How can students receive meaningful feedback on work in a math classroom that goes beyond getting the problems right or wrong? How do we assess and develop a student’s ability to meet hard-to-measure standards like understanding, problem solving, and communication? How might we make math rubrics that are helpful and friendly for our very young students but still keep a focus on content and process standards?
The curriculum writing team in Denton ISD tackled these challenging problems head on.
As the elementary math coordinator in Denton ISD, it is my privilege to get to work with the teachers that make up our math curriculum writing team.
Several years ago, we set out on a course to develop a guaranteed and viable curriculum for the teachers and students of Denton. Our journey began by studying Understanding by Design with Jay McTighe and Allison Zmuda, where we began the process of using backwards design to write curriculum. As we began writing units of study, it became clear that we needed to do a better job of articulating the criteria for success for teachers and students. Without clear criteria, we found that well meaning teachers often underemphasized the ever important process standards. So, we did as any believers in backward design would do: we studied our process standards deeply.
The process standards are critical to math instruction as they articulate how students acquire and demonstrate their mathematical understanding. These are the standards that help our students become mathematicians: not just the litany of content that must be taught over the year, but the processes that allow them to DO math.
With these process standards in mind, we worked together to generalize the big ideas of what we are looking for when teachers analyze student work. We came up with four general categories:
- Conceptual Understanding: because, outside of the process standards, there is quite a list of concepts students must learn
- Computational Accuracy: because often, accuracy matters in mathematics
- Problem Solving Process: because helping students get good at thinking is a worthy focus
- Communication: because more than half of our process standards address communication skills
Once we determined our big ideas, we set forth in trying to describe what quality looked like for each of these items. This proved to be messy work! We met some roadblocks, such as trying to tease out exactly what it means to prove students understand a concept separately from communicating about that same concept. These big ideas are so carefully woven together in real life it was tricky to separate them. Studying student work helped with that. We dug through, analyzed, talked about, and argued over several samples of low-, medium-, and high-quality math work, and it started to become clear what we are looking for in each of the categories.
Once we determined a product we could be proud of, we tested it with teachers and students and rolled it out to our 4th and 5th grade classrooms. With some minor adjustments, we were able to roll out a closely aligned version of the rubric to 2nd and 3rd grade teachers the next year.
Finally, we met our greatest challenge: making an analytic rubric that we use regularly with 7-11 year olds work with 5- and 6-year olds! Our curriculum writing team, with the help of a few more talented kindergarten and first grade teachers, labored over this task!
We decided that the students needed graphics to help them associate an image with the criteria. We also created a document to “bridge” the descriptors of quality with the actions students needed to take to create quality work. We determined there was a need for fluency to be added to the rubric as well. After all, some of the early mathematics curriculum requires students to know and be able to respond fluently such as identification of shapes, rote counting, and identifying coins. We used the images of the puzzle pieces coming together to represent the journey from beginning to meeting the standard. I love the imagery associated with putting a puzzle together. It communicates that learning takes work, and how we share evidence of our learning takes work! Our labor of love is shared with you below.
It has been exciting to see the rubric in action with our pilot classrooms. Students are beginning to use the language of the “bridge document” as they talk about their work. They are conferencing with their teachers and getting and giving feedback. Their work is beginning to improve in quality as the expectation has been made clearer to them. While we know there will be refinements to come, and we know we will never be finished with this work, it feels like we have taken a big step in the right direction for mathematics in Denton ISD.
Next year we will be able to say that every classroom, from kindergarten to grade 5, is using an aligned rubric to clarify goals, give feedback, and assess student work. Only time will tell the impact this has on student achievement. But I have to believe that good will come from this work. Tackling the challenge of creating a K-5 analytic rubric was tough stuff, but the payoffs are proving to be big for our teachers and students!
Math Rubric (K STUDENT)
| Learning Goals |
Beginning 1 |
Developing 2 |
Meets Standard 3 |
Understanding the Concept I can use what I know to grow my idea.What do I already know?How can it help me to build my idea?
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| Fluency
I can show fluency in _____________. How can I show what I know without difficulty? |
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| Accuracy
I can compute accurately. I am sure my answer is right! |
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| Problem Solving Plan
I can make a plan that helps me figure out the problem. What do I know? |
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| Communication
I can use math language to show my thinking. How do I use numbers, words, symbols, and pictures to show my thinking? |
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Math Rubric (K TEACHER)
| Learning Goals | Beginning
1 |
Developing
2 |
Meets Standard
3 |
| Understanding the Concept
What do I already know? How can it help me to build my idea? |
The student made connections using writing, words, pictures, or models that were irrelevant or confusing to the given concept. | The student made connections using writing, words, pictures, or models that were promising but needed guidance to build understanding of the given concept. | The student independently made interesting and appropriate connections or models to show understanding of the given concept. |
| Fluency
I can show fluency in How can I show what I know without difficulty? |
The student needed a lot of assistance and prompting to demonstrate fluency. | The student made small errors in demonstrating fluency (e.g., inconsistency, not realizing the error) | The student demonstrated fluency with ease — student accurately performed on cue without being stressed. |
| Accuracy
I can compute accurately. I am sure my answer is right! |
I attempted to complete the computation but got stuck even with teacher assistance. | I completed the computations but had mistake(s) in my work OR needed teacher assistance to get the right answer. | I independently completed the computations to get the right answer. |
| Problem Solving Plan
I can make a plan that helps me figure out the problem. What do I know? What do I have to figure out? |
The student didn’t understand the problem even with teacher assistance. | The student understood the problem but needed some teacher assistance to come up with an appropriate plan OR The student tried to set up the problem but used the wrong tool/strategy and/or gave up on the plan. |
The student independently understood the problem, figured out an appropriate plan, and checked to see if it was a reasonable solution. |
| Communication
I can use math language to show my thinking. How do I use numbers, words, symbols, and pictures to show my thinking? |
The student’s representations and/or explanation were inaccurate even with teacher redirection. | The student’s representations and/or explanation were mainly accurate with some minor errors in details and use of mathematical language. | The student’s representations and/or explanation were accurate and complete using mathematical language. |
Math Rubric – The Bridge Document (K)
| Learning Goals | What action do I need to take to reach my goal? |
| Understanding the Concept
What do I already know? |
I can use what I know to show my idea.
I can make a connection that shows my math understanding. |
| Fluency
I can show fluency in How can I show what I know without difficulty? |
I practice so the right answer just pops into my brain and out my mouth because I know it so well. |
| Accuracy
I can compute accurately. I am sure my answer is right! |
I can get the right answer without any help.
I am careful and make sure there are no mistakes. |
| Problem Solving Plan
I can make a plan that helps me figure out the problem. What do I know? |
I can identify what the problem is asking me to do.
I have a plan to help me figure out how to solve it. My answer makes sense. |
| Communication
I can use math language to show my thinking. How do I use numbers, words, symbols, and pictures to show my thinking? |
I can use math language, numbers, pictures, models, and symbols to show my thinking.
I can tell why I think my strategy and solution are correct. |
Math Rubric 3rd – 5th Grade Student
| Learning Goals | Beginning
1 |
Developing
2 |
Meets Standard
3 |
Advanced
4 |
| Demonstration of Conceptual Understanding
I can use what I know to grow an idea. What do I already know? How can it help me to build an idea? |
I attempted to demonstrate the concept but got stuck. Scaffolding or support didn’t help. |
I demonstrated some understanding of the concept. Scaffolding was needed and may have helped. |
I independently demonstrated a reasonable understanding of the concept based on patterns and connections I am making. | I independently demonstrated a deep and accurate understanding of the concept based on patterns and connections I am making. |
| Computational Accuracy
I can compute accurately. How can be sure my answer is right? |
I attempted to complete the computation but got stuck even with teacher assistance. | I completed the computations but had mistake(s) in my work OR needed teacher assistance to get the right answer. | I independently completed the computations to get the right answer. | I independently and efficiently completed the computations to get the right answer. |
| Problem Solving Plan
I can make a plan that helps me figure out the problem. What do I know? What do I have to figure out? Is this a reasonable solution? |
I didn’t understand the problem even with teacher assistance.
OR I tried to set up the problem but used the wrong tool or strategy and gave up. |
I understood the problem but needed some teacher assistance to come up with an appropriate plan
OR independently followed my plan but got an unreasonable solution. |
I independently understood the problem, figured out an appropriate plan, and made sure that the answer was expressed appropriately (ex. units of measure, fraction vs. whole number) | I independently understood the problem, figured out an appropriate plan, made sure that the answer was expressed appropriately , and checked to see if it was a reasonable solution. |
| Explanation of Strategy & Solution
I can use math language to show my thinking. How do I use numbers, words, symbols, and pictures to show my thinking? Does my answer make sense? Why do I think my answer is a good one? |
My explanation was inaccurate even with teacher redirection. | My explanation was mostly accurate with some minor errors in the use of mathematical language/logic.
OR When I tried to explain my thinking, I retold my process. |
My explanation was accurate using mathematical language and I told why I thought my strategy and solution was right. | My explanation was accurate, complete, and logical using precise mathematical language to explain what I did and why. |
We have just started our journey with UbD. At the same time we are working with an ELL consultant about what “evidence of student thinking” looks like. This is so helpful!