Fractions Mnemonic

My students can invert function and tell me for which values of x a geometric series converges, yet they can’t do squat with fractions. It’s not for lack of “teaching for understanding” – I made sure every step of the way was solid when we did fractions last year. But since then lack of practice and reflection has pushed the understanding and skills far back into awkward recesses of their long-term memory (at best!).

So I’ve been searching for a good mnemonic, and haven’t found anything that covers the whole topic. In high school, I too had trouble remembering the rules and often used the 1/2 fraction to remember. This week, I saw a student use it the same way I used to, without me or anyone else having shown him, and so I decided that together he and I could introduce this method to the rest of the class.

Here is the worksheet I’ve designed. Google Docs mangles equations, so download the file for best effect. Please offer suggestions for improvement. I’ll hand this out to the class on Monday.

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Why do students love this?

Last year, with my PreDP class (think Algebra I and II) I gave a class the Riddle of Diophantus and asked them to solve it. The students, with just a basic understanding of fractions, loved this problem and spent a good 40 minutes or so working on it. It allowed me to introduce algebraic expressions, substitution, and simple equations – as well as give the students so well-needed practice with fractions. This year, my colleague used it in her class with equal success. But what makes it work and engage students in a way that smaller problems rarely do?


Here are two versions:

“God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the measure of half his father’s life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the end of his life.” – from daviddarling

‘Here lies Diophantus,’ the wonder behold. Through art algebraic, the stone tells how old: ‘God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father’s life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.’ – from wolframalpha

Do you notice how the two versions differ? I don’t know if this is because of translation issues or if there were several versions around already when these puzzles were compiled by Metrodorus.


Anyway, last year I had read “The Teaching Gap” and was all into giving students complex problems to work on individually and then in groups. I’m still into this approach, but am getting lazier with finding these problems (the Nrich website really makes all excuses sound silly). The idea was that complex problems challenge students and are thereby both more intrinsically fun, and better for learning. 
This problem also has some other good qualities: it involves a translation from weird prose to math, which engages students who pride themselves on their language- and humanities-skills rather than math ability. And it’s worder like a riddle – which triggers the “I’ll beat it!” response in many people. It was also at the just right level of difficulty for my students. I find the hard part about finding/making problems is to gage the level of difficulty and the amount of scaffolding to provide. 


One thing I try to keep in mind is what my colleague J calls the Wax-on-Wax-off (think “Karate Kid) pedagogy. Get them with the rich and complex problem, then follow it with direct instruction or some other ways of organizing and summing up the important concepts and methods introduced through the problem. 
By the way, J really impressed me with this waxy-on-offy thing. He’s just finishing a ph.d in the education of natural sciences and maths, and until this year hasn’t spent much time teaching in “real” schools. Then off the bat he says something this awesome. Either he has some natural instinct for teaching, or it’s really possible to learn something of practical significance in teacher education schools. 

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Dealing with test results

This week, I’ve given three tests – a test in each of my three math classes (Math Studies, Math Standard and Math Higher). Inspired by all the brilliant SBG posts I’ve read lately, I want to find a good way to deal with the students’ results and reactions.


Some random student reactions to performing below their expectations:

– I can’t believe that was a 4 [barely passing, maximum grade is 7], I mean, I didn’t know anything.

– Can I retake the test?

– There’ll be many more tests, right, which will raise my average, right?

– Man, I really need to get my act together. 

– Will we be coming back to this material?

– What are we doing next? I want to be on top of it all the time from now on. 

and the heart-breaker:

– I don’t understand it! I did all the homeworks, helped other students, and then on the test I can’t remember it at all!

One student had her parent call the student’s mentor, to speak about “her teacher and her test”. That’s me and the test I gave them this morning, folks. And the parent called literally 5 minutes after the student had handed in the test and left the room. I wish this student had just come to talk with me directly. If it’s anything I’ve been proud, it’s that my students can confide in me. This year I’m trying on a tougher attitude and maybe it’s not working out well with some students.

So I’m figuring out how to deal with all these varied student reactions. For the Math Standards (think “honors”) and Math Highers (think “honors squared”) I’m settling for this strategy:

  1. Suggest to the students that if they are unhappy with their results, and think that they could perform better with more or different studying, then write a brief plan of what they will change from now on. 
  2. Suggest also that if they do not think that they are willing or able to reach a higher result through more or different studying, then they should switch to a lower level math than the one they are currently taking. Or accept that this is the kinds of results they’ll be facing.
  3. Provide the students with answers, but not solutions, to the test questions. 
  4. Give them as homework to figure out complete solutions to all questions. 
  5. Check their homework via Binder check as usual. 

The Math Studies students, however, cannot switch to any lower level. These are also students accustomed to lower results – in the sense that they still get frustrated, but no longer believe in their own ability to do better.
My strategy here feels a bit like flailing desperately and I wish I had something better up my sleeve:

  1. Make time to discuss with the class as a whole why their results are this abysmal (they are!). Acknowledge that they have been under tons of stress with other school work, and that they are upset and frustrated, and offer encouragement that they can do better (they can!) and that we’re in this together (we are! I’m just preparing them for the final exams.)
  2. Ask students to write down what they can do differently and what they wish I would do differently. 
  3. Ask them to talk with me privately about any concerns that they have that they don’t want to share with the whole class. 
  4. Then give them the answers to the test questions and give as homework to get the correct solutions. Instead of the re-test they’ve been asking for, do a binder check type of activity on this homework and if they do better, acknowledge that this is a sign of increased understanding of this topic. 

This should take care of all the “types” of students except the heart-breaker. I still don’t know what to do about her.

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What worked and what didn’t

This week, I’ve tried a few things with my two math classes (honors juniors and standard seniors). One, a scaffolded proof of the Sine and Cosine rules, went horribly for some very mysterious reasons. The other, a Binder check procedure I’ve copied from another blogger, is going very well.

What didn’t work: First, on Monday, the seniors were learning about the Sine and Cosine rules. I hate how their textbook presents these, like they are some inscriptions on stone-tablets, and even thinking about proving or questioning them is heresy. My students typically have the confused and bewildered “wtf” expression when they see such things, and last year I had a successful class with students in groups proving these rules with some little scaffolding from me along the way. Scaffolding such as “maybe pythagoras’ rule would help?” was enough, but the students took a full 90 minute class to complete the proofs, and this year I wanted to save time by providing more (much more) scaffolding.
So I made these worksheets. I’m trying to figure out how to include them in a post such as this, so bear with me while I experiment with different options.

Here is the file with the two worksheets, in pdf format.

I was particularly happy with having the students first try a specific case with rulers and all. Especially when it turned out that trying a specific case wasn’t all that easy for them. I think by actually measuring and trying it out, they gained a good understanding of what the rules actually say. I will keep doing this type of “physically trying the rule” for other trig or geometrical rules.

The rest of the worksheets however, went not as good, to put it very mildly.
Above all, the students (even the stronger students) immediately complained they didn’t understand what to do. I explained, and they moved one step along. Even though they were sitting in groups and usually have no difficulty discussing with each other, this time they didn’t discuss, they kept just calling me over and asking questions, and I had the impression they just wanted to be spoon-fed stuff. The breaking point was when one student wrote the Sine rule and when I asked how he arrived at it, he just said that it was obviously what I wanted them to arrive at so he figured why not just write it already?

Oh well. That’s when I gave up. The activity was taking too long time anyway and so I interrupted their “work” and, after I cursed at them for being so passive and not thinking for themselves (I’m not proud of this, but my relationship to these kids is good enough that it can withstand occasional lapses in judgement on my part), I quickly went over the proofs by myself on the whiteboard. The remaining ten minutes of class students worked on exercises, very quietly, and I demonstratively sat and played on my iPhone.
If you’re wondering, the following class (starting sequences and series) went great, and so I feel no lasting damage was done.

Now, this experience has me a bit disappointed and confused. I often work with this type of scaffolded investigative material, though rarely with proofs with this class, and most often it works very well – students enjoy the process and feel ownership over the formulas they are then asked to apply in various problems. Why didn’t this work? Was it too difficult, as my colleague J claims? Was it poorly formulated? Was it just a bad day for my students (and me, obviously)? Maybe it’s just the students’ expectations and interests? They are not taking math because math is fun, rather because they have to and this is the lowest level IB lets them get away with.
I’m going to try this same activity with my honors juniors in  a month or two, and will write more about it then.

What did work, or is starting to, is the Binder check procedure Sam wrote about a while back. I use it with my honors juniors, and yesterday was the second check (I plan on checking every two weeks or so). Results dramatically increased from the first check, as Sam said happened in his class as well, and I feel that this way of checking homework is giving me some great information about the students’ work between classes.
An important difference is that I do NOT grade homework. Because my students will be assessed ONLY on the final exam (in May 2012) and two investigative portfolios, neither the homeworks nor the test they’ll have on Monday actually counts for their final grades. I’m happy to say that the students know this, and that still they take both the binder checks and the tests very seriously indeed.

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Fumbling with inverses

My IB Mathematics course has started out well, with a successful introduction of functions, domain and range and composite functions. So maybe I started getting too cocky and didn’t spend enough time on today’s lesson – on inverses. Whatever the reason, it felt like hitting a wall. While students happily solved questions such as “what’s k(x) if h(x) = 5x and h(k(x)) = x?”, they could not even start on the very similar question “what’s the inverse of h(x) if h(x) = 5x+3?” Even with me explicitly explaining the connection, by definition, to the previous question, students were dumbfounded by these developments.

Now I’m the one who’s a bit stuck. Do I give an extra class on inverse functions – this would break my planning and mean I’m lowering my standards a bit. Or do I send them to the math study hall and go on with my planning, hopefully managing to fit in inverses now and then again? I’m leaning towards the latter but would really like to figure out what went wrong (and how to fix it). Maybe the language just got to abstract? Riley at Point of Inflection has a series of posts I probably should have read more carefully.

Ah well. “Where you stumble, there lies your treasure”, is this nice saying I’ve always liked. Maybe there’ll be something superbly useful in this mess somewhere.

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Standard Based Grading, the IB-system, and something in between

Since I started teaching two years ago, I’ve been teaching in part the Swedish (standard based by law) system and in part the IB (final exam) system. It’s therefore very interesting and funny for me to read about the issues US teachers are having with SBG, and the arguments people have against it (no, students do NOT come and demand a retest infinitely many times – they’re too lazy and anyway the teacher can limit the assessment opportunities). In this post, I’m going to share my experiences and thoughts about the two systems I know, and propose a compromise.

Standard based – well, let’s be honest: the difficulty of the course is set by the person setting, and interpreting, the standards. In most cases: a government body issues the standards, but the real legwork is done by the teachers. Now, like all teachers, I want two things: to keep standards high, so my students develop a lot of high quality knowledge, skill, and understanding, and to be a teacher whose students all earn A’s.

SBG is supposed to let me do this. In theory, I tailor the standards to my and my student’s interests and local needs and resources. Students are evaluated on knowledge and understanding, not attendance, homework, and other irrelevant stuff. And there is nothing, no bell-curve or other atrocious normalizing rule, that prohibits me from developing, and awarding brilliant learning in all students.

In practice, however, I find it very difficult to keep these two objectives (high standards and high student performance)  equally prioritized at the same time. It easily happens that I reinterpret a standard to accommodate the lesser than expected performance of the students. It happens in part because I can never be sure I’ve set the standards at a good level. It could be that I’m interpreting the state standards too harshly. Another reason is that I simply feel bad for the students when they perform poorly. Unfortunately, there’s a third reason: student accomplishments are tied to my feelings of self-worth as a teacher.

On a school and city level, there are more problems with SBG: when students choose schools, they look at average grades in the schools they are considering. When there is a competition for students, as there is right now in Sweden (city schools competing against each other and against private schools), it’s tempting for teachers and principals to lower the standards or simply award undeserved high grades. This has led to a formidable inflation of grades. While international comparisons show declined performance in mathematics among Swedish high school students, average grades has increased dramatically in math and many other subjects since SBG was introduced and made mandatory some 20 years ago.

In some ways, I therefore like the IB-system better. Students write a final exam, identical for all IB-students worldwide, and an external examinator marks it. The task of the teacher is to prepare the students for the exams, and so the teacher does not have to balance teaching and marking. Of course, it completely sucks that you have to follow a strict and in-flexible syllabus and just hope for the best when the exams come. And it’s quite stressful for the students to only have one chance of showing what they can do. But at least teacher bias does not affect student performance, teachers get more unbiased feedback on their teaching, and schools cannot raise grades to compete with other schools.

I think, what I’m proposing, is a system in between these two extremes: something where perhaps two teachers are working together (in the same or different schools), examining each other’s students. There is really nothing stopping me, and perhaps many other teachers, from working this way already. If teachers formulate the school interpretation of standards together, then they can design tests for each-other’s classes. It would limit, of course, the ways in which knowledge is assessed, but it would be increase fairness and provide opportunities for feedback and discussions between teachers.

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Bonding Day

Yesterday was my first Bonding Day. In my school, Bonding Day basically means teachers orchestrate a series of team-building activities for students, usually in some park in Stockholms. The purpose is for our 150 students to have fun together, and, by cooperate in teams, build friendships and a generally nice atmosphere school. 
The investment for this is as follows: teachers spend one and a half full day planning, and students miss one full day of classes during Bonding Day. 
As I enjoy the teaching part of being a teacher, I felt a bit frustrated that all this energy went into an activity that doesn’t directly further students’ understanding of school subjects. Even though I did not have to participate in the planning, I was indignant on behalf of my colleagues. But that was before Bonding Day. 
Yesterday morning, as students gathered in their respective groups, a slightly nervous suspense built slowly. Each teacher took a group to that teacher’s station, and then the fun started. My station was measurements, and it was hilarious watching these kids trying to figure out the different tasks. Also a bit abhorrent. Of the 11 groups that passed my station, only 3 could correctly identify North within 30 degrees West or East. I saw the groups struggle with leadership, and it was clear that those groups which featured cooperation succeeded best. 
At each station the kids got some money (school “money”) depending on how well they did the tasks. When all the stations were finished, we all ate lunch together and then it was auction time. 
Students selected one representative from each group to bid on cake decorations, for the cake competition which would finish the day. The auction was – well, as auctions go when the participants are teenagers holding fake money. I must remember to spend some time on auction-psychology during the Social Psych level of analysis this spring. 
The highlight of the day, however, was the actual cake-making. Somehow, what had started as a beautiful day with not a cloud in sight, suddenly turned into a grim raining and hailing inferno. It was cold, wet, and utterly great. Students gathered under umbrellas, completely engrossed in making and protecting the cakes.
After the competition, and the yumminess which followed, some groups performed a song their group had written during the day. They were all very good and received well-earned applauses. The idea was that the songs would be judged by a jury, and that together the song and cake contest would produce a winning team. Because of the rain and hail, however, teachers were disorganized in putting together a song-jury and the winning team became the team with the cutest cake. 
All in all, this was a fantastic day. Students were thinking, laughing, cooperating, arguing and trying new things. Teachers also enjoyed the day, and had many fun stories to tell each other afterwards over glasses of wine in the nearby bar. My school is known for having a great social atmosphere and close bonds between students and students, students and teachers, and teachers and teachers. I now think that that Bonding Day definitely is a major way in which we make these bonds happen. 
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