This year, as I have been in the past few years, I’ve been attempting to incorporate more writing in my math classes [note: Shelli found a post from 2009 I wrote on this endeavor]. It’s been extraordinarily enlightening, because what this has done is show me two things: (1) kids don’t know how to explain their reasoning in clear ways, and (2) I’m usually extraordinarily wrong when I think my kids understand something, and the extent to which I am wrong makes me cringe.
(wow, been too busy to shave, have we Mr. Shah?)
For the first point, I don’t actually do much. I ask them to write, they write, I comment. And we discuss (more at the start of the year, but I always let this go and I forget to talk about it a lot). In Algebra II, they get one or two writing questions on every assessment. And each quarter they had problem sets where they had to write out their thought processes/solutions comprehensively and clearly. Even though I didn’t actually do anything systematic and formal in terms of teaching them to write (mainly I just had them write), I can say that I’ve seen a huge huge improvement in their explanatory skills from the beginning of the year. What I used to get just didn’t make sense, honestly. A random string of words that made sense in their heads, but not to anyone reading them. But now I get much more comprehensive explanations, which usually include words, diagrams, graphs, examples. They aren’t usually amazing, but they’re not ready to be amazing.
For the second point, I realized that the types of questions that we tend to ask (you know, those more routine questions that all textbooks ask) don’t always let me know if a student understands what they’re doing. It just lets me know they can do a procedure. So, for example, if I asked students to graph , I would bet my Algebra II kids would be able to. But if I showed them the question and the solution, and ask them to explain what the solution to that question means, I would expect that only half or two thirds of the class would get it right. (Hint: The solution is the set of all points (x,y) which make the inequality a true statement.) They can do the procedure, but they don’t know what the solution means? That’s what I’ve found. And you know what? Before asking students to write in the classroom, I had deceived myself into conflating students being able to answer with a full understanding of 2-D linear inequalities. 
Before having students write, I actually believed that if I asked that question (“What does this solution mean?”), almost all the students would be able to answer it. (“Like, duh, of course they can!”) But since asking students to explain themselves, explain mathematics, I’ve uncovered the nasty underbelly to what students truly understand. The horror! The horror! But now that I recognize this seedy underworld of misconceptions or no-conceptions, I’ve finally been able to get beyond the despair that I originally had. Because now I know I have a place to work from.
The counterside to this point is that when kids do understand something, they kill it.
This simple question I made for my calculus students early in the year, and this student response, says it all. I have no concern about this kid understanding relative maxs and mins. No traditional question would have let me see how well this student knew what was up.
For me the obvious corollary is that: we need to start rethinking what our assessments ought to look like. If we want kids to truly understand concepts deeply, why don’t we actually make assessments that require students to demonstrate deep understanding of concepts? I am coming to the realization that the more we keep giving the same-old-same-old-assessments, the more we are reinforcing the message (implicitly) that we don’t reallyreally care to know about their thinking. We are telling our kids (implicitly) that we are content if they show their algebraic steps. But as I’ve noted, my big realization is that students performing those algebraic steps don’t necessarily mean that the student knows what they’re doing, or what the big picture is.
I don’t know have an example of what I think a truly ideal assessment might look like, but I do know it isn’t anything like I gave when I started off teaching five years ago (has it really been five years? why am I not better at this?), and I do know that each year I am slowly inching towards something better. Right now, my assessments are fairly traditional, but with each year, they are getting less so.
Sorry if I’ve posted something like this before. I have a feeling I have. But it’s what’s been going through my head recently, and I wanted to get it out there before I lost it.
 Another good illustration might be having students solve . Sure, they can get . But does doing that really mean they understand that whole “if you divide by a negative in an inequality, you switch the direction of the inequality” rule that has been pounded in them since seventh grade? Nope. The traditional questions don’t tend to check if the kids know why they’re doing what they’re doing.