Learning (by) Teaching

Flipped classroom and thatquiz

2012-01-30T19:53:00.001+01:00

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

2011-12-12T09:20:00.001+01:00

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

2011-11-12T13:12:00.000+01:00

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

2011-09-05T22:58:00.000+02:00

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

2011-09-03T11:44:00.001+02:00

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

2011-06-10T15:42:00.000+02:00

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

2011-03-22T00:13:00.000+01:00

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?

2011-03-21T23:02:00.001+01:00

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?

2011-03-19T19:06:00.001+01:00

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

2011-03-15T22:30:00.000+01:00

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

2011-03-13T22:24:00.000+01:00

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

2011-03-06T15:25:00.001+01:00

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

Groupthink about group work?

2011-02-22T23:20:00.000+01:00

My students always work together, in the sense that they discuss solutions whenever they get stuck, collaborate on investigations, and argue over concepts, methods, and proofs. They are (almost) always on task - and in the classroom buzz you can pick out exactly the kinds of things the above texts recommend: students comparing solutions, explaining to each other, arguing math...

In theory, this is great, and previously I've actively encouraged such cooperation. After all, what better way to learn to communicate math, sharpen argumentative skills, look at concepts and methods from different perspectives - well, y'all know the drill. Every modern text on teaching and learning mathematics seems to wet itself with excitement over group work. A classroom full of quiet students working alone with their (gasp!) textbook is, or should be, a remnant of older and more ignorant days.
But now I'm starting to change my mind. I see how students use each other as crutches, easy support instead of the harder work of figuring something out by oneself. Two main negative effects from this: students believe that they have mastered and understood something which they really haven't, and also that they are deprived of the chance to build thinking, memory and confidence by single-handedly struggling with math problems and concepts.

In an ideal world students would be doing this kind of individual work as homework. This is not an ideal world.

So I'd like to incorporate more of that quiet individual work in class, but at this point I'm not sure how to fit that in with the explorations we frequently have going on. It's a matter of priorities, I'm sure, but even without individual work we're struggling against the clock every lesson. Right now I'd just really like a structure that I can use for each (or at least most) lessons and which includes a brief warm-up review of the previous lesson, an exploration, discussion/summary, group practice, and individual practice. But unless these components on average take less than 10 minutes each (they don't), something's gotta give. It's a pretty nasty dilemma and I welcome any and all suggestions.

First attempts

2011-02-05T02:20:00.000+01:00

It's Friday night, and what better way to spend it than read through nrich and ncetm resources? (I don't know whether to take a lengthy think about my life or just be happy that I love my work).
Anyways, I found a few ideas that I've been meaning to try out in my teaching but haven't gotten around to just yet. With a trig unit coming up, and the ncetm resources being so very well-written and inspiring, I decided to jump in and design my own resource based on two (for me) novel ideas.

1. Always, never, sometimes true classifications: Supposedly great for promoting conceptual understanding and constructive discussion.

2. Matching activity: I like this one because it can so easily be differentiated by creating more options and/or leaving blanks.

Here is the actual file, which will probably be changed a bit before use in class on Monday.
Matching Basic Trig

(I do not know why Scribd likes to minimize letters and rotate pictures. The original document looks neater.)

Something I'm not very happy about is the increase in paper copies these kinds of activities seem to require. I'll probably project the classification task on the board instead, and am grateful for any other suggestions.

Testing, testing...

2011-02-04T20:47:00.000+01:00

This New York Times article can't be news to anyone by now, and I look forward to reading the details of the original study as soon as Science makes it online accessible to my university library. Of course this is nothing new. Francis Bacon, in the 1620s, said what dozens of research studies in the 1900s have confirmed:

"If you read a piece of text through 20 times, you will not learn it by heart so easily as if you read it 10 times, while attempting to recite it from time to time and consulting the text when your memory fails."

So I've happily been putting testing to more use with my students. Mostly, I'm using the opening and closing activities from Every Minute Counts, and the good news is that it's been very easy to do this in every class (both math and psych).

This is what it looks like right now:

Start of class: "put away notes and books and try your best to solve the problem on the board". Typically I'll have a basic problem that tests recall of previous lesson or homework. Sometimes I'll include another problem which opens up to whatever material we're doing the current lesson. As an example, last time with my Seniors, I started with the question "P(getting 2 sixes by tossing two dice once) = ?" and on the other half of the board had the question "P(getting two pink socks out of a drawer with 3 pink, 2 orange and 2 red) = ?" This led us straight into the distinction between dependent and independent events, and thus served two aims at once.

End of class: "list the main ideas from this lesson". After a few minutes they are allowed to compare their list with a classmate, and a minute or so later check their notes.

This has been incredibly easy. The main difficulty has been that students seem unused to, or unwilling, to let go of notes and book and classmate-support. I've spent a significant amount of minutes convincing them that this is a good idea, even if it feels frustrating to not remember everything you think you should. I'm usually strong at starting, weak at following up - so the fact that this is working and growing is a sign to me that this is worth pursuing and I hope that students will learn to test themselves while doing homework or revising as well.

How rigorous mathematics should be taught

2010-12-29T19:26:00.008+01:00

Following up on the discussion on f(t) about how to teach log laws, I'd like to share the primary school video I mentioned in the comments.
Actually, there are at least two videos, and this brilliantness is produced by always brilliant Deborah Loewenberg Ball.

1. Sean numbers*: Ball improvises a lesson about even and odd numbers, wherein her third-graders derive the precise definitions.

2. Betsy's conjecture: wherein third-graders explore proof.
You can also find transcripts and teacher notes for these videos.

I first read about them in this NYT article, which contains many other ideas worth thinking about.

*Side note: I LOVE how Ball engages her students by giving students' names to conjectures and numbers!

My intro to Logarithms

2010-12-22T12:48:00.001+01:00

This year was the first time I taught logarithms, and since logarithms is a tricky subject for many students (it was for me, for many years) I wanted to make sure to get it right from the start. In my school, it is very common that students decide to switch to the lower level math when they encounter logarithms. I was determined that this not happen this year.

There were several awesome resources I found when browsing the blogosphere, including James Tanton's approach of making a riddle and changing the name. I also loved Dan Greene's idea of having a symbol, L, instead of the word log.

However, the idea that really struck me was in a comment by Mr H to JD2718's post on logarithms. Mr H suggested that we start with logarithmic tables, and go from there, and that's kinda what I did.

Table of Bases and Powers

This is how the class actually went:

Give the class a few exponential equations which they can solve by converting to the same base. Intermix with these a few equations which cannot be solved that way. The students must graph or be stuck.
When a sufficient number of students are bewildered, give them the table above and ask them without further explanations if perhaps this table can be of some help. Let them figure out how to read the table and use it. Most of my students were able to figure it out quite quickly.
Ask students: what are you doing, how would you explain it to someone who missed today's class? Think pair share one minute per step.
Divide whiteboard in two, on one side write Logarithms and on the other write Square roots. Compare and contrast different features, like how both are something you do to something else, how square roots and squares are inverses, logarithms and bases are inverses (they "undo" each other if students haven't done inverse functions yet) and how they are used in specific examples.
Ask students to return to table and solve 10^x = 50. Show them that calculator can do it as well, but that for now base 10 problems are the only ones calc is useful for.
Give a few more exercises using table and calculator.
Show youtube logarithm song.

Homework: students received a link to a google-form to answer some questions about logarithms, so that I could check their conceptual understanding. Only about half of the students did this but their answers showed good understanding.

Next class, I started out by giving students this handout:

Using Logs for Effective Graphing

The students were quite impressed with this new tool to fit different size data on the same scale, and were even more intrigued once I showed them the xkcd graphs depth and height.

I also showed them pictures of the richter scale such as this one and students were intrigued and showed good understanding such as "Look, the Chile earth quake was about 10^3.5 times stronger than the Hiroshima bomb!". Music to my ears.

Then I introduced the laws of logarithms. I butchered that one. Check out Kate Nowak's recent post for a killer idea instead.

Next class, we looked at logarithmic functions. I started out by giving students a tiny review of inverses ("they undo each other, reflected around y=x, x and y are reversed") and then gave them this worksheet.

Logarithmic Functions

I really don't see the need to go into detail with log functions so I didn't.

Natural logarithm was introduced as inverse of natural exponential function and little else was done with it.

Test results were decent, no one treated logs as an object, and no one switched to the lower level. :)

Scratch

2010-12-08T13:34:00.000+01:00

A friend today told me about Scratch as a possible fun and simplish move towards that Conrad Wolfram vision of computational math in schools. I'm currently too groggy from post-surgery morphine to dig in and create any teaching material using scratch right now. Has anyone used this for teaching yet?

Friday article summary - what is "good" teaching?

2010-11-26T16:04:00.002+01:00

When Good Teaching Leads to Bad Results: The Disasters of "Well Taught" Mathematics Courses
Schoenfield, A. Educational Psychologist. (1988)

Original article is available here.

Judging from how often I've come across people referring to this article, everybody but me must have read it long ago. Probably during teacher training? Teacher education in Sweden is ridiculous and at least mine did not include any research articles of any kind. So I read this article today for my own sake, just to figure out what it is everyone else is talking about.

Quick summary

Background: When this article was published in the late 80s, there had already been substantial research done investigating the relationship between mastery of procedures and understanding of concepts and problems. For example, Wertheimer had shown how students proficient in the four arithmetic operations solved the silly long way questions such as (274+274+274)/3, and similar results were found in other areas of mathematics as well. Schoenfield and others had identified some beliefs that students hold about mathematics learning:

Belief 1: The processes of formal mathematics (e.g. 'proof') have little or nothing to do with discovery or invention. Corollary: Students fail to use information from formal mathematics when they are in 'problem-solving mode.'
Belief 2: Students who understand the subject matter can solve assigned mathematics problems in five minutes or less. Corollary: Students stop working on a problem after just a few minutes because, if they haven't solve it, they didn't understand the material (and therefore will not solve it).
Belief 3: Only geniuses are capable of discovering, creating, or really understanding mathematics. Corollary: Mathematics is studied passively, with students accepting what is passed down 'from above' without the expectation that they can make sense of it for themselves.
Belief 4: One succeeds in school by performing the tasks, to the letter, as described by the teacher. Corollary: Learning is an incidental by-product to 'getting the work done'"

Method: "When good teaching leads to bad results" was a case study of a class in a well-to-do suburban upstate New York school. Students in this school consistently scored very high on the New York Regents examinations, and the teacher of this class had a good reputation among students and colleagues. The class was very well run, with students spending a large percentage of time on-task and carefully following teacher directions. Schoenfield observed this class regularly during the year, and filmed the unit on geometric constructions which took place just before the Regents exams.

Results: Schoenfield mainly reports about the unit on geometric constructions, in which he observed how the teacher consistently emphasized accuracy and memory rather than understanding of why the methods of construction were successful. At no point were the students asked to, or encouraged, to apply what they had learned earlier in the year about proofs to understand and create their own constructions. The teacher was teaching for maximum performance on the Regents exams, in which questions likewise required no understanding or creativity in constructing.

Conclusion: Even very good teachers with strong students teach in such a way that the above mentioned beliefs are reinforced, and that a main reason for this is the nature of high-stakes assessments.

Some thoughts: obviously, high stake exams matter to how students and teachers approach learning and teaching. Because 80% of my students' marks are determined by their performance on the final IB exam, over which I have no influence, it is very tempting to follow the syllabus and exam format very closely. This is a double-edged sword: it keeps standards high, but narrowly procedure oriented. I'd like to include more conceptual questions on our in-class exams, but it just wouldn't fly with the students or, I think, school leaders.
Within these restrictions, however, there is a massive amount of freedom, only limited by the "contract".
Schoenfield also points out that many teachers and students use the textbook as an authority on what to learn and how to learn it. With courage and effort the textbook can be reduced to a helpful preparation for exams.

High stake exams are necessary - this I firmly believe from watching the inflation of marks and deflation of mastery and understanding among Swedish schools competing with each other. But the exams need to include questions that check conceptual understanding, in order to encourage teachers and textbooks teach accordingly. In the mean while, it is possible to teach that way anyway, and trust that the skills necessary for high marks on the exam will follow naturally.

The Great Divide

2010-11-25T22:25:00.001+01:00

An April post on Research in Practice caught my interest, and made it very clear what's been going on for me the last year and especially since I started daily reading all the great math teaching blogs out there:

There is a growing divide between my students' and mine perception of what good math teaching and learning looks like.

Simply put, I'm breaking the math-class contract.

Some examples of what I do, or try to do, that many students find weird and somehow wrong:

Rarely use the textbook in class
Give them open-ended questions
Ask them to explain concept and rules - not just apply them
Almost never give answers to problems, or if I do - don't give the solution
Encourage the students to discuss solutions and convince each other
Give them problems that require hours of work
Give them problems they cannot solve

Now, to be fair, my higher level kids are enjoying themselves plenty. Dissent is negatively correlated with achievement levels, which is kinda obvious really but also ironic considering how many of the lower level kids wouldn't even be so weak if only they had learned that math is about independent and creative thinking.
Weaker students need more structure and safety, but it traps them in a vicious cycle of low interest, low effort, and low results. What can be done, during the senior year of high school?

Teaching theories in psychology - how?

2010-11-20T15:09:00.000+01:00

In my teaching of psychology, I am keen that my students should have knowledge and understanding of the science behind the concepts, models and theories that I ask them to learn. I do this by giving the students tons of research study summaries, and then ask them to use this research to back up whatever they are arguing in their essays. This is going very well. The students enjoy the studies, are getting very proficient at evaluating them critically, and are recognizing the need for scientific support for whatever claims they are making. Here is the problem: the students are not learning theories.

I do describe the theories, I have them discuss the theories and evaluate them and explain them in their own words - but they are not retaining this information very well. I think the studies are more concrete, have a storyline, and are therefore easier to lodge and retrieve from long-term memory and so the students are focusing their energy on studies alone. How can I connect the theories to the studies in such a way that students will remember and appreciate the theories?

Here are some thoughts. I would really like to see suggestions from readers.

Include concepts and theories on a check-list of required knowledge for each unit.
Always have students write a paragraph summary about each theory before looking at the research studies associated with it.
Regularly use mind-maps with a central concept which branches to theories, which branch out to studies.
Require students to name and explain relevant theories in their essays, before going for the research studies.
Occasionally ask students to defend an argument just using theories, without studies.
Engage in discussion with students about how science works and that theories and studies are equally important to the construction and validation of new knowledge.
Ask the Theory of Knowledge teacher for help making students understand the importance of theory.

Of these, I'll try 1, 4, and 5 for now. There will be a second post about how it goes.

Friday article summary - concrete vs abstract?

2010-11-20T04:14:00.003+01:00

The Advantage of Abstract Examples in Learning Math
Kaminsky, J. et al. Science. (2008)

When learning new mathematical concepts, for example the concept of mathematical groups of three objects, is it better to start with a concrete example or with a generic representation? This is the question Kaminsky and colleagues at Ohio State University investigated in their 2008 article.

80 college students were divided in four groups, one of which received an abstract introduction while the others received one, two or three concrete examples (one of which is illustrated below as "Concrete A").

The other two concrete examples involved pizza slices and tennis balls, and were presented in a like manner. All concrete, and the one generic, representations were taught together with multiple-choice exercises. Thus, which method was used for teaching counts here as the independent variable.

The dependent variable was transfer: whether or not participants could answer multiple-choice questions (isomorphic to the questions used during learning of the examples) regarding the exercise below:

(This picture is actually from a different Kaminsky article,

"Do Children Need Concrete Instantiations to Learn an Abstract Concept?",

but she refers to it in the present article.)

The results were that just learning the one generic illustration gave superior learning to learning one, two, or three concrete examples.
Kaminsky et al. then conducted three more experiments. In experiment two, students were explicitly helped to find the analogies between the concrete examples. This did not affect transfer.
In experiment three, students were asked to reflect on any similarities in the concrete examples and write their reflections down. The results were bi-modal. About half the group achieved high transfer, while half performed as before.
In experiment four, half the students were taught only the generic representation, and half were given first a concrete example and then the generic representation. Transfer was higher for both groups compared to the students in the previous experiments, but even here the one generic representation gave better transfer than concrete-then-generic.
The results of experiment 1 and 4 are illustrated in the figure below:

Criticisms: critics have argued that Kaminsky's concrete examples were not concrete enough, which is strange because they seem to be precisely the kinds of examples math teachers use all the time. A more relevant criticism is that the transfer test (the children's game) was abstract rather than concrete, so it's no wonder that the students who learned the generic representation performed better. I find this criticism to be quite important. However, I think that it is reasonable to predict that results would have been very similar on a more concrete transfer test. Other critics have questioned whether results from college students are generalizable to children. The answer seems to be: yes. Kaminsky did a very similar study with sixth-graders and here are the results from that study:

"Do Children Need Concrete Instantiations to Learn an Abstract Concept?"

Kaminsky et al., available here

These results have been mentioned in popular media, the general public has reacted strongly and mostly negatively. It seems that the belief that concrete is better than abstract is very important to many people.

So what does this imply for teachers? Kaminsky et al. argue that giving generic representations is superior to giving concrete examples to introduce new concepts and methods. However, it is conceivable that transfer would be even better if the sequence was generic-then-concrete. The main idea seems to be that introducing concrete examples before the generic representation somehow locks the students into a restricted way of looking at the concept. Also, it may happen that the extraneous information in concrete examples distract students from focusing on the core concept. Most teachers have experienced that concrete representations help students understand a problem or concept, as is supported in the Kaminsky study on children. But we need to be careful and evaluate not just the direct and immediate gains in understanding, but also more long term mastery and transfer.

The original article, published in Science, is available here.

Challenge: Help me create a good task which tests abstract vs. concrete and I'll test it, on 40 students or so, and do all the rigorous stats and everything. I actually need to do an experiment anyways for my psychology course-work and this is the most interesting idea I've had so far.

Time

2010-11-14T17:00:00.004+01:00

Dan Meyer has his important ratio no. 1 which deals with the worth of instructional decisions as a ratio of instructional value and time expended in class (or out-of-class homework). It's student focused: the time expended is student time, and rightly Dan wants to maximize value per minute.

I think I need to develop my own important ratio, and it needs to deal with my time planning and evaluating lessons and student performance. It could go something like this:

BTW, if I told you how I finally got an equation into blogger it would go something like this.

So, thing is, every single one of my classes is unique. I never have the luxury that some other teachers seem to take for granted, that a lesson plan can be reused for several lessons the same day or week. On average I teach 15 hours per week, which would have been 18 if I had been working full time instead of just 80%. I have about 1 hour available for each hour of class time - this hour needs to include planning, preparing, printing, evaluating, feedbacking, marking, and getting to and from class. In reality, I work about 130% of what I get paid for.

This is pretty common for teachers in Sweden, although in larger schools sometimes teachers will have several groups who take the same course. Some schools, mostly charter schools, will make teachers teach 22 hours or more each week, and give extremely little time for planning. In those kinds of situations, trying to design WCYDWT and similarly ambitious lessons is just absurd. Which is why it's great that there is a growign library of these lessons online.

What's it like in the US and elsewhere?
How many hours do YOU teach, how many are unique, and
How much time do you have to prepare and evaluate each lesson?

Edit: I just googled "how many hours per week does a teacher teach?" and the answers from the US are saying about 25 to 34 hours per week. Is this true?!

Friday article summary - is cooperative group work bad?

2010-11-12T20:24:00.004+01:00

This term, I'm actually supposed to being taking a course which entails me to read and think about research articles in the teaching of mathematics. So to help me stay on track I'll use Fridays to read and comment articles.

First out:

Relationship between student and instructional factors and algebra achievement of students in the USA and Japan: an analysis of TIMSS 2003 data.

J. Daniel House and James A. Telese. (2008)

In this study, House and Telese aim to discover whether different student factors (such as self-confidence and interest in mathematics) and instructional factors (such as group work or alone work) correlate with achievement in the US and Japan.

To do this, the researchers used data from the 13 year-olds who participated in the TIMSS 2003 study, and carried out a statistical analysis of student responses and their subsequent test scores.

Results indicate that there are several differences in which factors are influential in Japan and the US.

In Japan, but not in the US, liking mathematics was found to be positively correlated with achievement. Also in Japan, but not in the US, spending class time doing mathematical computations without the calculator had a strong correlation with achievement - whereas the opposite relationship was found in the US.

Some similarities emerged as well. In both countries, students who made negative comparisons of themselves compared to other students tended to score lower. In terms of instructional strategies the results were more

surprising: in both countries cooperative learning was significantly negatively associated with achievement. This is against a background of other studies which have shown mixed and inconsistent results of cooperative learning.

Homework was positively associated with achievement - which is a kind of "duh" result - but working with "everyday applications of math" in class showed negative correlation with achievement.

Comments: It's not really clear to what extent we can infer causation and not just correlation. With attitudes, it's probably the chicken and the egg problem, but maybe that's what's happening in instructional strategies as well - with teachers lowering expectations and changing instruction strategies with weaker students.

Also, this article does not take into account the differences in weaker and stronger students in terms of what they need to develop skill and understanding. Maybe it's better for stronger students to work alone, but weaker, especially unmotivated students, need more group work?

In any case, these results offer reason to suspect that the current trends in collaborative learning need to be carefully evaluated. I suspect that what we are seeing is that many of the strategies modern teacher training encourages have the consequence of temporarily (in class) lowering the standards for student achievement. Instead of demonstrating understanding one by one - students are encouraged to cooperate, thereby masking their lack of understanding from both themselves and their instructor. On exams, when individual work is required, the students suddenly find themselves unable to solve the things they could do in class.

Formative assessment

2010-11-05T15:17:00.000+01:00

This week was a conference week, with several general and subject specific conferences for all city-employed teachers in Stockholm.
My knee-jerk attitude to these forced events is somewhere on the extreme end of the disgust-scale, but during the event it usually shifts to anger and then to desperation. Now that I have an iPhone things have gotten much easier.

This particular week, however, had at least one worthwhile presentation. Lena Göthe, the principal at a local school, talked about formative assessment, why and how to use it in secondary classrooms.

Why formative? Because it increases student motivation and provides more detailed and constructive feedback than summative evaluation. Lena refered to several research articles which claim that not only is detailed, formative feedback more helpful than summative points, scoring assignments (using points or numbers) is detrimental even if it accompanies detailed comments. It's not clear if that is because points encourages competition or because it shifts the attention away from the written feedback. (It is also not clear what populations of students this works for, and how, but let's suspend disbelief and try it out.)

How formative? Here is where it gets interesting. Obviously, teachers who are by law obliged to set grades need to use summative assessment at least sometimes. How can we also have time to give formative assessment? Well, we don't have to do it alone.
Formative assessment can happen in student meta-cognition, and in interaction with peers, teacher, and other people such as friends, family members and even strangers.
I'm trying to come up with some examples of how to work these resources for mathematics and psychology, and will share these ideas here:

Mathematics
Meta-cognition: exercises such as "explain this concept", "summarize the method you used", "what do you like best/dislike most?", "what's the muddiest point?" and other questions about concept, end result, and procedure activates student meta-cognition.
Using peers: students can analyze each other's work, for instance they could share and give feedback on the questions posed above under meta-cognition. Students can also check and feedback each other's solutions or together solve and/or discuss problems, solutions and concepts.

Psychology
Meta-cognition: evaluate own work against rubrics (potentially before handing it in), identify muddiest point, reflect on note-taking and other skills, identify what feedback the student wants from peer or teacher.
Using peers: evaluate and feedback each other's work, give each other advice, especially before handing it in for teacher summative assessment. Also, group discussion of feedback and strategies for improvement following summative assessment could shift student focus from points to comments and increase self-efficacy. Debates may also help students assess the strength and style of their arguments.

Giving effective feedback: What I will do, but didn't do with the tests I graded yesterday, is to hand back the assignments without marks - just written feedback - and get students to comment on this feedback before I take the assignments in again and mark them. It's also possible to give homework

Of course, summative feedback will always be there. It just doesn't have to be everywhere.
As always, I welcome whatever ideas or comments readers want to share.