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Antonio continues to work smoothly through even the harder homework problems. In fact, he hesitates less with harder concepts than he did with the simpler, foundational ones. That is less uncommon than most people realize. Getting started is the hardest part, and those who work past the difficulties catch wind in their sails.

In our ninth session together, Antonio and I work on a couple of problems from the previous week's problem set. We then cover complementary probability, which akin to our complementary counting lesson. Along the way we continue to find opportunities to apply problem solving techniques we previously developed.

Antonio is still soaking up notation, which he sometimes works with in a near-fluent way, but we sometimes have to talk through it. But we are now reaching problems where the notation itself is half the battle, which is often the time when learners make a more complete leap to fluency.

Finally we introduce probability problems involving continuous random variables. This is nothing like anything we have worked on in the past and will require Antonio to use his geometry brain.

A quick look at Antonio's transition over around nine hours of work together:

Hours 1 and 2: Reviewing late elementary school concepts fundamental to counting

Hours 8 and 9 (Parts VI and VII): Solving problems in discrete and continuous probability at a level seen in advanced high school or even a university classroom.

That's quite the progression!