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:

**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.**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**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.__not__to do a combined SL/HL course is that neither HL or SL students get the time they deserve.- 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.

### For year one:

### For year two:

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.