Common Core - #2 of Studying Structure - More on the Practices

In the first post of my Common Core series, I discussed the importance of really looking at the Standards of Mathematical Practices (SMP), because the title alone is not enough. You cannot assume that each title of the 8 SMPs gives enough information to truly know what students should be saying and doing to enhance their mathematical understanding. The descriptive narratives for each practice are necessary reading to clearly make good instructional decisions that will truly support student understanding of mathematics



My last post focused on SMP #4, Model with Mathematics. In this post, I'd like to choose another of the SMPs, and be a little more explicit in breaking down the structure to show how both the title and the narrative help inform instruction. Then, hopefully, you can do your own investigation of the remaining SMPs.  This is a great exercise to do with collaborative groups of teachers - maybe your next team/faculty meeting?! All of this is from work I have done being part of the International Fellows working with The Charles A. Dana Center, UT Austin.




I am choosing SMP #1, Make Sense of Problems and Persevere In Solving Them. Every teachers dream that students approach problems and understand them and not give up, right?!  But what does that look like in class? How do teachers support students so that they reach this point of being able to take a problem, make sense of it, and continue to try to solve it even when the going gets tough?  Here's where the narrative of the SMP comes into play - title is not enough, remember?!!

Narrative of SMP #1: Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

That's a lot of stuff going on there!  I have underlined some key components related to STUDENT actions.  Notice the narrative is about the students - what they should be doing, saying, considering. Taking this to the next step, reflect on students actions and write down (T-chart works) what students might be saying and doing if they are demonstrating this practice. Create a list, like the one below (this is NOT an exhaustive list, just some examples.  See posted pictures for more suggestions).
Notice, START WITH THE STUDENT!!

Students should saying/doing	Teacher should be saying/doing
-asking questions to their peers
-highlight key words or numbers to understand the problem
-making decisions about a plan to solve
-trying out solutions and CHANGING their plan if it is not working
-continually ask themselves “does this make sense”
-drawing pictures, getting tools (i.e. rulers, calculators, manipulatives) to help make sense and plan a solution
-checking their solution using a different method

Once you've generated several things students are saying and doing, THEN look at teachers actions. What you want students to do informs the actions of the teacher, so this step is done after getting a clearer understanding of the student actions.  Again, here's a list of teachers actions/behaviors, but it is NOT an exhaustive list.

Students should saying/doing	Teacher should be saying/doing
-asking questions to their peers	-provide a classroom environment that encourages conversation and student dialogue
-highlight key words or numbers to understand the problem	-teach strategies to help students identify key words, numbers and questions in problems
-making decisions about a plan to solve	-model and demonstrate various problem solving strategies
-trying out solutions and CHANGING their plan if it is not working	-Allow for productive struggle and act as a facilitator
-continually ask themselves “does this make sense”	-Asking students questions like "is there another way?" "Can you explain how you did this?"
-drawing pictures, getting tools (i.e. rulers, calculators, manipulatives) to help make sense and plan a solution	-Provide multiple resources
-checking their solution using a different method	-Encourage collaboration and sharing of solutions; require students to explain their thinking

This process, of really studying the structure of the SMPs, the title AND the narrative, and then focusing on student actions first allows for a clearer path of what teacher actions are necessary to help students achieve the SMP skills.

I encourage you to do this yourself, but better yet, do some collaborative learning with other teachers around the structure of the SMPs and you will get a better understanding of what students need in order to become the mathematicians and problem solvers that the Common Core Standards strive to achieve.

Next in my continuing series will be looking at the structure of the content standards - a much more complex structure than the SMPs.