The Five by Eight Card: A tool for observing
The “5 x 8 Card” is the product of user based design. The user is the school principal. The design targets the intersection of the principal’s experience and the educational purpose of developing the students’ expertise in mathematical practices defined in the CCSS Mathematical Practice Standards 1 through 8 (pages 6 -8, CCSS – Mathematics). Detailed empathy and user experience work led to design specs:
- Minimize reading: fit on 5 x 8 card (hence the name)
- Use concrete, catalytic ideas that excite action instead of trying to be comprehensive. Comprehensive requires too much text or, worse, very general, vague, boring language that inhibits rather than stimulates action.
- Aim for developing common taste in good instruction across actors with different responsibilities and expertise. Cannot assume mathematics expertise in principals.
- The 5x8 card is designed as an artifact that catalyzes shifts in classroom culture toward the CCSS practices
- Observation focuses on student actions, not the teacher. Teachers have many different ways to get students to act productively. It is student actions that lead to learning.
The 5 x 8 Card was not designed as a teacher evaluation tool. Rather, it is tool that focuses observers’ attention on what students are saying and doing so that their work (their thinking) can be at the center of educators’ discussions.
Components of the card are aligned with CCSS-M and promote a framework of equity by asking observers to attend to whether all students in a classroom have appropriate opportunities to development their mathematical understanding.
SERP team members have used the 5 x 8 Card to organize professional development for CCSS-M in combination with communities of practice for teachers, assistant principals, and principals. In those settings, the 5 x 8 Card provided a framework for gathering evidence of student thinking between meetings for discussion within the communities of practice, thereby supporting schools as they adopted CCSS-M.
- Using the
"5 x 8 Card"
- How to interview
your own students
Educators may download the card for use at their schools. Some educators have altered the card to match more directly the agenda at a particular school or in a particular district. This is fine. More information about the CCSS is available at www.corestandards.org.
Using Side 2 of the 5 x 8 card
LOGIC CONNECTS SENTENCES
|Evidence:||Students say a second sentence (spontaneously or prompted by the teacher or another student) to explain their thinking and connect it to their first sentence.|
|Standard:||Reason abstractly, construct viable arguments|
|Rationale:||The ultimate goal of CCSS-M is to promote student understanding of mathematics. A hallmark of understanding is the ability to use mathematical reasoning to construct an argument that defends one's position. A viable argument is constructed from a logical progression of statements. Brief, single-sentence student utterances most often are insufficient to reflect a logical progression of statements that forms a viable argument based on mathematical reasoning. Therefore, it is desirable for the teacher to pace questioning and responses and to facilitate student discussions in ways that allow students to follow up an initial statement with another in a logical series that forms a viable argument.|
REASONING DEVELOPS WHEN STUDENTS DEVELOP VIABLE ARGUMENTS
|Evidence:||Students talk about each other's thinking (not just their own).|
|Standard:||Critique the reasoning of others, model with mathematics|
|Rationale:||Students learn about mathematics by exploring their own and others' reasoning in problem-solving situations. Exploring the reasoning of self and others allows for flaws in thinking to be revealed and corrected. Opportunities for students to reveal their thinking and for their peers to evaluate and contribute to the improvement of student thinking can lead to stronger mathematical understanding.|
STUDENTS WRITE EXPLANATIONS
|Evidence:||Student work includes revisions, especially revised explanations and justifications.|
|Standard:||Constructing viable arguments, modeling with mathematics, attending to precision.|
|Rationale:||As students become more mathematically proficient and their reasoning skills increase they should be able to identify flaws in their own and others' thinking; thus prompting revision of thinking that leads to better problem solving.|
ACADEMIC SUCCESS DEPENDS ON ACADEMIC LANGUAGE
|Evidence:||Students use academic language in their explanations and discussions.|
|Standard:||Attend to precision, model with mathematics, look for and make use of structure, look for and express regularity|
|Rationale:||Mathematically proficient students understand and effectively use the symbol systems and vocabulary associated with mathematical modeling. Students can create precise arguments and reasoning when they use of academic vocabulary that is specific to the mathematics they are engaged with.|
|Evidence:||Students indicate that they believe they can learn to be good at math by learning more math and working hard to make sense of problems.|
|Standard:||Make sense of problems and persevere in solving them.|
|Rationale:||Mathematically proficient students frequently check to make sure their work makes sense when solving problems and correct their course when they realize they have made an error. They persist in the pattern of problem-solving and sense-making until they achieve a defensible solution.|
EVIDENCE OF EQUITABLE APPLICATION OF THE STANDARDS
English learners get time, encouragement, and support – from other students/teacher – in using academic language. Support includes scaffolds such as sentence frames, multiple choice oral responses, and reference to diagrams and other representations.
All students – regardless of gender, race, language background, or other student characteristics - have equitable opportunity to demonstrate thinking, critique thinking of others, and receive teacher support for CCSS-M based performance.
Teacher-student interviews allow us to understand student thinking in all stages of reading and attacking a problem. They can give us insights into student cognition along the spectrum of students. They can also help us build relationships with our students. Above all, teacher-student interviews are tools that help us grow as teachers. Interviews led our working group to surprising realizations about student thinking that we had not noticed in our individual classrooms. For example, from our collective interviews we gathered that students’ reliance on additive interpretations of proportional situations was widespread. The need to teach unit rate and other multiplicative interpretations also became apparent during interviews.
An important question teachers often ask when interviewing students is what role they themselves play in process. Should we offer clues? Let the student elaborate on incorrect approaches? Allow silence?
Below is some advice that might be helpful.
The Interview Process
The teacher first models the process by reading a problem and solving it as a think-aloud. Try an approach that doesn’t work and start again, saying “Oh! That didn’t work. I’ll try again.” Be sure to say some things that contradict each other, cross out some errors, and generally show how you puzzle out a math problem. Explain that the most important part is for you (the teacher) to understand how the student thinks about solving a problem, and where the mistakes and confusions may be, so that you can become a better teacher. For the student’s turn, follow the step below.
First Things First
Have student read the question twice: once aloud and then once silently (or twice aloud).
Early in the Interview
Prompt talk by asking what phrases in the text mean. Explicitly postpone attempts to find “the answer.” Focus on making sense of the situation first and then on considering the question.
Later in the Interview
Point out contradictions in the student’s thinking (gently - if this picture means so-and-so, what does this one mean?), ask for a diagram, ask him/her to explain obscure references (e.g., which, it, this), ask about overlooked phrases in problem, ask about relationships between quantities already identified.
After the Interview
Bring the recording and student work to the adult learning community, discuss interesting examples of student thinking, discuss implications.
Stages of teacher intervention during a student interview
Stage One: Initially prompt the student to talk without offering clues. Ask him/her to read the problem aloud, to share his/her interpretation of the problem and other thinking aloud, and to describe an approach he/she plans to use to better understand the problem or to attempt to solve. If possible, remain at this "Stage 1" throughout the entire interview.
- Can you tell me what the problem is asking you to do?
- What do you think you need to find out?
- Please say aloud, for the recorder, what you are doing and why.
- Can you identify the important facts in the problem?
Stage Two: IF stage one fails to be productive, then the teacher can begin prompting the student in ways that are related to the mathematics that the student has produced (not the problem itself). For example, the teacher may point to student work and ask for an explanation, ask the student to produce a diagram related to something he/she said or wrote, ask the student to identify the meaning of a variable, etc.
- What do those values represent?
- Why are you choosing that operation/process?
- Can you draw a picture/diagram that will show me what you just did?
Stage Three: IF stage one and stage two both fail to be productive, then the teacher can point to parts of the problems and ask the student to explain work he or she has generated.
- Where is this (phrase from problem) in your calculation?
- What about this part of the question?
- What do you already know about this problem?
When we assembled to listen to the recordings we collected, we often considered what transpired in terms of the states of making sense (see below). This helped us.
Consider where the student is in terms of the States of Making Sense. Doing so may help you understand where to try to take the student next.
EARLY STAGES No Clue/Unwilling to Try Senseless Calculation Understanding the Language (syntax) in the Problem Identifying Quantities MIDDLE STAGES Thinking Relationally Between Quantities My Way or the Highway
(unwilling to shift strategies)
Relying on Patterns Back and Forth
(towards capturing the problem’s structure)
LATE STAGES Checking for Reasonableness A-ha!
(realizing the mathematical structure of the problem)
Sample of audio recordings the SERP teachers collected: