Aryan seemed familiar with the derivatives and antiderivatives of some classes of functions, for example polynomials. He also seemed mathematically capable and strongly motivated to improve his understanding of the topic by systematic study (he mentioned that the approach in his class can be somewhat scattered).

Aryan's understanding of the conceptual and theoretical aspects of calculus is somewhat weak. For example, it was not entirely clear to him how the integral sign related to the definition of integration in terms of summed areas, or what the operation has to do with areas in general. There were also some weaknesses in more basic math knowledge, such as proficiency with geometric thinking, the formula for areas of trapezoid, and some algebraic properties of fractions, exponents, and negative numbers. Gaps in his proficiency with these topics made it difficult for Aryan to compute integrals cleanly and confidently: these concepts are used in both conceptual reasoning about and calculations of integrals, and uncertainties about such intermediate steps popped up many times. These gaps seems to be the result of somewhat lax classroom teaching standards that have aggregated over time, and are now holding back Aryan's achievement in topics such as calculus, which rely on having a strong vertical knowledge of prior topics. But Aryan seems to have strong mathematical capability and motivation to improve. So it should be straightforward for us to work through calculus and other year 12 topics systematically, while also reinforcing the gaps in prior knowledge and skills that pop up along the way.

Julian was curious about the concepts, and seemed to understand some of the basics more clearly after discussion. He also seemed to appreciate that such one-on-one discussion could usefully improve his knowledge in a more targeted way. This is a good sign that we can make progress in our tutoring.

Julian's level of engagement with the concepts is however a bit superficial, and much work must be done to raise the level of analysis that he can bring to math and science topics. For example, his calculations skills are quite weak. We will need to be more organised in the future, to ensure that there is adequate time to develop and practice these skills in relation to the expectations of his assessments (at least). Perhaps more importantly, his understanding of what qualifies as a causal explanation for physical or mathematical phenomena is more vaguely relational and less logically rigorous than desirable. We should work on these general thinking skills with regards to explanation and rigor as well.

There were a lot of subtle concepts in this lesson. The whole class was consumed by conceptual discussion and simple examples, with little time for calculation practice. But this discussion was very much necessary at the beginning of such a topic. The discussion seemed to be reasonably effective, though there is still a long way to go before his knowledge of Newton's laws could be regarded as solid. For example, Julian was able to correctly conceptualise force balance in the context of a thought experiment with no prompting.

As mentioned, the topic is new, large, and subtle, and there's a long way to go before Julian's knowledge is solid. For example, it's likely that there is some lingering obfuscation of the roles of force and inertia in Julian's thinking. We will need to probe this in future discussions. We did not yet have sufficient time or knowledge of his assessment needs to adequately practice calculation skills. So we need to plan and organise some time for this in future lessons.