, P., & Lind, A. (2009). Hur samstämmiga är svenska styrdokument och nationella prov med ramverk och uppgifter i TIMSS Advanced 2008. Stockholm: Skolverket.

Reinikainen, P. (2004). Explanatory variables of science achievement in Finland: Cultural approach. In C. Papanastasiou (Ed.), Proceedings of the IRC-2004. IEA International Research Conference (Vols. 1–4).

Kjaernsli, M., & Lie, S. (2002). Timss science results seen from a nordic perspective. In D. Robitaille & A. Beaton (Eds.), Secondary Analysis of the TIMSS Data (pp. 193-208). New York: Kluwer Academic Publishers.

Shen, C., & Tam, H. P. (2008). The paradoxical relationship between student achievement and self-perception: A cross-national analysis based on three waves of TIMSS data. Educational Research and Evaluation, 14, 87–100.

Skolverket. (1996). TIMSS 1995: Svenska 13-åringars kunskaper i matematik och naturvetenskap i ett internationellt perspektiv. Stockholm: Skolverket.

Skolverket. (2004). TIMSS 2003: Svenska elevers kunskaper i matematik och naturvetenskap i skolår 8 i ett nationellt och internationellt perspektiv. Stockholm: Skolverket.

Skolverket. (2008). TIMSS 2007: Svenska grundskoleelevers kunskaper i matematik och naturvetenskap i ett internationellt perspektiv. Stockholm: Skolverket.

Skolverket. (2009).   Svenska elevers kunskaper i TIMSS Advanced 2008 och 1995: en djupanalys av hur eleverna i gymnasieskolan förstår centrala begrepp inom matematiken. Stockholm: Skolverket.

Stigler, J. W. (1999a). The TIMSS Videotape Classroom Study: methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States. Washington, D.C.: U.S. Dept. of Education, Office of Educational Research and Improvement.

Stigler, J. W., & Hiebert, J. (1999b). The teaching gap: best ideas from the world's teachers for improving education in the classroom. New York: Free Press.

Välijärvi, J., Linnakylä, J., P., Kupari, P., Reinikainen, P., & Arffman, I. (2002). The Finnish success in PISA - and some reasons behind it: PISA 2000. Jyväskylä: Institute for Educational Research, University of Jyväskylä.

Scaffolding questions

I'm coming to believe that for a vast majority of (my) students, mastery is the key to motivation. Yes, I could try to include more inherently engaging activities such as wcydwt activities, but ultimately what will get the students to open their books at home is feeling confident about their own abilities to learn and do mathematics. As teachers, we can help build this confidence by proper scaffolding in class. Until now, I've used group work to provide some scaffolding, but now I'm trying scaffolding by ways of phrasing questions. Yesterday's class was about the derivatives of trigonometric, exponential, and logarithmic functions (only e^x and ln(x) are included in this course), so I'll use questions on this topic as examples.

1. Show that: Show that if f(x) = sin(x^2) + e^(sin(x)) then f'(x) = 2xcos(x^2)+cos(x)*e^(sin(x)).  This is an example of the "Show that A can be transformed by some relevant mathematical process into B" type of question.  Such questions have the advantage that students at least know where they are going, and just have to find a way to get there. Also,  the question itself provides immediate feedback on the accuracy of student work. Someone still has to ask the student to explain every step of their solution, but other than that the question is pretty self-contained.
2. Spot my mistake: I claim that if f(x) = ln(x^3) + e^(x^2)*cos(x) then f'(x) = 3/x - 4*e^(4x)*sin(x). What is my mistake? This is an example of the "Critically examine to find a mistake and try to understand what the person making the mistake was thinking."  Most of my students really struggled with this one because they did not recognizing the need for product rule. There were a lot of good discussions going on between students comparing their ideas of what might be wrong in the solution. To me, the benefits of this type of question are that they promote critical/logical meta-cognitive reasoning and discussion. 
3. Verify: Is it true that if f(x) = ln(sin(x^3)) then f'(x)=1/tan(x^3)? If not, explain what misunderstanding might have caused the mistake. This of course is very similar to the second type of question, but a little less scaffolded because the students are not told to expect a mistake. Once again, this gives rise to good discussions between students about whether the solution is correct, where potential mistakes are, and what misunderstandings/correct understandings give rise to this solution. 

All-in-all, I see these questions as best being used in sequence, though depending on the difficulty/novelty of the topic one or all steps may be skipped. After these scaffolded questions students should be ready for the classic "solve this..." exercises. Because the three question types mentioned above have other benefits than just providing scaffolding, it makes sense to me to intermittently use them for variation and to encourage reasoning and discussion even once students are confident with the less scaffolded exercises. 

Some questions that need to be answered: how do we ensure that students gradually let go of the scaffolding (their classmates, their books and notes)? How can scaffolding activities be combined/coexist with explorative investigation-driven open-ended work?

Tangents and normals - a peer scaffolding lesson

Yesterday we had a nothing-special lesson, with no great outbursts of creativity, which nevertheless went very well and is the kind of staple lesson-setup that requires very limited preparation..

The goal: students should understand the concepts of tangents and normals to a curve, and be able to calculate the equations of these lines.
It's not exactly brain surgery, but then I find students often get lost in questions about tangents and normals: they have a hard time connecting the many relevant concepts: derivative, gradient, equation of a line, constant term, perpendicular, negative reciprocal, etc. They start doing funny things, like plugging in values of x into the derivative function instead of the original function in order to find the corresponding value of y. They lose track of what they're doing.

So for this lesson (as for almost all topics in calculus) we used algebraic and visual representations throughout, in parallel. I find it really helps students understand and keep track of their work, and check whether their work seems reasonable.

I put a "do now" question on the board: what is the equation of the tangent to the function..." and gave them a simple cubic function. Five minutes to work, in pairs, and everyone had found the gradient of the tangent, and many had also found its y-intercept. Some got a minus sign wrong, and could quickly see on the graph (which stretched close to, but not including the y-intercepts) that they must be mistaken. Go-through together and everyone's on track.

Follow up: is there any other point on the graph of f that has the same gradient of tangent? Five minutes pair-work, and most students set up, and at least attempt to solve, the resulting quadratic equation. A few needed a hint, because they attempted to set the equation of the tangent equal to its own gradient... Sure sign they were having a hard time connecting the derivative function with the gradient of tangent. This will be solved once we do more work on using the derivative for graphing the function.
Go-through together, and we're fine.

My only act of "telling" during this lesson was when I introduced the concept of a normal as a line perpendicular to the tangent at a certain point on the graph. A few students recalled that perpendicular lines have gradients that multiply to -1, and we were ready to go. There was a bit of the "do we use the same point? where do I plug this in?" going on, but when I brought their attention back to the graph on the board, they answered their own questions easily.

So all in all - students helped each other learn about tangents and normals, they worked efficiently during the whole lesson, and seemed to understand and enjoy the topic.

Inferential statistics - main ideas

In my opinion, inferential statistics is one of the most important and most difficult to teach of all the topics in high school mathematics and psychology courses. I get to teach it twice, in math, where the focus (unfortunately) is to manually carry out the Chi-square calculations, and in psychology, where the emphasis is on understanding the need for the test and interpreting the results. Psych students even do their own experiment, where one of the things being assessed is their ability to set up, justify, carry out, interpret, and discuss inferential statistical analysis of their data. It's a challenging task for most students.

So far, I've taught inferential statistics every year, and never felt really satisfied with the outcome. Yes, my students can copy my example to obtain a test-statistic and compare it to the critical value in the book. Yes, they can even say "thus we reject the null hypothesis." But rarely do they demonstrate true understanding. This year's attempt to teach inferential stats failed, as usual. Students complained so much about their lack of understanding (I love when they do that) that I decided to give it another, serious, try. 

So for today, I considered the main difficulty in understanding inferential statistics. I think the main difficulty is understanding that random variation might create differences between groups that are due to chance. So I started with an object students know behaves randomly, a coin, and focused the lesson on the concept of random variation.

The setup: a normal coin, which I flip 15 times. Record number of heads and tails in a contingency table. 

Then I "bless" it. I make a show of it, concentrating hard and blowing on the coin carefully in cupped hands. 

Next I flip the coin another 30 times. 

It turned out that before the blessing, the coin came up 5 heads and 10 tails. After the blessing it came up 13 heads and 17 tails. Oh my, my blessing made the coin come up heads more than twice as much! Students immediately complained that I should take into account the different number of flips in each condition, thank you students. We calculated percentages. 33% heads without blessing vs 43% heads with blessing. 

Key Question: did my blessing work? Students were laughing at this, and suggesting wonderful things, like that the difference might be too small, and the sample too small, to be sure the results weren't merely due to chance. So how big should the difference be, for this sample, and how sure is "sure"? This led us into significance levels, and the need for statistical tests. We did a chi-square online (vassarstats is great for this) and when we saw that  the p-value was 0.75 we concluded that the difference in heads was most likely due to chance. We experimented with changing the data a bit, say what if there was 29 heads and 1 tail in the "blessed" condition? Students agreed that would be more convincing, and voilá the p-value was less than 0.001. 

That's nominal data. I also wanted students to experiment themselves, and to obtain ordinal data to use with a Mann-Whitney U-test. So I asked: "Are you telepathic?"

Students paired up. One student in each pair thought (but didn't speak) of a word, either "BIG" or "small". The other person then said a number, any number. The was tallied up in two columns according to the two words.  At the end, I picked the data of one pair of students and calculated the medians. Oh my - the median for BIG numbers was 87.5, compared to just 15 for the small numbers. Students thought about this, could they be sure their classmates were telepathic? We did a Mann-Whitney U-test online (thanks again, vassarstats) and found a p-value of 0.009. Students were impressed. We concluded that we can be at least 95%* (or even 99%) sure that this pair of students were telepathic, except...

What if there wasn't a random variation causing the difference in results? What if the variation comes from confounding variable within the experiment? Students were saying that maybe the girl thinking of the word somehow consciously or unconsciously signaled the word she was thinking. Someone said humans have a hard time being truly unpredictable and random. So we arrived at the conclusion that more evidence is needed, and that statistical tests can only (at best) rule out that the difference is due to random variation but that there can still be other threats to validity present.

Overall, I am very pleased with this lesson. I am happy I chose a coin, even more happy I chose ESP - something many students are naturally curious about and have already considered in a somewhat relevant manner. Many students told me later that they finally got the idea, that it made sense, that it was obvious that descriptive stats is insufficient to draw conclusions about data. They could even transfer their understanding to psychology, to explain how participants in an experiment might be randomly different from each other or even compared to themselves at an earlier point in time.  I am particularly happy with the telepathy-experiment. At first I thought that I should have made them flip a coin to decide what word to think about, to make it truly unpredictable, but because their choice of words wasn't perfectly random we had that very good discussion about internal validity and confounding variables which I think deepened students' understanding of the power and limitations of inferential statistical tests. 

Some changes I'll do for next time: provide each pair of students with a computer so they can do the test themselves. Spend more time working with hypotheses and writing up the results of the inferential test. I want them to say "therefore the difference between conditions is significant and we should reject the null hypothesis" so we should have spent more time saying, and thinking, about this statement and what it means.

*Yes, I know that this is an incorrect interpretation of significance level. I know, and it hurts me to teach it this way. But seriously I think I must, at least to begin with, because students are simply not ready/able/given enough time to fully understand the concept of significance according to the frequentist approach to statistics. I comfort myself with the thoughts that hey, priorities gotta be made, and that perhaps, if looked at from a Bayesian perspective, what I'm teaching my students actually makes sense. It's a hard decision, though. 

Perplexing!

I've recently had the opportunity to peruse a substantial amount of research articles about international differences in mathematics knowledge, as measured by TIMSS and PISA. I found some very interesting things in there, such as that amount of time spent on homework has a negative correlation to mathematics achievement both within and between nations. Meanwhile, frequency and effort put into homework has a positive correlation with mathematics knowledge. That's all good and great and I'm already changing how I talk to students about homework, but other results from the research studies are just bewildering:

1. If a student likes math, and believes in her own ability to do math, that's gotta mean the student is more likely to develop good understanding of math, right? Well, not really. Within nations, this relationship holds, and in some nations (such as Finland) the correlation is positive and quite high. But between nations, the relationship is actually the opposite: students in high-performing nations report that they like math less and consider themselves to not be good at it, compared to students in low-performing nations (Shen, 2008).
2. A student who is persistent with finishing tasks is likely to learn more math. That, by itself, is not weird. But Boe (2002) found that task persistance (as measured by the number of background questionnaire items answered by students in the TIMSS 1999 study) has a high correlation to mathematics achievement between nations, but less so between classrooms and very little between students. So nations in which students answered many of the background questionnaire items, which require no knowledge of mathematics or science, did better than nations in which students answered only a few of the questions. The correlation was around 0.75. On a student level, when comparing students within classrooms, the correlation was much lower. Overall, this "task persistance" variable seems to account for about 1/3 of the overall variation in results among students worldwide, and about half of the variation between nations. This is BIG. To my knowledge, no other variable has been found that explains so much of the variation. But what does it mean? Does it reflect cultural values of conscientiousness and long-term orientation (would explain why East Asian nations do so well)? Or is it that students who expect to do well on the TIMSS are more motivated to fill in the questionnaire? And WHY is the relationship strong at the nation-level but not student-level?  And why, given the stunning results, has this study been cited only a handful of times since it was published 10 years ago?
3. One reason that Swedish researchers are interested in the TIMSS background data is that Sweden has seen a dramatic drop from acceptable to outright poor results (still better than the US, though) from TIMSS 1995 to TIMSS 2007. "Why is this happening?" we're asking. Well. Hidden among the data are little-known figures such as these: Swedish 4th-graders receive almost 30% less mathematics teaching per year than the OECD average. 8th-graders receive 20% less than OECD average. Meanwhile the Swedish media and government has aggressively blamed teachers for the poor results. To be fair, there is not a strong correlation between amount of teaching hours and mathematics knowledge. Finland, for example, ranks very high but has the fewest teaching hours of all the participating nations. US, on the other hand, has plenty of teaching hours, yet achieves very low ranking. Yet it's difficult to ignore that East Asian nations and Russia, who always top the ranks-lists, not only provide students with a LOT more teaching (South Korea for example gives students 220 teaching days each year, compared with 178 in Sweden), but also in these nations many students go to after-school mathematics tutoring. 

That's it for now. 

Boe, E. E., May, H., & Boruch, R. F. (2002). Student task persistence in the Third International Mathematics and Science Study: A major source of achievement differences at the national, classroom, and student levels (Research Rep. No. 2002-TIMSS1). Philadelphia, PA: University of Pennsylvania, Graduate School of Education, Center for Research and Evaluation in Social Policy.

Shen, C., & Tam, H. P. (2008). The paradoxical relationship between student achievement and self-perception: A cross-national analysis based on three waves of TIMSS data. Educational Research and Evaluation, 14, 87–100.

Intro to calculus the graphing stories way

I think every IB teacher approaches calculus the same way: look, average rate of change. Look, instantaneous rate of change. Secant, tangent, gradient of tangent. Previous years I've tried to make this introduction come alive by giving students a function describing the distance fallen by a parachute-jumper, and having students discover themselves what the average speed would be, and the instantaneous speed at certain moments.
This has worked... well, let's just say students got the main ideas but not at an intuitive level, neither did they retain their discoveries more than a week or so. The whole thing seemed very artificial, pseudo-contextual.

So this year I decided to try graphing stories, choosing one about distance and one about speed. I downloaded graphing paper for graphing stories from Dan Meyer. I played the first video, asked students to graph it, and showed the answer. It was like lighting a fuse and seeing the classroom first fill with expectancy and then erupt in a constructive chaos of discussions about distance, displacement, speed, velocity, mathematical modelling, acceleration... "Look, the steepness is the same on the way from the camera as on the way back, so the guy who made the video assumes the speed of the dog was the same in both directions!" is one memorable quote.
It was a weird and funny experience to have the class completely ignore me and all my efforts to bring order into their discussions simply because they were so incredibly involved in figuring out how the video related to the graph and what we could infer from the graph about the beliefs of the grapher.

Then I showed the second video, about speed. Students immediately connected steepness or graph to acceleration and after a brief discussion when I asked "so what is the distance that the runner covered?" some students immediately replied that we must look at the area under the curve. I asked how we could calculate that area and students proposed dividing it into small sections, rectangles, trapezoids, triangles.

I believe that students developed a really strong intuitive connection to the main concepts of calculus, through their understanding of distance, speed, and acceleration. We will follow this up with more structured work using the graphing stories graphs to figure out approximations to average and instantaneous rates of change. That will take a full lesson. And then we're ready for functions and the limit definition of derivative.

It might seem like a long time (more than one week of class) to build up to the definition of derivative. In my opinion, conceptual understanding of derivative is necessary for everything that follows, especially when using derivatives (and second derivatives) to graph functions and for optimization problems. Without the conceptual understanding nothing else is going to make sense, will just be an imposing set of strange rules and lengthy procedures. With a solid understanding of derivative (and later integral) calculus comes alive, becomes beautiful, and leads students to ask, as one senior student just did: "What profession should I choose that lets me use calculus as much as possible?" :) :) :)

Most challenging: homework

Three and a half years into my teaching, I've now tried four ways of motivating students to do homework.

1. Hands off - my first year I was too overwhelmed with lesson planning and simply told the class "I expect you to master lesson content before the next lesson". Results in this class were the highest I've ever seen, but that may have other reasons than my homework (non-)strategy. 
2. Binder checks - this simply required way too much organization on behalf of myself and the students. 
3. Weekly quizzes - this works wonders in psychology, less so in mathematics. In maths, I'm concerned that weekly quizzes give some students weekly opportunities to fail, demotivating them further. 
4. Flipped classroom - didn't work, see previous post.  

I'm thinking in part that this is an uphill battle. In Sweden, attitudes towards children and childhood is very "let children be children" = much play, little work, minimum pressure. In a different culture maybe all the above strategies would work, maybe they wouldn't, I can't know. I do have the luxury of having students from all over the world in my lessons, and I've asked them about their previous school (and homework) experiences. Without exception, students from Russia and East Asia report doing much more homework (double or triple) in their previous countries compared to now in Sweden. 

They also report that teachers would: 

• assign much more homework, often give out exercise papers each lesson
• collect homework every lesson from every student
• mark every exercise from every student every day and give it back almost immediately (latest next day) 
• have almost no exams, letting homework be the assessment of choice

Considering that classes in East Asia are rarely smaller than 35 students, I have absolutely no idea how teachers find time for so much marking. Also not sure how students used to a more laid back system would react to a sudden change in this direction. Worth thinking about though. 

Flipped classroom - no more

It seemed like a good idea.

It worked for a while. How do I know? Because in class, more kids would take the initiative to come to me with questions regarding understanding rather than procedure. Because kids were getting group practice on sophisticated (more or less) questions done in class. Because I could see on thatquiz that students were doing their assignments and I loved seeing what was going well and needed re-teaching.

Then it didn't work anymore. Maybe it was in part because I was gone for a week and lost the connection with the class somehow. Students also say watching youtube videos was good (because you can replay) but also bad because you couldn't interact with the teacher to ask questions, etc. Of course, it's a much less social type of learning, and losing the social side can be hugely demotivational. The thatquiz homeworks, while done by many students, were sometimes done in something like 2 seconds per question - indicating that just maybe the students were copying other students' answers or in other ways avoiding actually thinking about the exercises.

The test results were horrible. We had a wonderful constructive chat afterwards, with the class unanimously agreeing to go back to the "old way", with going over new content in class and independent practice/review at home.

Even though it's disappointing that the flipped classroom strategy didn't work, I'm happy we tried it. It's this kind of experimentation that makes teaching fun, and the only way to eventually find more effective teaching strategies. If we hadn't tried it, I would always have wondered if we were missing out on something spectacular. Also, the kids learned a lot from this. Many of them still want me to post videos and they use thatquiz for extra practice with direct feedback, strategies they learned from our flipped classroom experiment.

Afterwards, I talked at length about why we tried flipped classroom (to get more time for problem solving / exercises in class because the students were not doing them at home). We talked about homework and how much they were doing. Very little, it turns out, and is rather unsurprising. We talked about if they want to do homework (unanimous yes) and how I can support them in doing it. I proposed, and the kiddos agreed, to try Mimi's strategy. In short, we agreed that the students will do at least 12 exercises from the book every week, I will collect them and choose one exercise per student to mark and give feedback on the overall level of difficulty of the exercises the students have chosen to do. Students seem enthusiastic about this idea, and I'm excited to try out yet another homework strategy. :)

 

Flipped classroom and thatquiz

My lovely class of math students is doing great work in class, but very little at home. So now's as good a time as any to experiment with something new (flipped classroom) and something old (thatquiz).

What we're up to:
Every assignment consists of a video (Patrick JMT has a wonderful collection) on some topic (sine and cosine rules, most recently) and some basics exercises. Correcting basic work is not my idea of fun or meaningful so I let students complete assignments on thatquiz. If you haven't seen or tried this absolutely magnificently wonderful site then you're in for a treat. It's free, no ads, very accessible and customizable, it offers direct feedback, marks all assignments for you, and you can make your own assignments or choose from hundreds of ready made ones. And it's constantly improving.

So anyways... in class, students may ask whatever questions they have about the topics covered in homework, but primarily class time is devoted to practice with exam questions, going over worked solutions, and laying the stage for the coming topics. Today, for example, we used basic trig in a guided investigation leading up to and including sine rule and trig area formula. Here's the file:

Right-Angled Trigonometry Practice and Development

I'm thinking hard about how best to use class time now that I can focus on exercises and problem solving. It's important for me to retain a constructionist approach at least in setting the stage for the new material which the students will meet in lecture-style videos in their homeworks. This is super exciting. More updates will follow.

Keeping it together

The textbook: first, let's learn about what a logarithm is, and fill in the blanks - a lot. Then, the laws, which we'll only learn if we practice simplifying meaningless expressions - a lot. Then, a bit about equations and application problems. Then change of base. Each in its own nice little sub-chapter. 

Me: Lets learn about logs, a'ight? What they are and ooh look they seem to obey a bunch of rules, I wonder if we can use that to solve equations and for applications?  

Bottom line is, it seems so utterly pointless to learn about logs, and log laws and then practice manipulating logarithmic expressions. Why would I  let my students wait to see the power of log laws in solving equations that previously left them dumbfounded? I'm even wondering, why teach them the change of base formula, when they can handle any equation just using basic log laws? If there ain't a need, why go there?

Useful things to do in class

I have the distinct sensation of landing in math teaching. It's so nice to feel like the years of wonderful but chaotic experimenting are finally manifesting into a set of strategies that I trust and feel comfortable with. Here are some things (in no particular order) we do in my math classes right now:

1. Do Now - every day, class starts with a Do Now, so students start doing math the moment they come in the door (best case scenario, in reality they still need some prodding to focus on the task). The Do Now task always recaps something the students learned the previous lesson, and serves as both retrieval practice, feedback for students on their level of understanding and skill and opportunity to correct misunderstandings.

What I like the most about the Do Now task is that it can often bridge the previous lesson to the current one.
For example, a recent Do Now task asked

"What are the x-intercepts of f(x)=x^2+3x+2? Discuss in pairs."

The previous lessons, students had solved quadratic equations. This current lesson, the objective was to graph quadratics from standard form. The Do Now bridged the lessons by giving students the opportunity to help each other apply their understanding of equation-solving to graphing.

2. Traffic-lights - these are my cheap-o "clickers". They are just laminated cards red on one side and green on the other. I tried them a month ago and loved them immediately, as do my students. Every lesson, during the Do Now, I hand out these cards. Students use them throughout the lesson to signal their understanding or need for further clarifications. When I explain something, I typically ask students a specific question such as "Are you able to explain every step of this solution to a classmate who is absent today?". If a student holds up red, I ask what step(s) need further clarification, and then ask a student who holds up green to explain those steps. I find that when students are made to take a stand like this, students who would otherwise pretend to understand do speak up. Also, during individual work, students turn their cards red side up if they want my attention.

3. Weekly homework-checks - this could be just looking through students notes during the Do Now, or even a mini-quiz integrated in the Do Now, or a much larger hand-in assignment. In whatever way I do it, I need to check student homework not because I'm so very interested in whether they did it, but because students are much more likely to do it if I check it. In my psychology teaching, every time I've done weekly homework checks, student test results have risen dramatically. Jury is still out on whether the same improvement is seen in mathematics, but I'm hopeful. I'm still working on a system to check homework comprehensively yet quickly.

4. Investigations, where appropriate - I've tended to overdo or underdo investigations in class. Now, I ask myself before each lesson whether the goals for that lesson are reachable by investigation within the time constraints of that lesson. Next lesson, the goal is for students to understand and apply function transformations. This is superbly appropriate for an investigation task, and most of the lesson will be devoted to it. Investigations are great, because of too many reasons to list here. However, investigations are time-consuming and don't always lead to the intended understanding and skill, so they need to be balanced with more structured teacher-led demonstrations.

5. Direct instruction, where appropriate - some goals, such as applying trigonometric relations or log rules, are more skill- than understanding-based and require lots of practice. In such cases, I prefer a structured teacher-led approach in which I or we together solve an exercise, then students practice individually or in small groups, and then the process repeats. This can also work well with less teacher scaffolding and more group-work, such as examining and evaluating solutions of problems.

6. Problem solving, where appropriate - sometimes, when solutions require many different steps, such as when students graph functions from first and second derivatives, or use derivatives in optimization problems, I prefer to put students in groups and have them figure out the solution to a problem themselves, before writing a structured summary on the board. I find that this type of scaffolding, students helping each other, helps students feel confidence and ownership of the solution method. If instead I was to simply present such a complex multi-step solution, students would be more likely to feel intimidated and to try to memorize all the steps instead of trust that they can construct a solution based on their understanding of relevant concepts.
I'd like to integrate WCYDWT style problems, but so far I have never felt that I have the time to do so.

7. Closing summary - OK, I'm not actually doing this one every time or even most of the time, but I'd really like to! When I do have the presence of mind to recognize that 5 minutes remain, I like to ask students to summarize either orally or in writing what they have learned during today's class. Sometimes, rarely, I use exit slips as well.

So - this is my math teaching in a nutshell. Lots of room for improvement still (and I hope I'll always feel that way!), but at least and at last there is some stability with strategies that promote quick and efficient feedback, confidence, understanding, mastery and fun.

As always, I welcome all feedback.

Challenges for the new school year

I'm really excited about two new responsibilities I have this year: starting a debate/speaking club, and heading my school's professional development through lesson visits program.

I've never done anything even remotely similar to the debate club before, but have been asking to do it all last year. Now I've got a few books and some videos and maybe can visit a nearby school and see how they do it... but mostly it'll be a trial and error work in progress.  I hope it will be tons of fun, as well as teach kids (and me) a zillion useful debate, speech, and argumentation skills.

Leading the lesson visits program is going to be awesome, and from the tens of lesson visits I did with my math colleagues last year, I think I'm on a somewhat solid footing. I firmly believe in this kind of professional development, and am very happy that my tiny school is finally committing to it. I'll be guiding my coworkers, sometimes coaxing and sometimes nagging them, to visit each other's lessons and then to talk about their experiences with other teachers. I have in mind some ideas all of which involve a framework for planning, visiting, and feedbacking the lesson. This book has some good suggestions. I know I've seen more great stuff on the blogosphere, and if someone'd care to point me in a promising direction I'd be most grateful.

Starting students on logical thinking

Every year, I make time in the first class of mathematics to introduce my students to the kind of thinking I expect from them throughout the year. That is, the inquisitive and logical thinking that'll prompt them to require and enjoy logical soundness in everything they are asked to learn in math class. I do this by showing them Lewis Carroll type syllogisms, such as this one:

• All babies are illogical
• Anyone who can handle a crocodile is not despised.
• Anyone who is illogical is despised. 

I ask the students to form conclusions based on these premises, and every year I find that students are completely stumped by this task. 

Most of them start by questioning the premises. 

"It's true that babies are illogical, but you can be a pretty horrible person and still be able to handle a crocodile..."

I can work with this. After all, it's important that the premises are sound, or else everything is on shaky ground indeed. However, what the students are clearly telling me that they are not able to either do, or understand, the task I am setting before them. In short, they are failing the standard Piaget formal operational stage test.

To be fair, more recent research (after Piaget's) is implying that a disappointingly small percentage (about 20%) of the adult population can do this kind of formal operational task. Also, to be fair, it's probably not that students or adults are "not able", but rather that they don't understand what's expected of them because of a lack of practice with these kinds of tasks. Still, when I showed this video to my psych class (16-17 year olds) a few years back, a large majority of students agreed with the boy instead of the girl. 

So, back to my first class lesson. Walking around the room, I explain to the students what they are asked to do. "Suspend reality for a moment, let's pretend that these premises are correct. Then what can you conclude?" Once students had understood the task, they could solve it with only a tiny bit more prodding.  

I then show them a simple equation, say x + 5 = 7. Students are happy to conclude that x = 2, but now I hope that they are understanding that they are making a logical conclusion based on a premise which might or might not be true. 

I follow this up with the best example of mathematical problem solving that I know of: a Sudoku. Not only is Sudoku great because students are making logical conclusions every step of the way, and in a way such that the reasoning is their own and the "logicalness" of it is readily visible to them, but Sudokus also illustrate many important principles about mathematical problem solving. I'll write about that later.

I think it's important that students get this initial "feel" for logical reasoning. In fact, I ask them to feel it - for me, when something is logically sound, I have this satisfied calm feeling in my stomach. And when it's not, there is a worry, almost like an unpleasant itch. I know some students feel this more than others. Last year, suddenly, a student in the regular (non-accelerated class) simply refused, for weeks, to learn the basic rule of differentiating polynomials because she had missed the class where it was explained why this rule works.  I've encountered this many times already, often in students who are considered to be "weak" at math. Sometimes, when these students' need for logical soundness has been satisfied, they become much "stronger". I like and respect this resistance to learn things without understanding them. 

Throughout the year, I'll be coming back to this initial lesson on logical thinking. I will frequently ask students to explain why something is true, and not just to show me that they know how it is applied. I hope my students will understand, and appreciate, that this insisting on logical reasoning and understanding is not something "extra", something added to the already considerable pressures of their studies. Instead, at least in my experience, 20 minutes of effort at understanding why something is true pays off in exam scores better than hours of practice with more or less routine exercises. After all, in order to fully understand why the rules of logarithms are the way they are, you must fully understand logarithms. In order to solve routine exercises by looking at solved examples, you need only be able to use a formula. 

Of course, this introduction to logical thinking leaves many finer aspects unexplored. When students were attempting to draw conclusions based on the above premises, many offered that "Anyone who can not handle a crocodile is despised" and other illegal moves. Ideally, I'd like to spend more time on these kinds of issues, and in the Mathematical Studies class there is even a section on logic which I really like. Also, students may not be aware what constitutes a logical conclusion in all cases. As Sue recently pointed out, many students are not able to distinguish between an example and a mathematical proof of Pythagoras rule. Such issues can be overcome with counter-examples, but also necessitate a discussion of the difference between inductive and deductive reasoning (and why deductive is so superior! :)). 

Overall, I'm happy with this start. We'll see how it plays out over the coming year. 

Random news

The school year was ending, has ended.  I haven't written anything for a long time, since these last couple of months have been all about exams and with very little actual teaching. Also, for the first time I've supervised a practice teacher - hey there T. S.! - and thereby had so many good discussions about teaching and learning that blogging seemed superfluous.  Definitely doing supervision again soon. 


Nevertheless, here is a brief update: 


IB exams are standardized, high stakes, and high pressure. I've been feeling very ambivalent about them, not least because I fear that those students who experience lots of anxiety in exam situations are not able to show the full extent of their understanding and skill in such situations. During the years, I have seen quite a few students freeze even during regular in-class exams - students have cried, left the room, or just sat though the full 2 hours and then handed in a blank paper. This time, however, I've noticed that these same students work through their difficulties eventually, and I'm starting to see that these high stakes exams provide a significant growing opportunity for my students. If nothing else, they are learning to perform under pressure. That just might come in handy later on.


The school system has been the focus of much media attention throughout the spring. Dagens Nyheter, one of the two largest daily newspapers, has features a series of articles highly critical of the developments in the Swedish school system these last 25 years. Those targeted in this series are school leaders, but above all unions, city and state government as well as university professors. The picture painted is one of tragic downfall of education quality as well as teacher's resources, status, and salaries.  Sigh. 
Then, soon afterwards, our minister of education Jan Björklund announces that from autumn teachers are obliged to include students' absence in the grades for every subject. Teachers have had no say in these developments, and the ruling contradicts the recommendations of every school agency that was assigned to investigate this proposal. It seems our current government is intent on following old-school US methods, while the US is (hopefully) already moving on. 


Lastly, my school is moving to a much more central Stockholm location. This means that we're likely to see an increase in students applying to the school, and perhaps will be better able to admit only those students who we feel are prepared for the IB program. I've been looking forward to this move for years - it shortens my commute from 90 minutes to 15 minutes each day.

Why teachers like me support unions

Why teachers like me...

What's a "teacher like me"? That would be a young third-year teacher with a masters' in mathematics, teaching mixed level courses at a small city school offering only the challenging IB programme to kids who generally are from low socio-economic groups in Stockholm, Sweden. I love my work, LOVE it, and spend way too much time striving to develop effective teaching based on sound research principles.

...support unions. 

I'm a member of one of Sweden's largest teacher unions, Lärarförbundet. Financially, and in terms of job security, there is no point in this membership - I have tenure and unions do not influence salary. Instead, the reason I support unions is because it is important to offer organized resistance to changes initiated, and sometimes even determined, by the many levels of school leaders.
Examples of such changes are increases in teaching hours as well as in administration and class sizes, all of which means decreases in the time we have to do a good job.

I am always astounded by how little non-teachers seem to be aware of how much work it takes to plan a good lesson. I have friends who teach at in universities, as teacher assistants; they have 4 hours for each hour of "lesson" (more like a seminar, really). Usually high school teachers have about one hour per hour of teaching, and this is supposed to cover preparing the lesson as well as marking any work collected from the students during the lesson. Teachers at lower levels typically have even less.
At the same time, teachers are blamed for not being good enough at explaining, engaging, motivating, fostering, caring, investigating, communicating with parents, cooperating with other teachers, organizing events, devising individual development plans, following new research, taking part in professional development, and documenting results. The expectations are wildly unrealistic given the constraints; as a result teachers get sick, get cynical, get divorced, or leave for something different. Meanwhile, what is happening with the students of these sick, cynical, sad and absent teachers?

These kinds of changes are happening in Sweden, where currently there is no limit on the number of hours a teacher can be ordered to teach. Some twenty years ago the average was about 14 hours (60 minutes) per week - I currently teach 18.5 hours and in other schools teachers teach up to 30 hours per week. We also don't have any restrictions on how many other tasks a teacher can be assigned, and there is no lower limit for how much time for planning and marking a teacher is entitled to. The only limit is when a teacher is assigned so much work that he or she becomes overworked and sick, and even then measures taken are to temporarily repair the damage, instead of permanently fix the situation.
Does anyone think this is reasonable? Does anyone think that teachers who are talented or lucky enough to have other options, will want to stay at a job such as this one?

This is the reason I support unions: by myself I can achieve little to improve even my own situation, and much less anyone else's. Together we can resist and maybe even reverse some of the developments which over the last couple of decades have undermined the high quality in education that we as teachers, as students, and as a community, are striving to achieve.

What is this job?

I just saw the final episode of a Swedish TV-show called Class 9A.  The idea is very interesting: a school is in trouble because the teachers and principal are not doing their jobs well, and one ninth-grade class in particular is singled out to receive help from expert teachers. These experts come a few times per week during one term (August through December) and coach the regular teachers while at the same time teaching the troubled class. This is the second season of this show, the original aired a few years ago in a different city and school than the current season. Interest for this show has been very high, probably in part because it ties in well with the current political emphasis on school reform.

The team of expert teachers - come to save class 9A at Mikaelskolan

One prominent feature that distinguishes the expert teachers from the regular teachers is the amount of effort the experts put into each student. For instance, at one point Stavros, the expert math teacher, spends 2 hours giving one student private tuition and in the previous season of this show, one teacher fetched one student from home every morning to help the student come to school. Overall, the experts imply that one reason things are not going well is because teachers are not doing enough for the students.
In general, it's not difficult to understand that these expert teachers are probably given MUCH more time and many other resources not usually afforded to regular teachers. I understand that what they are doing is not realistic for regular Swedish teachers who, on average, have a little more than one hour for preparing (and marking) each class.
But then, how can you tell what amount of effort is realistic, how much is enough?

Another issue, related to the first, is what it is we are aiming for. In this show, the goal is phrased exclusively as "getting all students to pass and be admitted to high school". The principal says this, the experts say this, the regular teachers say this. Yet, surely this is absurd.
First of all, education for me is about learning, not passing. I would find my job utterly meaningless if I thought of it primarily in terms of getting students to pass, or even getting students to achieve high marks. I teach because I love and believe in the power of education - because education opens a window on understanding this world we live in, and gives tools which increase ones chances of leading a rich and meaningful life, and because it is a privilege to see and assist students' growth.
However, there is a second problem with the goal that the point of school is to get students to pass: in Swedish school systems the teachers themselves are setting the grades. Guess what happens if the teacher is aiming to pass as many students as possible? They pass. And then they come to high school and I wonder what on earth they have been doing for nine years when in tenth grade they cannot even do multiplication with negatives.
This becomes especially ludicrous in the context of Class 9A, because it is set like an experiment and everyone is asking all the time "how is this going? are we seeing any results from this awesome and costly and highly publicized intervention we're conducting?"  There is a reason scientific experiments are often conducted blind. Researcher expectancies can have a tremendous effect, and doubly so when the researchers are the ones evaluating their own work and under great pressure to succeed.
Yet passing rates and grades and standardized tests remain the most clear cut and simple way we have to measure and compare learning. So how can we formulate goals that do justice both to education for its own sake, and to the reality of grades and test-scores?

What do you make of this?

On the subway, just now, I overheard two high school students talking about their math teachers:

- ...the old one? did you have him? He was f-ing horrible.  

- I heard so. 

- yeah cuz he gave us these difficult problems and no one got them and one time I was working on a different problem and he asked if I could do the one on the board and I said no and then he said, like in front of the whole class, "oh that means you have a lot to revise!".  

- jerk 

- yeah... 

(pause) 

- but you know then we had a different teacher and she was awesome! if someone didn't get something she was like "OK let's do this again!" and she explained it and was really patient. It was great.  

- yeah there are so few teachers like that, who can get contact with their students, no wonder at the end of high school everyone is so tired of studying.

Overhearing this exchange was really awesome for me, because it gets right to what I've been most concerned with all year: the differences in what students want, and what I want.

Because what these students are saying, what I think most of my weaker math-students would say if I asked them, is that teachers that give a lot of help, teachers who guide students through problems, teachers who make mathematics simple, are the good teachers. Teachers who give challenges, and then expect students to stick with the problem until they solve it, are the bad teachers.
Meanwhile, what I want is to engage students in thinking, to show them that they can do math more or less on their own, form their own conjectures and prove them, too. I hope that this way math comes alive and students develop interest and confidence as well as understanding and skill.

This is nothing new, and part of the invisible contract which Ben Blum-Smith wrote about beautifully in this post a year ago. But I do wonder about the effects of this discrepancy in teacher and student attitudes.
Could it be that these students are right? That, for them at least, and at this late stage in their relationship with mathematics, it is better to provide a crazy amount of scaffolding and to sacrifice ideals of creativity, fun and even deeper understanding for the benefit of getting the students to feel confident and safe in math class.
This has been my main question this year and I still don't know how to answer it.

Developments in Sweden

I haven't been writing about all the strange changes in education policy that are taking place in Sweden because the audience of this blog is mostly from the US and, frankly, I just assumed y'all don't give a damn.
But yesterday something happened which is just too weird for me not to comment on it.  A debate article calling for more direct instruction was published in our largest newspaper, DN.
The author? Our secretary of education Jan Björklund.

A little bit of history: 
Contrary to popular belief, Sweden is not very centrally controlled, and education has, since the early nineties, been controlled and paid by individual cities. We have, also since the mid-nineties, a large number of charter schools which operate under laws more lax than perhaps anywhere else in the world. The government occasionally updates the main law of education ("skollagen") and published standards and grading criteria for each course, but since the early nineties government has actively avoided regulating or even advising teachers on how to teach, and even what to teach. Now, however, everything is changing.
I'd say it all started  when Alliansen, our right-wing coalition (our right is still USA left) entered power with the explicit goal of getting rid of the wishy-washy education ideals that the left-wing had established over an (almost uninterrupted) reign of many decades. These wishy-washy ("flummiga") left-wing policies (transmitted to teachers not through laws, but rather through teacher training institutions) had long been emphasizing the individual freedom and responsibility of the students, the importance of social goals in schools, intrinsic motivation, and an avoidance of grades and (in the most extreme instances) homework. In Sweden, children still do not receive grades until 8th grade. For teachers, all this translated into trying to get students to lead their own learning while at the same time creating learning opportunities individualized for each student.

These kind of education policies have fallen into disrepute in part because of convincing research reports that teachers have been unable to implement individualized teaching the way left-wing policy makers have envisioned. Instead, teachers have approached the ideals of student responsibility and individualization with the kind of laissez-faire leadership which is actually the absence of leadership. Students have been assigned project-based group-work with very little structure, or just told to do individual work in the textbook. As a result weak students, left to their own devices, became even weaker. For this and other reasons (charter schools causing the inflation of grades, as an example) Sweden's results in PISA and TIMSS decreased substantially.

So when the right-wing coalition said they wanted to move in a very different direction, emphasizing structure, clear standards, and discipline, many teachers, parents and concerned old-timers felt that finally politicians were talking sense about education. It's likely that this convinced many voters to vote for the right-wing coalition for the first time in their lives. Recently, however, there is an increased awareness that while it certainly appears that Björklund is very sensible in these matters, this is only relative to the utterly unrealistic idealism that the left-wing coalition has been advocating for years. In fact, Björklund, maybe because of his military background (he was never a teacher), is looking for simple and rigid old-school solutions which may solve some problems, but are likely to cause others.

Some reforms which are taking place this year:

• Grades (letters from F to A)  from 6th grade. 
• More specific national course plans for all national courses. Schools will no longer be required to produce their own interpretations of national course plans. 
• New grading system (our third in twenty years), in which it is more difficult to achieve the higher grades. It will be written in prose, however, and it is a challenge and a mystery to most teachers how to use this system for actual grading. 
• Students in vocational high school programs will no longer be eligible for university education, although the schools will be obliged to offer students additional courses  for eligibility should the students wish so. 
• Teachers will be required to obtain  a teacher "licence". This is just a word and a document, it does not require any more training or proof of competence than has been required up to now. Yet the unions and the government are sure that this piece of extra bureaucracy will magically increase the status of the teaching profession ("after all, doctors are licensed practitioners, and they have high status and salary"). I think this is a stunning case of correlation mistaken for causation, and I cannot for the life of me recall why, a year ago, I thought this was a brilliant idea. 

And finally, yesterday, Björklund writes that teachers must reclaim the position of authority in the classroom. 

He announces that the government will soon pass a law which states that students have the right to receive continuous and active support from the teacher through structured teaching. This could of course mean many different things, but Björklund explicitly states that the teacher should address the whole class as one group ("undervisning i helgrupp"). He also suggests that the teacher should maintain an active dialog with each student and together investigate different questions and problems. He does not seem to be aware these two suggestions are mutually exclusive; one reason why teachers often divide students into groups is because it is impossible to give each student in a large group sufficient attention and feedback. 

I am, however, not unhappy with this most recent development. For too long has direct instruction had a bad rep, and for too long have teachers struggled to teach every students according to that student's individual needs (and then felt guilty over not being able to meet such an overly ambitious ideal). Good direct instruction has it's place in teaching, as one instrument among many others. In general, what I hope Swedish teachers will take away from all this is that teaching is leading learners in learning, and that even though leading can mean very different things depending on whether one is lecturing or organizing group-work, it is the responsibility of the teacher to maintain control of the learning opportunities presented in class to all students.

How, and how not, to explain graphing trig functions

Graphing trig functions should be relatively easy for students who have already mastered general function transformations with quadratics and exponentials.  Nevertheless, there are so many steps involved that I think an emphasis on procedure and practice is justified in this case. With that in mind, I aimed to build on students' prior understanding of transformations while giving them an outline for how to graph trigonometric functions of the form f(x) = A*sin(B(x-C))+D. 
I completely butchered the first lesson on graphing trig function and promised my students (all patience and humor to the end of that horrid, horrid hour) to make amends by presenting the procedure crystal clear the following lesson.

What I learned from the failed lesson:

• Save your geogebra files before moving into the classroom, as the computer can and will have a fit and shut down unexpectedly (bye bye 30 minutes of work) when you plug in the projector.
• When creating a nice mnemonic for how to graph these functions (I came up with BC AD - Before Christ, Anno Domini, get it?) go ahead and check first that this is actually a reasonable way to graph these functions. BCAD isn't, in the sense that it doesn't allow you to place points each step of the way. 
• If something isn't working, stop doing it. I actually persisted for 40 minutes giving students one retarded way after another to graph trig functions, when I should have assigned some other work and taken a few minutes to think things through. It's a testimony to the loveliness of my students that they persevered and, when at last I came to my senses and called the whole thing quits, laughed with me (and, admittedly, at me). 

In my defense, when I immediately after the lesson googled how to graph such functions... NOTHING came up. Most websites and videos explain A, B, and D - but skip the C. I sat down with my colleagues and we  came up with the following method:

• D gives principal axis, so make a line there.
• A gives amplitude, so make lines at D+A and D-A. Now we know the range of the function. 
• C is the phase shift. For sine, put a point at (C, D). For cosine, put it at (C, D+A). 
• B is the frequency, and 2*pi/B is the period. Figure out how long one period is and place a point at (C+period, D) for sine or (C+period, D+A) for cosine. 

Now draw a full period of the function between the points you've made.

Mnemonic? I'll ask the kids to make one up. All my ideas involve dog ate cat bones and dingo ate chubby baby.

I'm afraid the lesson itself won't be any more exciting than demonstration and practice. But I think sometimes (especially after the havoc I put them through last time) demonstration and practice is what the kids want most of all. For homework, I give them a modeling activity involving the movement of the sun in three arctic cities.

Features of trigonometric functions

This Monday, a very basic investigation of trig functions/reinforcement of function transformation skills with a standard/honors junior class. I was toying with the idea of just throwing a matching activity to the students, but I'm still new to using those and besides they take so much work cutting and organizing.
The activity below is very characteristic of how I teach and what I'm aiming for is student ownership (discovery, confidence, etc) and connection to previous materials.
Students will be doing this in pairs, each pair doing either the sine or cosine activity. Afterwards, pairs will combine into groups of four so that each group has one pair which has done the sine and one which has done the cosine. They will then compare and discuss the question at the bottom.

Any suggestions for improvement are, of course, welcome.

Investigation Transformations of Trigonometric Functions