Common Core State Standards for Math (CCSSM) were a good thing for math education. I cheered their adoption because to me, these standards supported what I had been practicing and preaching, both as a teacher and a professional development provider. But - even though I have spent the last several years studying and coaching others in implementing the Common Core in a variety of ways, I just recently learned more to solidify, for me, their power. It was sort of an ah-ha moment: if I, who consider myself very familiar with the CCSSM, just learned something new, imagine how those who have spent little, if any, time with these standards (i.e. parents, news reporters, politicians) could benefit from looking a little more deliberately at the standards instead of relying on the rhetoric or a cursory glance to make their decisions?!
I've been working with The Charles A. Dana Center at UT Austin supporting math teacher training. As part of this endeavor, I participated and delivered some training that focused on looking at and understanding the structure of the CCSSM and alignment of the standards (within grade and vertical). My huge take-away was how important this process is for any educator who is implementing the CCSSM. This in depth look is something I think is missing in much of the PD that surrounds these standards, which might explain a lot of the confusion about how to teach and help students understand the standards and practices, which in turn leads to the confusion among parents, the media and politicians.
My plan here is to write a couple posts about some of the ways looking deeply at the standards can bring clarity and understanding to how they work. More importantly, it emphasizes the importance of seeing the Math Content Standards and the Standards of Mathematical Practice as equally important in helping students learn, understand, and apply mathematics. You cannot look at isolated standards or practices or problems and hope to know how the standards and practices work to BUILD mathematical competence over time - (which, unfortunately, seems to be what is happening - looking at isolated components versus the overall picture).
This post is going to focus on the structure of the Standards of Mathematical Practice. Pretty simple really - there are 8 practices, and their structure contains the title and then the narrative description. The problem here being the title is what most educators focus on, and the title DOES NOT give you enough information to really know what students need to do and what educators need to do to support students. You really need to read the description to get a better sense of what students, as mathematicians SHOULD BE SAYING AND DOING, which then informs what resources the teacher needs to provide and what the teacher should be saying and doing.
I am guilty of assuming the title of the practice was enough. I will use Standard of Mathematical Practice #4 as my example. The title of this practice states that students should be able to:
Model with mathematics.Seems pretty simple. On its surface it seems to imply using models - i.e. manipulatives or physical objects to help students understand math - which is wrong, or rather very incomplete thinking. Here's the narrative description of this practice:
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.I've highlighted just a few key phrases that show what this standard really means is students can USE mathematics to model and explain problems, which is NOT the same as using manipulatives to learn mathematics, which in my experience is what many educators think this practice means. What the practice says is students can approach a real-world problem and APPLY and use some type of mathematical model (i.e. equation, function, geometric figure) as a way to explain the problem and and come up with a solution. If you continue reading the narrative, you will see that it references students ability to make assumptions, approximate solutions, interpret their results and create mathematical models that help them solve the problem. It's a lot more descriptive than just the titl, and something all educators need study and understand so that they are not misinterpreting what the titles alone might be saying.
The pictures to the right are of some of the work I did with teachers that shows their end-result of studying structure of the practices. These t-charts show an understanding of how the narrative informs both students and teacher actions (sorry, they are a bit out of focus!)
The point I am making here is that the title is not enough - the narrative is more important than the title because it gives expectations for what students should be saying and doing, which in turn informs what the teacher should be saying and doing. The structure of the practices, if looked at as a whole, provides information to help teachers create a classroom that supports student learning - you need to go beyond the title.