Trying to find math inside everything else

Posts tagged ‘standards of practice’

SoP Portfolios

When we learned about planning back in grad school, we were told that if something is important, you need to assess it; if you don’t assess it, then it’s not important. When the Common Core Standards came out, most math teachers were very excited about the Standards of Practice. The problem was, of course, that the Standards of Practice are hard to assess. So most standardized tests that use the Common Core don’t assess them, which of course means they don’t get implemented in the same way as the content standards. The Standards of Practice are important to me (though I frame them as the Mathematical Habits of Mind), and that means I need to find a way to assess them. But I’ve never had a really good way to test them before – I was always kind of making it up as I went along. Now, though, I think I’ve hit on something now that really works.

After seeing Ashli’s video about not putting grades on papers, I stopped doing it this year – but having the grades still be there in the online gradebook wasn’t quite what I wanted to do, and it became very hard to keep track things, especially because classwork and homework were what I used to measure the Standards of Practice. This semester I gave written feedback on all the assignments that I’ve given but recorded no grades – not even in my own gradebook – the only thing I kept track of was if something was incomplete or missing.

If we decide that every assignment is a formative assessment, we can’t possibly grade it as students are learning the material. So instead each assignment is like a first draft (or second or third) and students can read the feedback that I gave and make changes in order to improve their work. Come the end of the marking period (or eventually the end of the semester) students create a portfolio of their work. They don’t need to include everything that they’ve done but rather a representative sample that shows that they apply the Standards of Practice/Habits of Mind as they work.

The portfolio has a cover sheet (shown below) that that asks them to reflect on what habits they have used in their mathematical work this semester.

They have to find evidence of their own habits in their work and write a few sentences citing that evidence. I gave suggestions of which assignments might be easier to find evidence of those habits in. And they only had to include work that they cited as evidence as part of the portfolio.

To get us started we read through the rubrics that I created for the habits and created posters of what those habits might look like when doing assignments. We have them hanging on the walls of my classroom – that way I can referr to them easily when something comes up (such as when we worked on Des-man, I tried to emphasize the tinkering nature of the process).

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The first portfolios coming in have been graded and some of them were stellar and others need some work, but it was the first time and they are not really used to this whole reflective idea. But I have noticed that most of my students have started to use that vocabulary more and have become more aware of the kind of things that they are expected of them in the long-term, not just immediate math facts. I think if I start this from the beginning next year it’ll create a really great culture of thinking using the habits and the standards of practice.

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assessment

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Habits of Mind, Standards of Practice

For the past three years, I’ve loosely organized my classroom around the Mathematical Habits of Mind which I first read about in grad school at Bard. I would give the students a survey to determine which habits are their strengths and which are their weaknesses, group them so each group have many strengths, and go from there. Last year I even used the habits as the names of some of my learning goals in my grading.

As I was planning for this year and the transition to the Common Core, I was thinking about how to assess and promote the Standards of Practice. And I realized that they are very similar to what I was already doing with the Habits of Mind. In fact, having a habit of mind would often lead to performing a certain practice! In that way, the SoP are actually the benchmarks by which I can determine if the habits of mind are being used.

Let me demonstrate:

Students should be pattern sniffers. This one is fairly straight-forward. SoP7 demands that students look for and make use of structure. What else is structure but patterns? Those patterns are the very fabric of what we explore when we do math, and discovering them is what leads to even greater conclusions.

Students should be experimenters. The article mentions that students should try large or small numbers, vary parameters, record results, etc. But now think about SoP1 – Make Sense of Problems and Persevere in Solving Them. How else do you do that except by experimenting? Especially if we are talking about a real problem and not just an exercise, mathematicians make things concrete and try out things to they can find patterns and make conjectures. It’s only after they have done that that they can move forward with solving a problem. And if they are stuck…they try something else! Experimenting is the best way to persevere.

Students should be describers. There are many ways mathematicians describe what they do, but one of the most is to Attend to Precision (as evidenced in things like the Peanut Butter & Jelly activity, depending on how you do it.) Students should practice saying what they mean in a way that is understandable to everyone listening. Precision is important for a good describer so that everyone listening or reading thinks the same thing. How else to properly share your mathematical thinking?

Students should be tinkerers. Okay, this one is my weakest connection, mostly because I did the other 7 first and these two were left. But maybe that’s mostly because I don’t think SoP5 is all that great. Being a tinkerer, however, is at the heart of mathematics itself. It is the question “What happens when I do this?” Using Tools Strategically is related in that it helps us lever that situation, helping us find out the answer so that we can move on to experimenting and conjecturing.

Students should be inventors. When we tinker and experiment, we discover interesting facts. But those facts remain nothing but interesting until the inventor comes up with a way to use them. Once a student notices a pattern about, saying, what happens whenever they multiply out two terms with the same base but different exponents, they can create a better, faster way of doing it. This is exactly what SoP8 asks.

Students should be visualizers. The article takes care to distinguish between visualizing things that are inherently visual (such as picturing your house) to visualizing a process by creating a visual analog that to process ideas and to clarify their meaning. This process is central to Modeling with Mathematics (SoP4). It is very difficult to model a process algebraically if you cannot see what is going on as variables change. To model, one must first visualize.

Students should be conjecturers. Students need to make conjectures not just from data but from a deeper understanding of the processes involved. SoP3 asks students to construct viable arguments (conjectures) and critique the reasoning of others. Notable, the habit of mind asks that students be able to critique their own reasoning, in order to push it further.

Students should be guessers. Of course, when we talk about guessing as math teachers, we really mean estimating. The difference between the two is a level of reasonableness. We always want to ask “What is too high? What is too low? Take a guess between.” Those guesses give use a great starting point for a problem. But how do you know what is too high? By Reasoning Abstractly and Quantitatively, SoP2. Building that number sense of a reasonable range strengthens our mathematical ability. We need to consider what units are involved and know what the numbers actually mean to do this.

What we do, or practice, as mathematicians is important, but what’s more important is how we go about things, and why. A common problem found in the math class is students not knowing where to begin. But if a student can develop these habits of mind, through practice, that should never be a problem.

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assessment, schooling, theory

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