A parent asks: "What are the jobs after your upper El student knows basic operations in fraction, decimal, and integer computation? I’m most interested in ones I might be able to DIY at home. Thanks so much!"
First, I want to be clear that basic operations with decimals and so on aren't necessarily a prerequisite to these other types of work. Students can begin exploring many other aspects of math even before they have a solid grasp on these topics. Another thing to consider is how deeply your child understands these basic operations. It's not uncommon that children can follow the process for solving different operations, but still get thrown off by a problem or story context that requires thinking about the operation in a different way, even for children who have learned operations using Montessori materials. This is a big topic, and I'm not going to dive into it right now, but you've inspired me to explore it more in another blog post soon.
So here is a brief list of topics to explore, along with some comments and resources for some of them.
Squares, cubes, and powers of numbers
Work on squares, which particularly focuses on dividing squares into binomials, trinomials, and other multinomials, is pretty easy to adapt to graph paper, and an older elementary child will likely prefer working this way. Here is a blog post I wrote some time ago that will walk you through the concepts. I think DIYing the material we use for cubing would be extremely difficult, and the material is quite expensive, so I would leave this work aside for the time being.
The powers of numbers material may also be a bit hard to DIY, but you could explore the concept using any small cubes, as long as they're all the same size. Start with two cubes; this is 21. Now double that to make four cubes. Now you have 22. Double that again and you have eight cubes, which is 23. Wash, rinse, repeat. If you have a set of Golden Beads or base-ten blocks, you also have powers of ten. A ten bar is 101. A hundreds square is 102. A thousand cube is 103. A unit is 100. A fun question for your child: what would 104 look like? 105?
Algebra
While children are hopefully thinking about algebraic reasoning all through their elementary years, upper elementary is a great time to explore in more depth. To my mind, there are really three key topics that will lay the foundation for a solid understanding of algebra: equations, growing patterns, and graphing.
The real key to understanding how to work with equations is understanding the idea of balance. We have Montessori lessons that use bead bars for these concepts, but you can also literally use balances. Here is a website full of wonderful puzzles that quite literally explore the concept of balance. There is nothing wrong with simply working within the website, using paper to experiment with different solutions, but you could also explore making some of these mobiles for real. A good place to start is to consider--with your child!--what materials you would need to correctly weight the mobile. What I would not do in the beginning is worry about formalizing any of the concepts into algebraic notation. This can come later.
To explore patterns of growth, start with visual designs. You can find hundreds of them on Fawn Nguyen's Visual Patterns site, which is brilliant. The idea is to look at the first few steps of a pattern and consider how you see it growing (graph paper and colored pencils highly recommended!). Then decide what the pattern will look like after some large number of steps. At first, you might try to figure out what the pattern will look like after ten steps, so your child can work their way up from step one all the way to step ten, but eventually you want to challenge them to find what the pattern will look like after a much larger number of steps, say 43, or 100, to encourage looking for a general rule for the pattern.
There are many, many activities you can find online to explore graphing. For a Montessori approach, take a look at Mike Waski's Algebra for the Adolescent. It's quite an expensive book, as it's a complete teaching album for middle and high school algebra, but it's excellent. Despite the "adolescent" part, there is a great deal that's useful for older elementary students.
I also highly recommend The New York Times' feature What's Going On In This Graph?. Each week, they choose a graph from a NY Times article for students to think about, then they get an expert from the American Statistical Association to share some statistical ideas relevant to the graph. I think they are on a break for the summer, but I imagine they will start again soon. As these graphs deal with current events, I'd encourage you to pick and choose the graphs you want your child to study, based on your best judgement about what is appropriate for them.
I hope these ideas give you a good place to get started! Remember that in elementary, the biggest goals are exploration, curiosity, and comfort with mathematical ways of working. Middle and high school teachers can take that foundation and help their students formalize it, but a child who is afraid or turned off by math will need to do a lot of unlearning before they can really learn math as an adolescent. I'm excited to hear in the comments how it goes and what other questions you have, and to hear what ideas other parents and guides have.