I have been working as a math tutor (alongside my Montessori work) for 20+ years, and I see the same pattern over and over: a conscientious student has gotten straight-As in math all through school before suddenly hitting a wall in Algebra 2 or Precalculus and getting a B or a C on their first test. Their parents call me asking for help. When the student and I start discussing their work, it turns out they've gotten good grades because they are hardworking students who do what's asked of them, not because they had any deep understanding of the math. Now the work has gotten hard enough that some of those gaps in deep understanding are coming back to trouble them. In these cases, we go on a bit of a mining expedition to find those gaps, and almost without fail, it turns out the student doesn't have a deep and flexible understanding of either multiplication or fractions.
Being able to perform a multiplication computation using only the standard (or standardish) algorithm is actually a fairly useless skill, especially when most of us carry calculators around in our pockets. It's even more useless if you don't understand what's going on under the surface. Our Montessori materials are, fortunately, brilliant at demonstrating what is really going on with this algorithm. It's less of a given that children can recognize when to use multiplication in different contexts, can think of multiplication in flexible ways, or can make reasonable estimates.
Similarly, our Montessori lessons do an excellent job of demonstrating what's going on under the surface with fraction operations, but they don't necessarily connect these ideas well to other contexts or representations of fractions. Fractions are a slippery concept, because you have to keep in mind both the parts and the whole and recognize that they form a relationship. This gets even more slippery when you start thinking about multiplication and division, because then you have to start thinking about parts of parts and parts of wholes and keep straight which is which.
So of course, now you're saying, "That's nice, but how do I know whether my child really understands basic operations and what questions do I ask?" Thankfully, the internet is your friend here. There are many, many wonderful teaching resources online, so allow me to curate a few for you. In all of these cases, assessment and learning can be rolled into one problem. Share the problem with your child and let them work on it. If they can solve it, wonderful! If it's a good problem, your child has just demonstrated deep understanding. If they can't solve it, then you know you need to work on the topic some more. (Shameless plug: if you now need to do some math teaching and you have no idea what to do, you might want to sign up for my coaching for parents service.)
One last caveat: "To clarify, there is a big difference a small computation error and a teachable, interesting, juicy mistake." (Zager, p.57). Computation errors are what most of us are trained and primed to notice, so it's what we worry about, but the truth is, computation errors are the boring bit. They happen. Conceptual errors are where we want to focus our attention. If your child can correct an error after you simply point out that the answer is incorrect, you haven't found the interesting conceptual misunderstandings.
So what are some worthwhile resources?
First and foremost, I would begin with number talks. These are best in small groups, but you can do them one-on-one. They're fun, simple to do, and can be done anywhere. The basic idea of a number talk is to take a mental computation problem and try to find as many ways to compute it as possible. Here is an excellent video demonstrating number talks specifically at home:
YouCubed, where this video came from, has an excellent page on number talks at home, with many additional resources. In fact, I highly recommend exploring the entire YouCubed webpage, particularly the YouCubed at Home section.
Another excellent resource is Robert Kaplinsky's Depth of Knowledge matrices. I believe these are intended to help teachers judge the sophistication of thought required by different problems, but you can also use the problems directly from the matrices as a simple form of assessment. I think they are particularly good for assessing "computational number sense", i.e. how well do children really understand the properties of different operations. You can find many, many more of these "fill the gaps" type problems at Robert Kaplinsky's other website: Open Middle Math. (The "middle" refers to the fact that these problems have a correct answer but many, many ways to get there, not to middle school.)
Another way to support sense-making (as it's called in the ed biz) is to always, always, always insist that your child estimate a reasonable answer, and then compare their answer to the estimate at the end to see if it's reasonable. I call this the "smell test": does the answer "smell" right? I will have more thoughts on the topic of estimation in the next few weeks. In the meantime, a great resource for developing estimation skills by using knowledge of the actual world is Andrew Stadel's Estimation180 site.
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