## Overview

This was my third time around teaching the first year of the IB mathematics SL sequence. It was different from my previous two iterations given that this was not an SL/HL combined class. This meant that I had more time available to do explorations, problem solving sessions, and in-class discussions of the internal assessment (also called the exploration). I had two sections of the class with fourteen and twenty students respectively.

I continued to use standards based grading for this course. You can find my standards (which define the curricular content for my year one course) at this link:

IB Mathematics SL Year 1 – Course Standards

## What worked:

• My model of splitting the 80 – 85 minute block into twenty minute blocks of time works well. I can plan what happens in those sub-blocks, and try as hard as I can to keep students doing something for as much of those as I can. The first block is a warm-up, some discussion, check in about homework or whatever, and then usually some quick instruction before the next block, which often involves an exploration activity. Third is summary of explorations or the preceding activities, example problems, and then a fourth of me circulating and helping students work.
• Buffer days, which I threw in as opportunities for students to work on problems, ask questions, and play catch up, were a big hit. I did little more on these days than give optional sets of problems and float around to groups of students. Whenever I tried to just go over something quick on these days, those lessons quickly expanded to fill more time than intended. It took a lot of discipline to instead address issues as they came up.
• I successfully did three writing assignments in preparation for the internal assessment, which students will begin writing officially at the beginning of year two. Each one focused on a different one of the criteria, and was given at the end of a unit. Giving students opportunities to write, and get feedback on their writing, was useful both for planning purposes and for starting the conversation around bad habits now.

I had rolling deadlines for these assignments, which students submitted as Google Docs. I would go through a set of submissions for a class, give feedback to those that made progress, and gentle reminders to those that hadn’t. The final grade that went into PowerSchool was whatever grade students had earned by the end of the quarter.

The principle I applied here (and one to which I have subscribed more fervently with each year of teaching) is that my most valuable currency in the classroom is feedback. Those that waited to get started in earnest with these didn’t get the same amount of feedback as students that started early, and the quality of their work suffered dramatically. I’m glad I could have the conversations I had with students now so that I might have a chance in changing their behavior before their actual IA is due.

An important point – although I did comment on different elements of the rubric, most of my feedback was on the criterion that titled the assignment. For example, in my feedback I occasionally referenced reflection and mathematical presentation in the communication assignment. I gave the most detailed feedback for communication, and graded solely on that criterion.

These were the assignments:

• I budgeted some of my additional instruction time for explicit calculator instruction. I’ve argued previously about the limitations of graphing calculators compared to Geogebra, Desmos, and other tools that have substantially better user experiences. The reality, however, is that these calculators are what students can access during exams. Without some level of fluency accessing the features, they would be unable to solve some problems. I wrote about this in my review of the course last year. This time was well spent, as students were not tripped up by questions that could only be solved numerically or graphically.
• Students saw many past paper questions, and seem to have some familiarity with the style of questions that are asked.

## What needs work:

• I’ve come to the conclusion that preemptive advice is ineffective. “Don’t forget to […]” or “You need to be extremely careful when you […]” is what I’m talking about. It isn’t useful for students that don’t need the reminder. It doesn’t help the students that don’t have a context for what you are telling them not to do, not having solved problems on their own. I have found it to be much more effective to address those mistakes after students get burned by them. Some of my success here comes from my students subscribing to a growth mindset, which is something I push pretty hard from the beginning. Standards based grading helps a lot here too.
• I desperately need a better way to encourage longer retention of knowledge, particularly in the context of a two year IB course. I’ll comment more on this in a later post, but standards based grading and the quarter system combined were factors working against this effort. I did some haphazard spaced repetition of topics on assessments in the form of longer form section two questions. The fact that I was doing this did not incentivize enough students to regularly review. I also wonder if my conflicted beliefs on fluency versus understanding of process play a role as well.
• Students consistently have a lot of questions about rounding, reporting answers, and follow through in using those answers in the context of IB grading. The rules are explicitly stated in the mark schemes for questions – answers should be reported exactly or to three significant figures unless otherwise noted. The questions students repeatedly have relate to multiple part questions. For example, if a student does a calculation in part (a), reports it to three significant figures, and then uses the exact answer to answer part (b), might that result in a wrong answer according to the mark scheme? What if the student uses the three significant figure reported answer in a subsequent part?

I did a lot of research in the OCC forum and reading past papers to try to fully understand the spirit of what IB tries to do. I’d like to believe that IB sides with students that are doing the mathematics correctly. I am not confident in my ability to explain what the IB believes on this, which means my students are uncertain too. This bothers me a lot.

• Students still struggle to remember the nuances of the different command terms during assessments. They also will do large amounts of complex calculations and algebraic work in site of seeing that a question is only two or three marks. There is clearly more work to do on that, though I expect that will improve as we move into year two material because, well, it usually does. I wish there was a way to start the self-reflection process earlier.
• Students struggle to write about mathematics. They also struggle with the reality that there is no way to make it go faster or do it at the last minute without the quality suffering. I still believe that the way you get better is by writing more and getting feedback, and that’s the main reason I’m glad I made the changes I did regarding the exploration components. That said, students know how to write filler paragraphs, and I call them out on filler every single time.
• We spent a full day brainstorming and thinking about possible topics for individual explorations. Surveying the students, only four of them are certain about their topics. The rest have asked for additional guidance, which I am still figuring out how to provide over the summer. I think this process of finding viable topics remains difficult for students.

## Conclusion

I’ll be following these students to year two. We have the rest of probability to do first thing when we get back, which I’ll combine with some dedicated class time devoted toward the exploration. I like pushing the probability and Calculus to year two, as these topics are, by definition, plagued by uncertainty. It’s an interesting context in which to work with students in their final year of high school.

## 2015-2016 Year in Review: IB Mathematics SL/HL

This was my second year working in the IB program for mathematics. For those that don’t know, this is a two year program, culminating in an exam at the end of year two. The content of the standard level (SL) and higher level (HL) courses cross algebra, functions, trigonometry, vectors, calculus, statistics, and probability. The HL course goes more into depth in all of these topics, and includes an option that is assessed on a third, one-hour exam paper after the first two parts of the exam.

An individualized mathematics exploration serves as an internally assessed component of the final grade. This began with two blocks at the end of year one so that students could work on it over the summer. Students then had four class blocks spread out over the first month of school of year two two work and ask questions related to the exploration during class.

I taught year one again, as well as my first attempt at year two. As I have written about previously, this was run as a combined block of both SL and HL students together, with two out of every five blocks as HL focused classes.

## What worked:

• I was able to streamline the year 1 course to better meet the needs of the students. Most of my ability in doing this came from knowing the scope of the entire course. Certain topics didn’t need to be emphasized as I had emphasized in my first attempt last year. It also helped that the students were much better aware of the demands of higher-level vs. standard level from day one.
• I did a lot more work using IB questions both in class and on assessments. I’ve become more experienced with the style and expectations of the questions and was better able to speak to questions about those from students.
• The two blocks on HL in this combined class was really useful from the beginning of year one, and continued to be an important tool for year two. I don’t know how I would have done this otherwise.
• I spent more time in HL on induction than last year, both on sums and series and on divisibility rules, and the extra practice seemed to stick better than it did last year in year one.
• For students that were self starters, my internal assessment (IA) schedule worked well. The official draft submitted for feedback was turned in before a break so that I had time to go through them. Seeing student’s writing was quite instructive in knowing what they did and did not understand.
• I made time for open ended, “what-if” situations that mathematics could be used to analyze and predict. I usually have a lot of this in my courses anyway, but I did a number of activities in year one specifically to hint at the exploration and what it was all about. I’m confident that students finished the year having seen me model this process, and having gone through mini explorations themselves.
• After student feedback in the HL course, I gave many more HL level questions for practice throughout the year. There was a major disconnect between the textbook level questions and what students saw on the HL assessments, which were usually composed of past exam questions. Students were more comfortable floundering for a bit before mapping a path to a solution to each problem.
• For year two, the exam review was nothing more than extended class time for students to work past papers. I did some curation of question collections around specific topics as students requested, but nearly every student had different needs. The best way to address this was to float between students as needed rather than do a review of individual topics from start to finish.
• The SL students in year two learned modeling and regression over the Chinese new year break. This worked really well.
• Students that had marginally more experience doing probability and statistics in previous courses (AP stats in particular) rocked the conditional probability, normal distribution, and distribution characteristics. This applied even to students who were exposed to that material, but did poorly on it in those courses. This is definitely a nod to the idea that earlier exposure (not mastery) of some concepts is useful later on.
• Furthermore, regarding distributions, my handwaving to students about finding area under the curve using the calculator didn’t seem to hurt the approach later on when we did integration by hand.
• This is no surprise, but being self sufficient and persevering through difficult mathematics needs to be a requirement for being in HL mathematics. Students that are sharp, but refuse to put in the effort, will be stuck in the 1-3 score range throughout. A level of algebraic and conceptual fluency is assumed for this course, and struggling with those aspects in year one is a sign of bigger issues later on. Many of the students I advised this way in year one were happier and more successful throughout the second year.
• I successfully had students smiling at the Section B questions on the IB exam in the slick way that the parts are all connected to each other.

## What needs work:

### For year one:

• I lean far too hard on computer based solutions (Geogebra, Desmos) than on the graphing calculator during class. The ease of doing it these ways leads to students being unsure of how to use the graphing calculator to do the same tasks (finding intersections and solutions numerically) during an assessment. I definitely need to emphasize the calculator as a diagnostic tool before really digging into a problem to know whether an integer or algebraic solution is possible.
• Understanding the IB rounding rules needs to be something we discuss throughout. I did more of this in year one on my second attempt, but it still didn’t seem to be enough.
• ### For year two:

• Writing about mathematics needs to be part of the courses leading up to IB. Students liked the mini explorations (mentioned above) but really hated the writing part. I’m sure some of this is because students haven’t caught the writing bug. Writing is one of those things that improves by doing more of it with feedback though, so I need to do much more of this in the future.
• I hate to say it, but the engagement grade of the IA isn’t big enough to compel me to encourage students to do work that mattered to them. This element of the exploration was what made many students struggle to find a topic within their interests. I think engagement needs to be broadened in my presentation of the IA to something bigger: find something that compels you to puzzle (and then un-puzzle) yourself. A topic that has a low floor, high ceiling serves much more effectively than picking an area of interest, and then finding the math within it. Sounds a lot like the arguments against real world math, no?
• I taught the Calculus option topics of the HL course interspersed with the core material, and this may have been a mistake. Part of my reason for doing this was that the topic seemed to most easily fit in the context of a combined SL/HL situation. Some of the option topics like continuity and differentiability I taught alongside the definition of the derivative, which is in the core content for both SL and HL. The reason I regret this decision is that the HL students didn’t know which topics were part of the option, which appear only on a third exam section, Paper 3. Studying was consequently difficult.
• If for no other reason, the reason not to do a combined SL/HL course is that neither HL or SL students get the time they deserve. There is much more potential for great explorations and inquiry in SL, and much more depth that is required for success in HL. There is too much in that course to be able to do both courses justice and meet the needs of the students. That said, I would have gone to three HL classes per two week rotation for the second semester, rather than the two that I used throughout year one.
• The HL students in year two were assigned series convergence tests. The option book we used (Haese and Harris) had some great development of these topics, and full worked solutions in the back. This ended up being a miserable failure due to the difficulty of the content and the challenge of pushing second semester seniors to work independently during a vacation. We made up some of this through a weekend session, but I don’t like to depend on out-of-school instruction time to get through material.

Overall, I think the SL course is a very reasonable exercise in developing mathematical thinking over two years. The HL course is an exercise in speed and fluency. Even highly motivated students of mathematics might be more satisfied with the SL course if they are not driven to meet the demands of HL. I also think that HL students must enjoy being puzzled and should be prepared to use tricks from their preceding years of mathematics education outside of being taught to do so by teachers.

## Playing with Transformers & Building an AC to DC Converter

My one remaining IB Physics HL student was alone in class today, so I made a quick switch on plans to do one of the required activities in the syllabus: building a full-wave rectifier circuit. This was also an opportunity to play with transformers (no, not that kind of transformer) which I presented last class as a really useful application of electromagnetic induction:

I did almost nothing during the whole class period, aside from giving instructions and asking questions. With wires, resistors, and other circuit parts all over the place, this was not a well organized lab, but that doesn’t matter. Here’s what we did:

• Measure the AC voltage and frequency coming out of the supply
• Connect the supply to one of the transformer coils, and place the other coil inside. Measure the voltage and frequency on the secondary coil. Comment on the number of coils on the primary and secondary. Observe the effect of inserting an iron core into the transformer.
• Predict what would happen if the two coils were switched, then measure to compare with prediction.
• Build a half-wave rectifier circuit with a diode and resistor using the secondary coil as a supply. Measure the voltage wave form.
• Build a full-wave rectifier circuit, and measure the voltage signal across a resistor. Check this out:

This yielded a nice ripply DC signal:

• Connect a capacitor in parallel with the resistor.

As we had hoped, the ripples are almost gone!:

In the final step, I connected a 5V regulator to the output and showed that this would be a way to make a cell phone charger if we had the right USB connector.

This is quite possibly the most satisfying and successful ‘let’s build something’ lab I’ve done with students. It fit neatly within a class period, including the clean up time. Granted, things would be different if there were multiple groups going through the process, but I’ll just ignore that fact because it’s a Friday.

## 2014-2015 Year In Review: IB Physics SL/HL

This was my first year teaching IB Physics. The class consisted of a small group of SL students with one HL, and we met every other day according to the block schedule. I completed the first year of the sequence with the following topics, listed in order:

### Semester 1

1. Unit 1 – Experimental Design, Uncertainty, Vectors (Topic 1)
2. Unit 2 – Kinematics & Projectile Motion (Topic 2.1)
3. Unit 3 – Newton’s Laws (Topic 2.2)
4. Unit 4 – Work, Energy, and Momentum (Topic 2.3)
5. ### Semester 2

6. Unit 5 – Circular Motion, Gravitation, and Orbits (Topics 6.1, 6.2)
7. Unit 6 – Waves and *Oscillation(Topic 4, AHL Topic 9, *AHL Engineering Option Topic B3.1,3.2)
8. Unit 7 – Thermal Physics (Topic 3, Engineering Option Topic B2)
9. Unit 8 – *Fluid Dynamics (Engineering Option Topic B3)

For the second semester of the course, there was at least one block every two weeks that was devoted to the HL student and the HL only content – the SL students worked on practice problems or other work they had for their IB classes during this time. Units 7 and 8 were concurrent, so the HL student had to work on both the thermodynamics content and the fluid dynamics content together. This was similar to how I did it previously while teaching the AP physics B curriculum.

One other fact that is relevant – none of my students are native speakers of English. More on this later.

## What worked:

• The growth students made during the year was significant. I saw students improve in their problem solving skills and their organization in the process of doing textbook style assessment problems.
• I learned to be honest about the IB expectations for answering questions on assessments.In the beginning, I tried to shield students from questions that combined conceptual understanding, computation, and complex language, often choosing two out of the three of them for any one question that I either wrote or selected from a bank. My motivation was to isolate assessment of the physics content from assessment of the language. I wanted answers to these separate questions:
1. Does the student understand how the relevant physics applies here?
2. Does the student understand how to apply the formulas from the reference table to calculate what the question is asking for?
3. Can the student process the text of the question into a physics context?
4. Can the student effectively communicate an answer to the question?

On official IB assessment items, however, this graininess doesn’t exist. The students need to be able to do all of these to earn the points. When I saw a significant difference between how my students did on my assessments versus those from IB, I knew I need to change. I think I need to acknowledge that this was a good move.

• Concise chunks of direct instruction followed by longer problem solving sessions during class worked extremely well. The students made sense of the concepts and thought about them more while they were working on problems, than when I was giving them new information or guiding them through it. That time spent stating the definitions was crucial. The students did not have a strong intuition for the concepts, and while I did student centered conceptual development of formulas and concepts whenever possible, these just didn’t end up being effective. It is very possible this is due to my own inexperience with the IB expectations, and my conversations with other teachers helped a lot to refine my balance of interactivity with an IB pace.
• Students looked forward to performing lab experiments. I was really happy with the way this group of students got into finding relationships between variables in different situations. Part of this was the strong influence I’ve developed with the Modeling Instruction curriculum. As always, students love collecting data and getting their hands dirty because it’s much more interesting than solving problems.

## What needs work:

• My careless use of the reference sheet in teaching directly caused students to rely excessively upon it. I wrote about this previously, so check that post out for more information. In short: students used the reference sheet as a list of recipes as if they provided a straight line path to solutions to questions. It should be used as a toolbox, a reminder of what the relationships between variables are for various physics concepts. I changed this partly at the end of the year, asking students to describe to me what they wanted to look for on the sheet. If their answer was ‘an equation’, I interrogated further, or said you aren’t about to use the reference sheet for what it was designed to do. If their answer was that they couldn’t remember if pressure was directly or inversely related to temperature, I asked them what equation describes that relationship, and they were usually able to tell me.
Both of these are examples of how the reference sheet does more harm than good in my class. I fault myself here, not the IB, to be clear.
• The language expectations of IB out of the gate are more of an obstacle than I expected at the beginning of the year. I previously wrote about my analysis of the language on IB physics exams. There tends to be a lot of verbal description in questions. Normally innocuous words get in the way of students simultaneously learning English and understanding assessment questions, and this makes all the difference. These questions are noticably more complex in their language use than that used on AP exams, though the physics content is not, in my opinion, more difficult. This is beyond physics vocabulary and question command terms, which students handled well.
• Learning physics in the absence of others doesn’t work for most students. Even the stronger students made missteps working while alone that could have been avoided by being with other students. I modified my class to involve a lot more time working problems during class and pushed students to at least start the assigned homework problems while I was around to make the time outside of class more productive. Students typically can figure out math homework with the various resources available online, but this just isn’t the case for physics at this point. It is difficult for students to get good at physics without asking questions, getting help, and seeing the work of other students as it’s generated, and this was a major obstacle this semester.
• Automaticity in physics (or any subject) shouldn’t be the goal, but experience with concepts should be. My students really didn’t get enough practice solving problems so that they could recognize one situation versus another. I don’t want students to memorize the conditions for energy being conserved, because a memorized fact doesn’t mean anything. I do want them to recognize a situation in which energy is conserved, however. I gave them a number of situations, some involving conservation, others not, and hoped to have them see the differences and, over time, develop an awareness of what makes the two situations different. This didn’t happen, partly because of the previous item about working physics problems alone, but also because they were too wrapped up in the mechanics of solving individual problems to do the big pciture thinking required for that intuition. Group discussions help on this, but this process is ultimately one that will happen on the individual level due to the nature of intuition. This will take some time to figure out.
• Students hated the formal process of writing up any parts of the labs they performed. This was in spite of what I already said about the students’ positive desire to do experiments. The expressions of terror on the students’ faces when I told them what I wanted them to do with the experiment break my heart. I required them to do a write-up of just one of the criteria for the internal assessment, just so they could familiarize themselves with the expectations when we get to this next year. A big part of this fear is again related to the language issue. Another part of it is just inexperience with the reality of writing about the scientific process. This is another tough egg to crack.

There was limited interest in the rising junior class for physics, so we won’t be offering year one to the new class. This means that the only physics class I will have this year will be with the same group of students moving on to the second year of IB physics. One thing I will change for physics is a set of memorization standards, as mentioned in my post about standards based grading this year. Students struggled remembering quick concepts that made problem solving more difficult (e.g. “What is the relationship between kinetic energy and speed?”) so I’ll be holding students responsible for that in a more concrete way.

The issues that need work here are big ones, so I’ll need some more time to think about what else I will do to address them.

## IB Mathematics HL: Vectors & Planes

There’s nothing big to report here, but I did want to share a really successful approach I put together relating vectors and planes. This is a required topic for the IB HL Mathematics curriculum. All of the textbooks I looked in did a fairly theoretical analysis of Cartesian and vector forms for planes from the start. I wanted to present a lesson that gave students a bit more intuition about the concepts involved, and then get to the mathematical vocabulary when needed.

Vectors and Planes

These notes were created live during class using OneNote. I don’t intend these notes to replace the textbook, but I do want them to serve as the ‘residue of logic’ that we used during the lesson so that students can go back and review them to remember the key ideas. I have a small group, so we can sit around a big table and work together. There’s lots of conversation between us and between students when I set them loose to do an exercise.

All of the students demonstrated good understanding throughout the lesson in the problems I gave. The students that did the homework immediately after the lesson did well on a subsequent quiz. The student that didn’t, well, didn’t. No surprise there.

## Dot Circle – An Introduction to Vectors

After learning from Jessica Murk before our spring break about the idea of revising mathematical writing in class, I decided to try it as part of an introduction to the fourth topic in the IB Mathematics curriculum: vectors. The goal was to build a need for the information given by vectors and how they provide mathematical structure in a productive way.

I started by adapting Dan Meyer’s activity here with a new set of dots.

I asked all students to pick one dot, and then asked a student to give the class instructions on which one they picked. They did a pretty good job with it, but there was quite a bit of ambiguity in their verbal descriptions, as I wanted. This is when I sprung Dan’s helpful second slide that made this process much easier:

Key Point #1: A common language or vocabulary makes it easy for us to communicate our ideas.

I then moved on to the next task. Students individually had to write directions for moving from the red dot to the blue dot. I gave them this one to start as a verbal task, but nobody was willing to take the bait after the last activity:

Fair enough.

I then gave one of the following images to each pairs of students, with nothing more than the same instruction to write directions from the red to the blue dot.

Here is a sampling:

• Move across 5 dots on the outermost layer counter-clockwise, with the blue dot at the bottom of paper (closest to you)
• Move 7 units to right, and move about (little less) 3 units up so that the blue dot is right on the vertical line
• Fin the dot that is directly opposite to the red dot that is across the diagram. Once there, move down one dot along the outermost layer of dots.
• Stay on the circle and move right for five units
• Move from coordinate $latex \frac{7 \pi}{6}$ to the coordinate of $latex 2 \pi$ on the unit circle.

After putting the written descriptions next to the matching image, students then rotated from image to image, and applied Jessica’s framework for students giving written feedback for each description they saw.

Here is some of the feedback they provided:

Then, without any input from me, I had students sit down and each write a new description. Just as Jessica promised, the descriptions were improved after students saw the work of others and focused on what it means to give specific and unambiguous directions.

This is where I hijacked the results for my own purposes. I asked how the background information I gave helped in this task? They responded with:

• Grid/coordinate system in background of the dots
• Circle connecting dots – use directions and circles to explain how to move
• Connected all dots – move certain number of ‘units’

One student also provided a useful statement that the best description was one that could not be misinterpreted. I identified the blue dot as (3,0), and asked if anyone could give coordinates for the red dot. Nobody could. One student asked where (0,0) was. I pointed to some other points as examples, and eventually a student identified the red dot as (3,8). Another said it could also be (3,-5). I pointed out that if I had asked students to plot (3,-5) at the beginning of the class, the answer would have been totally different.

This all got us to think about what information is important about coordinates, what they tell us, and that if we agree on common units and a starting point, the rest can be interpreted from there. This was a perfect place to introduce the concept of unit vectors.

We certainly spent some time wandering in the weeds, but this ended up being a really fun way to approach the new unit.

If you are interested, here is the PDF containing all of the slides:
Point Circle

## Qatar Airways and the IB Mathematics Exploration

It isn’t always a common occurrence to have a distance and bearing to a particular location, but given my choice in airline for winter break, I had exactly that. So during our first class back from the break, I asked students to figure out where I was when I took this picture:

Students scrambled to open up various online maps and make sketches. The students settled on a range of answers. Then I showed them this:

We had a quick discussion about assumptions. Then students looked again and talked to each other while revising their answers. Once they were satisfied with their answers again, I shared the correct answer.

This led into a nice discussion of the mathematics exploration project that is submitted as the internal assessment of the IB mathematics courses. The students know that I take pictures and videos of this sort of thing all the time – it’s a habit instilled in me by someone we all know. The students said though that they don’t usually see math in the things around them, which is a problem given that the math exploration is supposed to come from them.

My recommendation, which comes partly from Dan’s suggestions, is just to start. I told the students that any time they see something interesting or beautiful when they’re walking around, to take a picture of it to review later. With some time between seeing it and reviewing it, they should ask themselves why it interested them. What is it that makes the picture beautiful? Are there patterns? Is it organized in an interesting way? I also shared my RSS reader on Feed.ly and how I save articles that interest me and tag them accordingly. This is how I find interesting ideas to share with the class – they should do the same to figure out what might be a good source of material for their work.

We have had several discussions in class about what this exploration will be about, but the emphasis has really been on something that interests them. Having students be curators of their own ‘interesting-stuff’ collection now seems the most obvious way to get them started.

## Semester in Review: Combined IB SL/HL Mathematics Class

This past semester was tough. I’ve always taken on more than I probably should, but I hit my limit and need to change things around.

The biggest element of the challenge came from my combined IB mathematics standard-level/high-level class. It’s a given that this combined SL/HL situation isn’t ideal. My internet searches on ways this has already been done successfully haven’t yielded much, aside from people saying that this is a bad idea. Another important given, however, is that I had input into the schedule building last year, and saw that our size and staff prevents this from being done any other way. The way I see it is as a design challenge: given that SL and HL students are working in the same room, what do I do to optimize that time?

My lesson planning process has been pretty consistent over the past few years. My learning standards for each group are based in the curriculum documents provided by IB. These are pretty solid documents in mathematics, and I don’t tend to struggle there. Some standards are common to SL and HL, with HL specific additions added to my standards descriptions. The HL specific standards have also been pretty easy to parse out of the documents.

From the standards, I piece together pacing based on my experience and knowledge of my students. I’ve been teaching them all for the past two years at least, so I feel comfortable knowing how to push them. For each lesson, I curate a set of problems as the benchmarks, work out the prerequisite skills, and then figure out what students are doing at each stage of the class. At this point (which is usually about forty-five minutes in), I identify how direct instruction fits into the sequence, if it needs to be there.

This is where the separation between the groups gets tricky. I don’t always have the SL students necessarily do the same warm up questions as the HL. The HL students might be given one basic problem, and another that forces them to figure out a need for a new method, find patterns, or attempt to generalize based on observations. While the HL students do this, I am debriefing with the SL students, giving them a mini lesson on the objectives of the day, and then setting them off to do some practice. This frees me to work with the HL students, give them a mini lesson on their objectives, and then get them working in a team. I circle back to the SL and work with them wherever needed.

I let the HL students flounder a lot when they are working together. That productive struggle leads to a need for me to come in and nudge them in the right direction with the right question or observation. In a perfect world, I don’t need to nudge and the students figure it out themselves, but the dense reality of the curriculum doesn’t allow for too much discovery.

This process, on the whole, is exhausting. It’s only one of my classes to prepare on any given day, though the block schedule gives a lot more flexibility to do this than if I only had 45 minutes. Out of necessity, I can’t spend time fixated on the perfect pivotal questions. While it is easier to do teacher centered instruction, the planning I used to do just isn’t practical. I do a lot less instruction and a lot more throwing my students into problems and cleaning up issues along the way. Students wanting a clear set of instructions from me aren’t getting them, which admittedly bugs me sometimes. In the long run, these students are spending more time figuring things out on their own and talking to each other, which makes me feel better about the situation. I just wish I was a better curator of materials to make this more smooth for students.

I have identified some ways I plan to change things around for second semester and lighten the load. The curation piece is the big one. Choosing good problems for each group to work on together is the most important element of that work. My direct instruction is then focused on leading students through the tough parts of the thinking process, and then getting out of the way to let them finish the job. The downside to this is that the completeness of my class notes decreases, but I’m not convinced students look back at those notes frequently anyway. There is a lot of good material available online to help students through the basic skills, and my time might be better spent finding and collecting that content for students to work through on their own.

I also feel the need to improve the quality of my interactions with each group. This is especially difficult when I am switching gears so quickly. Some SL or HL discussions don’t fit neatly into a twenty minute interval together while the other group is working. I’ve decided that two blocks of every two week cycle (five blocks total) will be HL specific time. This means the SL students will have time to work on their own and help each other, and I can spend longer intervals of time working specifically with HL students on their exclusive content. The SL students undoubtedly have work to do for their other IB courses, and have expressed an interest in having time to work. The dedicated HL time will also mean the time spent with SL students and on common content becomes more streamlined and focused.

I’m always looking for ways to improve my workflow, so your suggestions are, as always, very welcome.

## Analyzing IB Physics Exam Language Programmatically

I just gave my IB physics students an exam consisting entirely of IB questions. I’ve styled my questions after IB questions on other exams and on homework. I’ve also looked at (and assigned) plenty of example questions from IB textbooks.

Just before the exam, students came to me with some questions on vocabulary that had never come up before. It could be that they hadn’t looked at the problems as closely as they had before this exam. What struck me was that their questions were not on physics words. They were on regular English words that, used in a physics context, can have a very different meaning than otherwise. For these students that often use online translators to help in decoding problems, I suddenly saw this to be a bigger problem than I had previously imagined. An example: a student asked what it meant for an object to be ‘stationary’. This was easily explained, but the student shook her head and smiled because she had understood its other meaning. On the exam, I saw this same student making mistakes because she did not understand the word ‘negligible’, though we had talked about it before in the context of multiple ways to say that energy was conserved. Clearly, I need to do more, but I need more information about vocabulary.

It got me wondering – what non-content related vocabulary does occur frequently on IB exams to warrant exposing students to it in some form?

I decided to use a computational solution because I didn’t have time to go through multiple exams and circle words I thought students might not get. I wanted to know what words were most common across a number of recent exams.

Here’s what I did:

• I opened both paper 1 and paper 2 from May 2014, 2013, 2012 (two time zones for each) as well as both papers from November 2013. I cut and pasted the entire text from each test into a text file – over 25,000 words.
• I wrote a Python script using the pandas library to do the heavy lifting. It was my first time using it, so no haters please. You can check out the code here. The basic idea is that the pandas DataFrame object lets you count up the number of occurrences of each element in the list.
• Part of this process was stripping out words that wouldn’t be useful data. I took out the 100 most common words in English from Wikipedia. I also removed some other exam specific words like instructions, names, and artifacts from cutting and pasting from a PDF file. Finally, I took out the command terms like ‘define’,’analyze’,’state’, and the like. This would leave the words I was looking for.
• You can see the resulting data in this spreadsheet, the top 300 words sorted by frequency. On a quick run through, I marked the third column if a word was likely to appear in development of a topic. This list can then be sorted to identify words that might be worth including in my problem sets so that students have seen them before.

There are a number of words here that are mathematics terms. Luckily, I have most of these physics students for mathematics as well, so I’ll be able to make sure those aren’t surprises. The physics related words (such as energy, which appeared 177 times) will be practiced through doing homework problems. Students tend to learn the content-specific vocabulary without too much trouble, as they learn those words in context. I also encourage students to create glossaries in their notebooks to help them remember these terms.

The bigger question is what to do with those words that aren’t as common – a much more difficult one. My preliminary ideas:

• Make sure that I use this vocabulary repeatedly in my own practice problems. Insist that students write out the equivalent word in their own language, once they understand the context that it is used in physics.
• Introduce and use vocabulary in the prerequisite courses as well, and share these words with colleagues, whether they are teaching the IB courses or not.
• Share these words with the ESOL teachers as a list of general words students need to know. These (I think) cut across at least math and science courses, but I’m pretty sure many of them apply to language and social studies as well.

I wish I had thought to do this earlier in the year, but I wouldn’t have had time to do this then, nor would I have thought it would be useful. As the semester draws to a close and I reflect, I’m finding that the free time I’ll have coming up to be really valuable moving forward.

I’m curious what you all think in the comments, folks. Help me out if you can.

## Releasing my IB Physics & IB Mathematics Standards

Our school is in its first year of official IB DP accreditation. This happened after a year of intense preparation and a school visit last March. In preparation for this, all of us planning to teach IB courses the next year had to create a full course outline with details of how we would work through the full curriculum over the two years prior to students taking IB exams.

One of the difficulties I had in piecing together my official course outline for my IB mathematics and IB physics courses was a lack of examples. There are outlines out there, but they were either for the old version of the course (pre-2012) or from before the new style of IB visitation. The IB course documents do have a good amount of detail on what will be assessed, but not the extent to which it will be assessed. The math outline has example problems in the outline which are helpful, but this does not exist for every course objective. The physics outline also has some helpful details, but it is incomplete.

The only way I’ve found to fill in the missing elements is to communicate directly with other teachers with more experience and understanding of IB assessment items. While some of this has been through official channels (i.e. the OCC forums), most has been through my email and Twitter contacts. Their help has been incredible, and I appreciate it immensely.

At the end of the first semester for Mathematics SL, Mathematics HL (one combined class for both), and Physics SL/HL (currently only SL topics for the first semester), I now have the full set of standards that I’ve used for these courses in my standards based grading (SBG) implementation. I hope these get shared and accessed as a starting point for other teachers that might find them useful.

For my combined Mathematics SL/HL class:
Topics 1 – 2, IB Mathematics SL/HL

For my combined Physics SL/HL class:
Topics 1 – 2, IB Physics SL/HL

The third column in these spreadsheets has the heading ‘IB XXXX Learning Objective’ – these indicate the connection between the unit standard (e.g. Standard 3.1 is standard 1 of unit 3) to the IB Curriculum Standard (e.g. 2.3 is Topic 2, content item #3). Some of these have sub-indices that correspond with the item in the list of understandings in the IB document. IB Mathematics SL objective 1.3.2 refers to IB Topic 1, content item #3, sub-topic item #2.

If you need more guidance there, please let me know.

## If you are a new IB Mathematics/Physics teacher accessing these…

…please understand that this is my first year doing the IB curriculum. There will be mistakes here. In some cases, I also know that I’ll be doing things differently in the future. If these are helpful, great. If not, check the OCC forums or teacher provided resources for more materials that might be helpful.

## If you are an experienced IB Mathematics/Physics teacher accessing these…

…I’d love to get your feedback given your experience. What am I missing? What do I emphasize that I shouldn’t? What are the unspoken elements of the curriculum that I might not be aware of as a first year? Let me know. I’d love it if you could give me the information you wish you had (or may have had) to be maximally successful.

I’ve benefited quite a bit from sharing my materials and getting feedback from people around the world. I’ve also gotten some great help from other teachers that have shared their resources. Consider this instance of sharing to be another attempt to pay that assistance forward.