Looking for Standards in the Mathematics Classroom
What is the 5x8 Card?
The 5X8 Card is a tool to facilitate the observation of what students are saying and doing in
classrooms that reveals their underlying mathematical thinking. Elements of the card are aligned with
the Common Core State Standards (CCSS) in Mathematics and promote a framework of equity by
asking observers to attend to whether all students in a classroom have appropriate opportunities to
develop their mathematical understanding.
Standards for
Mathematical Practice
Observing for Student
Vital Behaviors
The 5x8 Card is not…
The 5x8 Card is not designed as a teacher evaluation tool. Rather, it is a tool that focuses the
observers’ attention on what students are saying and doing so that their work (their thinking) can be
at the center of educators’ discussions. The student vital behaviors listed are not intended as the
definitive or comprehensive list. Rather, they are a sample set of specifics: student moves and
interactions that begin to train the observer’s eye and ear. Used in this way, with the focus on the
developing eye and ear of the person observing student learning (the teacher, for example), the 5x8
Card can result in a dramatic shift in the quality and quantity of data the senses take in. Small-grained,
detailed information about student thinking prompt a different kind of conversation – with colleagues,
or with students – and can lead to decision-making and action more responsive to the immediate
learning needs of students.
Version 4: 3-5-12
This compendium was co-developed within OUSD in partnership with SERP and Lawrence Hall of Science
For more information: serpmedia.org/5x8card/
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Looking for Standards in the Mathematics Classroom
Phase 1: Learning to hear and see the Standards of Mathematical Practice
and student vital behaviors.
This tool allows teachers and other educators to:
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develop the teacher’s or observer's eye/ear to see and hear students’ mathematical thinking,
talk, and action (production)
focus on gathering evidence of student thinking through careful observation and sense-making
of the vital behaviors necessary for learning mathematics
build a stronger practice of evidence gathering as the basis for understanding teacher decisionmaking and providing students with feedback – (affirming the mathematical disposition and
vital student behaviors at the heart of the Standards for Mathematical Practice)
Evidence Gathering: examples of student artifacts
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Student responses on performance tasks
Student work products
Video and audio of students working on math
Student interviews (by adult or other students)
Written explanations: including diagrams, sketches, symbols, and words
Spoken explanations
Phase 2: Giving assets-based feedback to students based on evidence.
How can teachers, school- and district leaders begin to act on the evidence and information gleaned
from observing students in the midst of their learning? One way is to include students themselves in
the process.
Collectively, math class has a lot to do with learning the moves of a young mathematician –
developing the disposition and the vital behaviors at the heart of the Practices. Students need to
internalize these behaviors, and to do so, will need frequent opportunities to recognize, label, discuss,
imagine, practice, and reinforce these mathematical behaviors for themselves. An essential catalyst in
this process is the feedback they get from a knowledgeable adult.
Even as teachers are learning to provide feedback that advances students’ learning, participation from
a teacher colleague, an administrator, or even a visitor to the classroom, can send strong and
productive signals to students about the skills and disposition most valued, and about the
developmental nature of the learning that needs to take place in the classroom. Students benefit
particularly when they hear these same messages from someone other than their teacher, who is
there every day. [excerpt from assigning competence – goes here?]
Teachers also benefit from hearing the same messages about student learning, delivered differently.
Any two observers will see/hear different things. Their interpretations too will vary. But perhaps
most significantly, how we talk to students about what we see and hear in their mathematical
contributions and thinking will vary in interesting and useful ways. Opportunities for sharing in this
process of giving students feedback, then, become high-leverage for moving a school’s culture and
practice towards the Mathematical Practices.
Version 4: 3-5-12
This compendium was co-developed within OUSD in partnership with SERP and Lawrence Hall of Science
For more information: serpmedia.org/5x8card/
2
Looking for Standards in the Mathematics Classroom
Giving assets-based feedback to students based on evidence: sample statements, stems
[examples don’t cover the vital student behaviors sufficiently… needs work]
 This class seems to be getting very good at ___________. Here are several examples I observed:
 Here is one quote that seems typical of the way you are learning to push each other’s thinking:
 There isn’t much more important in math class than learning to ask good questions. Here are the
questions I heard today:
 I noticed when I was listening to this group… That worked well, from what I observed. I think it worked
well because you are learning to explain your thinking and build off each other’s ideas.
 At our school, we are all learning to make strong arguments – in math class, English class, even
in electives and PE. I heard an argument today that tells me your skills are getting stronger! A
favorite example? I got to hear…
 (Name), your ability to justify your answers seems to help you understand the math more
deeply, but clearly is pushing the thinking of your partner/group as well. Well done.
 Early in the visit I heard a student say (or ask) _____________. What I noticed is that over
the next 20 minutes this is what seemed to happen:
That is the power of a good
idea, or a good question. That’s exactly what will help….
 If the math is too easy, the learning won’t stick. I notice there’s a real struggle, a productive
struggle, that tells me you are learning the importance of perseverance. The mathematics
learning will stick. And you’re developing the skills that guarantee graduation and success in
college and career. Here’s an example I just observed:
 It is so important that every day you notice your own learning. Revise your thinking. Notice
the change. I think at the beginning of the class I saw/heard students thinking _________, but
now we have evidence that they understand ________. Here’s my evidence:
 What do you want me to look for when I stop by this class another time?
 I noticed that many students in this class are good at asking questions (for example). What
does your teacher do that helped you get good at this skill?
 The best learning is when students ask each other lots of questions and can use a second
sentence explanation – even when the teacher isn’t around. Who is getting good at this?
What’s helping you? I would love to come back and take a look at how this is going.
 You and your teacher will soon be able to make the claim that all 6th graders at our school are
able to construct viable arguments to support their own thinking and critique the reasoning of
others. Here is the evidence I observed today:
Think about what you need next
to keep getting better at explaining your own thinking or responding to your classmates.
Version 4: 3-5-12
This compendium was co-developed within OUSD in partnership with SERP and Lawrence Hall of Science
For more information: serpmedia.org/5x8card/
3
Looking for Standards in the Mathematics Classroom
Phase 3: Helping students learn to recognize and discuss the Mathematical
Practices and vital student behaviors.
Math class can sound and look very different when the instructional focus calls on students to defend
their ideas, apply their skills to novel problems, and explain the mathematics underlying procedures
they are familiar with. All students need on-going opportunities to develop these skills. When
introducing new behaviors, it is important to see the developmental process for the group. Students
need opportunities to label and discuss new behaviors. They need help recognizing when new
behaviors occur, and to be able to label these behaviors and discuss them in an objective way.
Students need opportunities to practice, and benefit from reinforcement when these behaviors occur.
Strategies for helping students recognize and discuss Mathematical Practices and vital student
behaviors:
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Student version of the 5x8 Card
Self-assessment tool
Mathematics reflection prompts for students
Participation Quiz or Participation Skill-Builders (with 1-2 focus skills)
Phase 4: Giving feedback to a teacher after observing students with the 5x8
Card.
Teachers may be accustomed to having visitors in their classrooms, where following the visit there is
feedback or a debrief discussion. The 5x8 Card has primarily been a vehicle for training the eyes and
ears of the observer (could be the teacher, her/himself!) Recognizing and sharing back specific
moments in the learning is one way to give to feedback, but there are other ways to look more
closely and then share the teacher moves that seem to allow and encourage students to be engaged
in important mathematical work.
Student moves generalizations. Culture of rigor, inquiry.
Antecedents – teacher moves that mattered.
Questions about the learning and teaching.
Menu of next step considerations.
Version 4: 3-5-12
This compendium was co-developed within OUSD in partnership with SERP and Lawrence Hall of Science
For more information: serpmedia.org/5x8card/
4
Looking for Standards in the Mathematics Classroom
5x8 Evidence-Gathering Card: Detailed Descriptions
Logic connects sentences: Practices 1, 2, 3, 6.
Student Vital Behavior: 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.
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: Practices 1, 2, 3, 6, 7, 8
Student Vital Behavior: Students talk about each other's thinking (not just their own).
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: Practices 1, 2, 3, 4
Student Vital Behavior: Student work includes revisions, especially revised explanations and
justifications.
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.
Version 4: 3-5-12
This compendium was co-developed within OUSD in partnership with SERP and Lawrence Hall of Science
For more information: serpmedia.org/5x8card/
5
Looking for Standards in the Mathematics Classroom
Academic success depends on academic language: Practices 3, 6
Student Vital Behavior: Students use academic language in their explanations and discussions
(spontaneously and/or prompted by the teacher or other students.)
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.
----------------------------------------------------------------ELLs produce language: Practices 1, 2, 3, 6
Student Vital Behavior: English learners get time, encouragement, and support – from other students
and/or teacher – in using academic language in English or in their home language. Students are
familiar with, and take advantage of language support scaffolds such as sentence frames, multiple
choice oral responses, and reference to diagrams and other representations.
Rationale: English learners can participate in mathematical discussions. They may use imperfect
language or language that is from their everyday lives and experiences. Students will need support in
connecting their ways of talking from every day and move towards more academic ways of talking.
----------------------------------------------------------------Believing (that you can get better at math by learning) motivates
Student Vital Behavior: Do students believe that they can learn to be good at math by learning more
math, by working hard, and persevering to make sense of problems? Or do students think they
cannot change how good at math they are?
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.
----------------------------------------------------------------Equity (The foundation for the above)
Student Vital Behavior: Which students are participating? (e.g. boys more than girls, the same few
students, ELL and special ed students?) Are they volunteering? Called on to do math? Talking about
math in their group? Off task? All students ask math questions.
Rationale: 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 Common Core Standards based performance.
Version 4: 3-5-12
This compendium was co-developed within OUSD in partnership with SERP and Lawrence Hall of Science
For more information: serpmedia.org/5x8card/
6
Looking for Standards in the Mathematics Classroom
How does the 5x8 Evidence-Gathering Card relate to the 4 Claims in
Mathematics Summative Assessment?
There are four major claims found in the SMARTER Balanced Assessment Consortium assessments
of the Common Core State Standards (CCSS) for Mathematics.
Claim #1 - Students can explain and apply mathematical concepts and interpret and carry out
mathematical procedures with precision and fluency. (concepts & procedures)
Claim #2 - Students can solve a range of complex well-posed problems in pure and applied
mathematics, making productive use of knowledge and problem solving strategies. (problem
solving)
Claim #3 - Students can clearly and precisely construct viable arguments to support their
own reasoning and to critique the reasoning of others. (communicating reasoning)
Claim #4 - Students can analyze complex, real-world scenarios and can construct and use
mathematical models to interpret and solve problems. (modeling and data analysis)
The 5x8 Evidence-Gathering Card allows math teachers and math leaders to focus on Claim #1 by
attending to students’ progress to algebra. The specific mathematical concepts and procedures will not
be apparent to principals who do not have strong content and pedagogical background in
mathematics.
Claims #2-4 are somewhat found in the 5x8 Evidence-Gathering Card. Looking for specific student
work (oral or written: including symbols, diagrams, charts, models, etc.). These tasks are moving
toward rich tasks and/or mini-projects.
Version 4: 3-5-12
This compendium was co-developed within OUSD in partnership with SERP and Lawrence Hall of Science
For more information: serpmedia.org/5x8card/
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