Wednesday, July 18, 2018

Concept or Context?

I was recently had the opportunity to work with a group of teachers on creating a vertical articulation for grade K-6 on Measurement, Data, Probability & Statistics. I had given them packets that included the introductory paragraphs from the original CCSS document, the critical focus areas from the AZ Mathematics Standards and the Arizona Math standards.

While circulating I listened in on a conversation between two teachers that were studying the information for grades K-3. They were sharing their concern that more emphasis was not placed on time and money in the early grades. One of the teachers is an ESS teacher. She shared that in her self-contained classroom they dedicate a significant portion of time learning about money and time. She described these ideas as "major life skills." The third grade teacher sitting at the table agreed that it is vital for all students that they have a strong understanding of time and money in order to be successful as adults.

I observed as they looked back through the standards for evidence that more time should be dedicated to these two topics. I stood back as they searched the standards and the paragraphs.

The review confirmed what they had initially thought, time and money are only mentioned in two short standards in first, second and third grade. They also noted that these clusters were all labeled as "Supporting Clusters" in the AZ math standards.

As their dissatisfaction began to lean toward outright disgust, I stepped in and asked them if time and money are mathematical concepts or if they are contexts?

After some discussion they were able to come to the realization that time and money are a context.

Next we looked through the standards and identified where the context of time and money could be used to teach and assess the content of the standards. As we did so, the big aha moment came. They excitedly reached the conclusion that just because time and money was not mentioned in more standards, it did not prohibit them from using time and money contexts to help students understand the concepts and thus increase their understanding and skill with time and money.

I think that it is important for us to consider what contexts we should be including in order to teach the content of our grade level. What contexts can we use to help our students be more prepared for life outside of school?

Tuesday, July 3, 2018

Establish Mathematics Goals to Focus Learning

If you have read NCTM's "Principles to Actions," then you are familiar with the Mathematics Teaching Practices. These practices, "provide a framework for strengthening the teaching and learning of mathematics. (NCTM, 2014, p.9) If you are not familiar with "Principles to Actions" and the Math Teaching Practices, I suggest you look into them. They really do have the power to transform your teaching.

Although all of the Math Teaching Practices are important, I find the first practice, "Establish mathematics goals to focus learning," to be the most important for me to be mindful of when planning a lesson. While important, it is also the most challenging portion of the planning process. Here are the steps I take.

Step 1: Read the Standard
I mean, REALLY READ THE STANDARD! The standards are very dense and can be difficult to make sense of. I read them closely and look for parts. Lets look at a 6th grade standard.

Understand the concept of a ratio as comparing two quantities multiplicatively or joining/composing the two quantities in a way that preserves a multiplicative relationship. Use ratio language to describe a ratio relationship between two quantities. For example, "There were 2/3 as many men as women at the concert.”

I see two parts. 1) Understand what a ratio is and that it can be extended. 2) Describe ratios using appropriate ratio language (this includes written formats). At first glance, this seems pretty straight forward. Caution is needed here as understanding ratios and seeing the multiplicative relationships within them can be difficult. So although it is only 1 sentence in the standard there is a lot of intricacy in what students need to understand. However, this blog is about writing mathematical goals, so I am not going to dig deeper into that but does illustrate my point about really reading and thinking deeply about the mathematics that is the focus of the standard.

Step 2: Consult the AZ Performance Level Descriptors & Item Specifications
You can access these documents here for grades 3-5, here for grades 6-8 and here for End of Course standards (Alg. I, II, and Geometry). These two documents have become "go to" resources for me. They clearly define what students should be able to do and give some insight into what is basic understanding and/or skill and what is full understanding and/or skill. They provide information about number sets that should be included as well.

From the Performance Level Descriptors we see the following information.

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As I read across the columns, I can begin to get a sense of what the standard is expecting students to not only do, but what the understanding expectations are. Generally, there are multiple ideas that students are to understand but I can begin to determine where my focus will lie. What sticks out to me is that students are to "connect between representations for ratio situations."

Here is the item specification information for the standard.

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This document contains a lot of information that is helpful in thinking about the mathematical goal. There is a lot of emphasis on using language and describing. I also notice what representations students will be expected to use in the "Sample Task Demands" section. For this standard, tape diagrams and double number line graphics are specifically mentioned. This can also be helpful as you think about the mathematical goal. The "Recommended Math Practices" can also be considered when writing math goals.

Step 3: Consult other resources if needed
At this point, I might refer Acheive the Core's Coherence Map which can help when building on previous understanding or preparing for future understanding and/or skills. I will continue to consult resources if I am unclear on the standard.

Step 4: Write the mathematical goal
I will then take a stab at writing a mathematical goal. For this standard, there are a lot of mathematical goals that are appropriate, depending on the experience that students have with ratios. If students have little experience with ratios I might start with "students will understand that a ratio can be a part to part relationship or a part to whole relationship." Or, "students understand that a ratio is a multiplicative relationship and that this relationship can be extended to larger or smaller quantities." If I decide to focus on representations, the math goal might be "Students will understand that double number lines and tape diagrams can be used to extend ratios."

Once I have my math goal, I can begin to build an effective task based lesson. The goal keeps me on track and ensures that I assess for the correct understanding during the lesson. I choose better task, ask better questions and am ultimately a better teacher when I use this process.

I'd love to hear what your thoughts are and what process you use to write math goals in the comments.

Principles to Actions: Ensuring Mathematical Success for All. NCTM, National Council of Teachers of Mathematics, 2014.