sonia dupree, sr. administrator for high school math anna ... · o 17 of 22 classrooms (77%) showed...
TRANSCRIPT
Sonia Dupree, Sr. Administrator for High School Math
Anna Jackson, Coordinating Teacher for High School Math
Wake County
What is implementation? A specialized set of activities designed to put into practice an
activity or program of known dimensions.
How do I know when I have reached full implementation? Full Implementation is reached when 50% or more of the
intended practitioners, staff, or team members are using an effective innovation with fidelity and good outcomes.
What work is to be done once we have full implementation? The Stages are dynamic within organizations such as schools
and clinics, moving back and forth among Stages as personnel and circumstances change. Understanding Stages is important so the work of Implementation Teams can be matched to the Stage of the provider organization.
‘11-12 ‘12-13
Implementation Team Principal
Assistant Principals
Department Chairs
PLT Leaders
Workshop
Attendees/Curriculum Writers
Implementation Resources
Content Specialists
Implementation Team
Teacher Leaders
Training Opportunities
Curriculum Support
Materials
Content Wikis
Researched Practices
You and your math team, those
sitting at your table, would like to
complete a common observation
in an effort to norm your
observation language, practices,
and expectations.
In addition, you would like to use
this as an opportunity to identify
strengths in your team, areas in
which the team may need to
improve, and determine the
collective knowledge of the team.
Meet Keith Walker. Mr. Walker is a Math I teacher at
your school. You and your team will review the
lesson he delivered and his responses to some
specific questions.
Use your observation tool
to record the elements
you found during the
observation.
You may use the second
page of reflection
questions to record
information that will be
helpful as you have a
conversation about
student learning in Mr.
Walker’s classroom
1. Complete the observation tool independently.
2. If you need help understanding an element,
please use the Observation Help Guide.
3. Choose a table facilitator.
4. Seek agreement on each item to determine
what the team did or did not see in the
observation. (Use the help guide when your
group gets stuck and use evidence to
determine whether the element was evident
in the observation.)
5. Record these findings on the observation
poster.
6. Create two questions that you would ask Mr.
Walker to intentionally help him grow as a
teacher.
7. Identify one way in which Mr. Walker could
improve this particular lesson.
8. Post your observation tool, questions, and
improvement on the wall.
Where are we the same?
Where are we different? (What evidence did
you use to make your decision)
What commonalities do we see in the
questions?
What commonalities do we see in the
suggested improvements?
How could you use a process like this with
your department?
WCPSS Secondary Math Walkthrough Checklist
Teacher_______________________________________ Date of Observation___________________
Room #__________ Course__________________________________________ Time_________________
Unit/Lesson/Topic___________________________________________________________________________
Classroom Environment Seating:
Singles
Pairs/Trios
Groups
Seating Orientation:
Students face towards teacher
Students face towards each other
Room Arrangement:
Inhibits student interaction
Allows student interaction
Facilitates student interaction
Classroom displays:
Learning aids, concept-related items
Ongoing activities, projects
Examples of student work
Student recognition
Applications, careers
Racial, cultural diversity
Extracurricular opportunities
Lesson Overview:
Written objectives
Written agenda
Assignments posted
Classroom Culture Major activities of teachers & students:
Lecture/note-taking, teacher-led demonstration
Class discussion, small group discussion, student presentation/board work
Hands-on activities following a set of specific steps
Hands-on activities with open-ended instructions/latitude to decide steps
”Seatwork”: reading text, working on worksheet, questions, problem set
Processing: represent/analyze data, find patterns, write/reflect on learning
Assessment: test/quiz, performance task, questioning to assess learning
Using Discourse
Teacher-- Students
Students--Students
Both
Collaborative culture
No collaborative culture
Some evidence of collaborative culture (e.g. group roles defined)
Evidence of collaborative culture
Collaborative norms clearly defined
Technology
Teacher-driven Student-driven Lesson Enhanced
Technology used: iPad/iPod Document Camera Computer Projector Calculator Interactive Board Other
Standards for Mathematical Practice* 1. Make Sense of Problems and Persevere in Solving Them
Students are reasoning, thinking, and/or proving their answers.
Students are demonstrating a structured approach to problem solving.
Students check the reasonableness of their answers.
Students collaborate to understand the approaches of others.
3. Construct Viable Arguments and Critique the Reasoning of Others
Students are engaged in mathematical discourse.
Type(s) of discourse:
paraphrase agree/disagree
explain add on
verify question each other
Students are making and testing mathematical conjectures.
Students are making arguments to defend their reasoning.
Students understand and evaluate arguments of others.
4. Model with Mathematics
Students are using mathematical models as evidence to support problem solutions.
Models may include:
Drawings
Manipulatives
Symbols
Graphs/Tables
Students are using and/or sharing multiple representations (verbal, graphical, tabular, algebraic).
Comments
* Incorporating the Standards for Mathematical Practice will lead to increased levels of rigor in the classroom. These 3 standards were chosen for focus because of their high leverage.
SAMPLE HIGH SCHOOL MATH WALKTHROUGH VISIT SUMMARY
**Greeting**
We observed 18 of the 22 members of your department This number includes one student teacher. Due to time constraints, we
were unable to make it to the remaining teachers’ rooms and chose not to visit the half-time teacher or the Gradpoint teacher.
PLUSES (GOOD THINGS WE OBSERVED HAPPENING!)
Most of the teachers demonstrated a high level of energy and enthusiasm for the math they were teaching
Teachers appeared to have good rapport with their students as evidenced by students responding and interacting positively to
their teachers
In most classrooms, students are seated in arrangements that support interaction (pairs, trios, groups); however, the students
are mainly facing toward the teacher
Many teachers engaged students by using whiteboards (hand-held or on the wall) or a document camera to let students
display solutions to problems, rather than teachers working the solution for students (this saves time, gets students engaged,
and values their work)
16%
79%
5%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
Groups of 4 or more Pairs/Trios Singles
Seating Arrangement
13%
87%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Facing other students Facing teacher
Seating Orientation
A few teachers used contextual problems that were interesting and engaging for students
We saw evidence of the Standards for Mathematical Practice in several classrooms:
o 17 of 22 classrooms (77%) showed evidence of SMP #1 Make sense of problems and persevere in solving them.
o 15 of22 classrooms (68%) showed evidence of SMP #3 Construct viable arguments and critique the reasoning of others.
o 12 of 22 classrooms (55%) showed evidence of SMP #4 Model with mathematics.
The following types of math discourse were observed:
DELTAS (THINGS TO IMPROVE ON)
The majority of lessons observed were Lecture/Demonstration
Most of the questions being asked are on the “knowledge” or “remembering” level of bloom’s taxonomy.
Most questions were not directed to an individual student. Therefore, understanding by all students was difficult to ascertain.
0
2
4
6
8
10
12
14
16
18
agree/disagree question eachother
verify explain paraphrase
Nu
mb
er
of
Cla
ssro
om
s O
bse
rve
d In
Type of Math Discourse
69%
6%
25%
0%
10%
20%
30%
40%
50%
60%
70%
80%
Lecture/Demonstration Student-led investigation Teacher-led investigation
Type of Lesson
Teachers are doing way too much of the talking (and the work)!
Too many problems used were “naked math” with no context; sometimes this is needed, but context increases engagement,
depth, problem solving skills, comprehension, etc.
We saw lots of worksheets with too many problems on them – lots of repetition and little depth or context that would involve
real life problem solving.
There is little to no use of technology to support instruction.
RECCOMENDATIONS
Questioning Techniques:
Use the Question Analysis tool (or something similar) that was introduced at the Common Core Summer Training and revisited
at the October math department team leaders’ meeting. In this activity, all questions that a teacher asks during a class period
are scripted by someone else. Then the teacher uses Bloom’s revised Taxonomy to analyze the type and level of questions
being asked. Once the “baseline” is established, teachers work together to develop and employ higher level questions as part of
their lesson planning. The question analysis can be repeated to track improvement. We are available to help with this!
Decrease the amount of choral response/”all-call” questions. After posing a question, give an appropriate amount of wait-time
and then call on a particular student to answer. An easy way to call on students randomly is to use popsicle sticks with students
names on them that you draw from a jar.
Math Discourse:
Take advantage of grouping structures to promote student math discourse. Have students do more “Think-Pair-Share” or “Talk
to a Shoulder Buddy” type activities as a first step.
Work on establishing group culture with roles and norms
When planning lessons, incorporate opportunities for math discourse. Use rich task problems that lend themselves to deep
discussion. Use fewer problems, in context whenever possible, that you can explore deeply. Also, as part of your lesson
planning, develop higher-level questions that you plan to ask students. (This way, you won’t have to come up with them on the
spur of the moment.) We have a list of question stems that can be used as a starting point.
Try to incorporate at least one student-led investigation per unit, increase the number of these activities over the course of the
semester.
79%
11% 11%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
Teacher-Students Students-Students Both
Type of Verbal Interactions
Work Smarter, Not Harder
Let students work out and explain problems for the class, instead of the teacher. Some ideas:
o as students arrive to class, assign them a problem from homework to display on the board (or document camera); ask
them to explain it to the class (or ask another student to explain what is written, getting the original student to clarify
as needed – this helps with communicating a solution process); if the student says they don’t know how to do it, tell
them that it’s their responsibility to find someone who can help them and that they are still required to explain the
problem; (you may want to strategically assign difficult problems)
o use the same process outlined above for classwork that needs to be discussed; circulate as students work on problems
and assign problems to students to display and explain
Utilize PLTs to create more investigative type lessons. Let students discover math instead of lecturing them.
**Closure**
Sonia Dupree, Senior Administrator for High School Mathematics
Anna Jackson, Coordinating Teacher for High School Mathematics
WCPSS Secondary Math Observation Tool
Teacher________________________________________ Date of Observation____________
Time_____________ Room #____________ Course_________________________________
Unit/Lesson/Topic_____________________________________________________________
Classroom Environment
Seating:
Singles
Pairs/Trios
Groups
Seating Orientation:
Students face towards teacher
Students face towards each other
Room Arrangement:
Inhibits student interaction
Allows student interaction
Facilitates student interaction
Classroom displays:
Learning aids, concept-related items
Ongoing activities, projects
Examples of student work
Student recognition
Applications, careers
Racial, cultural diversity
Extracurricular opportunities
Lesson Overview:
Written objectives
Written agenda
Assignments posted
Classroom Culture
Major activities of teachers & students
Lecture/note-taking, teacher-led work
Class discussion, small group discussion, student presentation or modeling
Hands-on activity following specific steps
Hands-on activity with open-ended instructions/latitude to decide steps
“Seatwork”: reading text, working on worksheet, questions, problem set
Processing: represent/analyze data, find patterns, write/reflect on learning
Assessment: test/quiz, performance task, questioning to assess learning
Using Discourse
Teacher-- Students
Students--Students
Both
Collaborative culture
No collaborative culture
Some evidence of collaborative culture (e.g. group roles defined)
Evidence of collaborative culture
Collaborative norms clearly defined
Technology Teacher-driven Student-driven Lesson Enhanced
Technology used: iPad/iPod Document Camera Computer Projector Calculator Interactive Board Other
Researched Best Practices
Old skill/information/spiraling
New skill/information
Similarities & Differences
Summarizing /Note-taking
Reinforcing effort
Homework & practice
Nonlinguistic representation
Setting objectives/feedback
Generating/testing hypotheses
Cues, questions, organizers
Standards for Mathematical Practice*
1. Make Sense of Problems and Persevere in Solving Them
Students are reasoning, thinking, and/or proving their answers.
Students are demonstrating a structured approach to problem solving.
Students check the reasonableness of their answers.
Students collaborate to understand the approaches of others.
3. Construct Viable Arguments and Critique the Reasoning of Others
Students are engaged in mathematical discourse.
Type(s) of discourse:
paraphrase agree/disagree
explain add on
verify question each other
Students are making and testing mathematical conjectures.
Students are making arguments to defend their reasoning.
Students understand and evaluate arguments of others.
4. Model with Mathematics
Students are using mathematical models as evidence to support problem solutions.
Models may include:
Drawings
Manipulatives
Symbols
Graphs/Tables
Students are using and/or sharing multiple representations (verbal, graphical, tabular, algebraic).
Comments
* Incorporating the Standards for Mathematical Practice will lead to increased levels of rigor in the classroom. These 3 standards were chosen for focus because of their high leverage.
WCPSS Secondary Mathematics Observation Tool The Look-Fors : What They Mean and Why They Matter
Classroom Environment
What It Is What It Isn’t Why It Matters
Seat
ing
Arrangement of desks
Ability for students to engage with each other in learning tasks
Ability for students to see key information
Single: Student desk does not touch other desks
Pairs/Trios: Student desk touches 1 or 2 other desks
Groups: Student desk touches other desks in groups of 4 or more. This includes o seminar discussion circles o horseshoe and U arrangements o other arrangements with contiguous
desk placement
For Observation purposes, it does not evaluate
teacher-chosen v. student-chosen seating
purposeful seating v. random seating
fixed room structures beyond the teacher’s control, such as door, built-in cabinets, lighting, or support columns.
Marzano, High-Yield Strategy 6: Cooperative Learning. Research shows that socially-constructed learning has an effect size of .73. (note: an effect size of .50 is considered medium)
Marzano, High-Yield Strategy 3: Reinforcing Effort—Effect size .80. Peer evaluations and Self evaluations of effort require that students be aware of their own and their peers’ relative effort. Seating arrangements have a direct effect on this.
Marzano, High-Yield Strategy 7: Providing Feedback –Effect size .61. Formative peer and teacher feedback can be given only when peers and teachers are able to see and evaluate student attempts at mastery.
Lev Vygotsky’s Zone of Proximal Development rests on the awareness that there is a difference between what students can demonstrate independently and what they can demonstrate with the help of others. Thus, seating that maximizes peer support is crucial.
Room arrangements themselves can be the trigger for the teacher to design a variety of student-centered learning tasks (McCorskey and
McVetta, 1978)
Room arrangements influence teacher and student beliefs about the role of the teacher in the learning environment.
Flexible room arrangement allow teachers to adjust for a variety of learning tasks (Professional Learning Board synopsis)
Seat
ing
Ori
enta
tio
n Placement of desks and which way
students face
Orientation towards one focal point, such as the front screen or teacher lectern OR
Orientation that allows multiple focal points, depending on the learning activity
Ro
om
Arr
ange
men
t
Includes teacher choices of: o Placement of student desks o resource area(s) o teacher location(s) o materials and technology
Includes effect on movement and purposeful student interaction.
Inhibits means the room arrangement is fixed in a way that it deters students from interacting in meaningful ways for learning tasks.
Allows means the room arrangement is flexible enough for students to interact if needed.
Facilitates means the room arrangement is fluid and inviting for meaningful student interaction during learning tasks.
Cla
ssro
om
Dis
pla
ys
Easily seen displays especially designed to add energy and awareness to the learning environment.
Learning Aids: maps, word walls, concept posters, graphic organizers, skill reminders, and other easily seen supports for student learning.
Ongoing Activities and Projects: large and easily seen assignment specifications, models, anchors, exemplars, skill supports, milestones, benchmark displays
Examples of Student Work: examplars of admirable student work such as essays, tangible products, models, printouts of digital presentations.
Student Recognition: Data charts, Wow! Boards, extra-curricular awards and recognitions, for example
Applications, Careers: Easily seen representations of discipline-specific applications of content-area learning in the 21st-century world of work
Racial, Cultural Diversity: Representations of many types of students, their heritages and backgrounds, and their interests and values found in classroom displays. This includes holiday displays.
Extra-curricular opportunities: Notices of clubs, organizations, and other opportunities that support content-area learning, development of social and cultural capital, and leadership development, such as Debate Club, Robotics Club, Student Council, to name a few.
Static, never changing from Day 1
Trite, stale posters with little connection to particular needs of the students
Out-of-date information and representations, such as maps with last decade’s borders, or outmoded formats for resumès, or representations of careers that no longer exist.
Hard-to-find, hard-to-see displays, unless clearly grouped and labeled as a classroom exhibit area
Student models and exemplars that clearly are from students of bygone years; torn, musty, faded examples of student work.
In order to make a difference in student learning and disposition, displays must have a meaningful connection to the curriculum.
(Marlynn Clayton, Classroom Spaces that Work, 2002)
Purposeful displays that include all students in some way send important messages to students: o The teacher values what students do o This is the students’ classroom as much as it is
the teacher’s classroom o In this classroom, students share their
learning with each other and get feedback from each other.
(Mike Anderson, Classroom Displays, ASCD , 2011)
Many researchers are looking at the factors of student engagement and dividing them into three areas: cognitive domain, emotional domain, and behavioral domain. Purposeful classroom displays affect student learning directly by activating the cognitive and emotional domains.
(Richard Jones, Strengthening Student Engagement, 2008)
Marzano, High Yield Strategy 3: Reinforcing Effort and Providing Recognition—Effect size .80. Displays of student work are central to providing recognition and demonstrating to students that teachers value their efforts.
Marzano, High Yield Strategy 5: Non-linguistic Representations – Effect size .75. Maps, charts, graphic organizers, photographs, pictures, concept maps, and other non-linguistic representations are all types of displays that can affect the cognitive domain of student engagement and learning.
Marzano, High-Yield Strategy 9: Questions, Cues and Advance Organizers—Effect size .59. Teachers who use purposeful displays to pique interest, pose real-world problems, activate prior experience/knowledge are using Marzano’s 7th high-yield strategy.
Marzano, High-Yield Strategy 1: Identifying similarities and differences—Effect size 1.61. This Super Yield Strategy is easy to achieve with charts showing similarities and differences among unit and course concepts.
Less
on
Ove
rvie
w
Written objectives: Learning objectives for the day written in colloquial language that students can understand. These may be written as learning targets. These may be what students will demonstrate by the end of the class period.
Written agenda: The order of events and activities for the class period. Even better: include the expected time each event/activity will take. Even better: format the class agenda as a business meeting agenda would be, including the Who and the Why.
Assignments posted: Even in this day of teacher websites with postings and updates for assignments, having the assignments posted on the board is important to student learning. Include purpose, product, and due date. Even better: have the assignment handout and rubric of skills posted nearby adjacent to the assignments.
Written objectives that are copied straight from the CCSS or teacher’s manual; that use educational jargon, abbreviations, and acronyms; that are only for a full unit or course; that are “canned” or “stock” objectives. Written agenda that is a generic/ happens-every-day listing. Assignments posted that are hidden or unreadable from student desks
Marzano, High-Yield Strategy 7: Setting Objectives and Providing Feedback—Effect size .61. Objectives and Agendas set purpose for students for the class session.
Marzano, High-Yield Strategy 9: Questions, Cues and Advance Organizers—Effect size .59. Objectives and agendas are advance organizers. Objectives, agendas, and assignments posted are all cues for students.
Marzano, High-Yield Strategy 4: Homework and Practice—Effect size .77. Assignments posted reminds students of the homework and practice opportunities they have.
Agendas help student make smooth transitions between parts of the lesson. In addition, it adds to a stable and orderly classroom routine. (Melissa Kelly, Steps to Starting a Class Off Right. )
Agendas that conform to a business-meeting format send the nonverbal message that learning is the students’ jobs, that the classroom time is purposeful, and that all students have a role in the culture of the classroom.
Not only do Agendas help students know what to expect, but they also are a visual cue to teachers as they manage the pacing of instruction and learning.
Classroom Culture
What It Is What It Isn’t Why It Matters
Maj
or
Act
ivit
ies
of
Teac
her
s an
d S
tud
ents
Lecture/note-taking, teacher-led demonstration Classroom is teacher-
centered at this point. Students are doing one or all of the following: listening to teacher-talk; copying or summarizing notes; watching the teacher show something.
Lecture/note-taking, teacher-led demonstration NOT Student-generated meaning-making.
Mini-lectures are valuable for giving direct information that student may need for background information or as the basis for a problem-solving task. Because research shows that students must make their own meaning in order to learn, lecture on its own has little impact on student learning of concepts or skills.
Class discussion, small group discussion, student presentation/board work Students are at the lead during this activity. Students are talking with each other about a course concept.
Class discussion, small group discussion, student presentation/board work NOT Teacher asking questions and calling on students, or a one-to-one conversation between teacher and one interested student.
Socially-constructed learning yields high retention rates because students are getting immediate feedback on their ideas from people who matter to them—their peers. Highly interactive discussions or student presentations place students in the role of peer-teachers, which places higher value for the student on the need to understand and communicate concepts to peers. Research for peer teaching routines in multiple studies throughout the 1990’s and 2000’s indicate a high return on critical thinking skills and reasoning.
Hands-on activities following a set of specific steps Students are working together or independently to replicate a procedure or process previously presented.
Hands-on activities following a set of specific steps NOT Students watching one student do the whole process.
Adding to research supporting Constructivist theories of learning, 2013 research from Stanford University reports that students learn concept knowledge best if they explore hands-on projects prior to reading or watching videos on the topic.
Research stretching back 20 years or more shows
Hands-on activities with open-ended instructions/latitude to decide steps Students are following a “design cycle” of thinking steps to generate original procedures, processes, or products.
Hands-on activities with open-ended instructions/ latitude to decide steps: NOT prescribed steps leading to a well-structured or expected outcome.
that problem-solving skills used for solving predictable outcomes are important. But on their own, these are not sufficient for solving problems in open-ended, multimedia problem-solving environments.
Using design cycles, such as the STEM Design Cycle or the IB/MYP Design Cycle, formalizes the steps that effective critical and creative thinkers use to “play with” open-ended problems with multiple possible solutions—such as those in life or on assessments that judge thinking more than discrete facts.
”Seatwork”: reading text, working on worksheet, questions, problem set
Independent or paired learning, in which the student is figuring out his own responses, with or without minor help.
”Seatwork”: reading text, working on worksheet, questions, problem set NOT group discussion or problem-solving; NOT guided practice or structured scaffolds.
Independent work is another piece of the learning-strategy puzzle. It is not to be used to the exclusion of socially-constructed learning, but it is valuable in its own right. Without independent work, students will not be able to assess the levels their own skills and knowledge independent of supportive prompts. Independent work, monitored by the teacher-coach, trains students to play an active role in their own learning through self-awareness of their learning needs. Self-direction, strategic planning, and heightened self-motivation are the desired learning habits that emerge as outcomes.
Processing: represent/analyze data, find patterns, write/reflect on learning Individual, paired, or group thinking in which students use higher-order thinking skills to manipulate concepts and make new meaning for themselves. Student are engaged in higher-order thinking when they visualize a problem by
making a graphic representation of it;
separate relevant from irrelevant information in a problem; seek reasons and causes; justify solutions; see more than one side or solution or process of a problem; weigh sources of information based on their credibility; reveal and explain assumptions in reasoning; identify bias or logical inconsistencies (source)
Processing: represent/ analyze data, find patterns, write/reflect on learning: NOT lower-order thinking, such as memorization, explanation to understand, or rote application to well-structured problems. NOT copying notes or information
In “Education and Learning to Think” (National Research Council, 1987), an early important consideration of higher order thinking in math and science, higher order thinking is categorized as responding to “non-algorithmic” (open-ended) problem solving, involving complex and multilayered ideas, requiring nuanced judgments, and applying multiple criteria to new and unpredictable situations. Why is this important? There is well accepted research from multiple sources that show us using higher order thinking in learning tasks has these payoffs for students:
deeply embedded and long-remembered content knowledge
strong transference of knowledge and skills to new contexts (such as ACT, EOC)
direct and effective application to new situations
Assessment: test/quiz, performance task, questioning to assess learning
May be observed as independent or group structure. May be formal test or informal ticket-out-the-door. May be summative or formative. Key: the purpose of the activity is to assess student understanding.
This is NOT:
Simply grading
An activity to provide or use new information
Input
Assessment is not an end in itself, but rather a means for educational growth. Assessment works best when it is integrated into the learning-and-teaching cycle, an ongoing part of learning rather than an “event”. Assessment is most effective as an agent for growth when it is designed to assess the multidimensional integration of knowledge and skills developed over time.
Usi
ng
Dis
cou
rse
Discourse is lengthy and extended discussion about concepts, problems and skills related to course standards. Discourse occurs when learners are faced with open-ended problems, with multiple possible solutions, with ideas that even experts in the field might disagree on, with arguments/counterarguments, and with other types of non-structured response.
Discourse is NOT Q and A
Quick answers to pointed questions
Definite right-or-wrong answers
Teacher and a single student discussing
Unrelated to course standards
Strategies that embed discourse in the learning process prompt students to “produce” as they learn. This “output” gives students direct information about their learning: they articulate understanding, discover holes in their knowledge, justify ideas, and construct meaning from texts. As such, discourse leads to the development of reading comprehension, critical thinking skills, argument, and even personal voice. “…[S]tudents can and will internalize thinking processes experienced repeatedly during discussions.” (source)
Co
llab
ora
tive
Cu
ltu
re
Evidences of a well-established collaborative culture: Students transition quickly and smoothly
when prompted to begin collaboration.
Student movement and talk is purposeful and focused on the group learning task.
Students do not need the teacher to explain/re-explain the assignment, but instead use each other for questions and answers.
Students are clearly adept at using a rubric as a guide for the finished product and for self- and peer-assessment of mastery.
Students are eager to begin the task.
Students are trained to take on social roles, either formally or informally.
Teacher’s role is Coach and Facilitator, not Tutor or Demonstrator.
Teacher moves consistently from group to group, listening for misconceptions and monitoring for understanding.
Teacher does not give students answers, but offers suggestions for improvement, asks leading questions, or provides additional resources.
Students want the teacher to stop talking so that they can begin working.
Not a collaborative culture: Students balk at self-
directed learning tasks.
Students require many teacher prompts before they transition to collaborative structure
Student movement and talk is erratic and random. Focus is not consistently on task.
Students need the teacher to explain/re-explain the assignment.
Students ignore the rubric or don’t understand how to use it. Students do not self-assess as they work.
Students do not work smoothly as a team, often overlapping their jobs and roles.
Teacher’s role ends up being Tutor for a few students.
Teacher does not monitor or coach students.
Of Marzano’s 9 high-yield strategies, Cooperative Learning yields a high effect size of .73. Students who experience this learning strategy have a 27% percentile gain simply because of this strategy.
A collaborative culture, notes Marzano, trains students for “the real world” through these 5 features:
“Positive interdependence” – learning is a “we’re all in this together” experience
“Face to face promotive interaction”—we all are part of each other’s learning community, providing feedback, struggling together in supportive ways, and celebrating the Aha’s
“Individual and group accountability”—each of us is valuable and needed to the success of the whole
“Interpersonal and small group skills” – which include building trust, effective communication, resolving conflicts, taking leadership
“Group processing”—each of us is part of the solution for effective thinking and production
Marzano recommends using a variety of criteria in grouping students, using “base” or “home” groups for long-term learning; formal groupings for specific purposes and tasks; and informal groupings for quick-thinks or check-ins.
Some educators recommend using the terminology of “learning teams” rather than “group work” as a way to influence student understanding about the purpose of working cooperatively with peers.
Digital Technology
What It Is What It Isn’t Why It Matters
Teacher-Driven
Teacher has control of the digital technology. Students watch the teacher demonstrate or use the technology.
Interactive for students.
As mentioned as part of many other effective teaching criteria, it matters whose mind is making the meaning out of the information. If teachers are controlling the digital technology, it is just like teachers holding the pencil while students try to answer a question or solve a problem. Students need to have ownership of the tools that promote the learning they need to make. At the same time, though, students need to know that teachers are monitoring usage and success, and that they are right there as a coach or trainer if the student starts down the wrong path of thinking.
Student-Driven
Students have control of the digital technology. Teachers serve as coaches and facilitators.
Teacher sitting at her/his desk grading papers while students work.
Lesson Enhanced
The lesson creates the purpose for the digital technology used. The technology does not create the need for a lesson. Digital technology is used as a natural part of the way lessons are delivered in the classroom.
A special event.
Students need to develop the skills to become skillful and creative users of the tools being used in the current culture—both in school and beyond. The many forms .
Researched Best Practices
What It Is What It Isn’t Why It Matters O
ld S
kill/
Info
rmat
ion
/Sp
iral
ing
Arranging the learning and teaching cycle to enjoy a students’ “comfort zone” (see Vygotsky) before introducing new material
Arranging the learning and teaching cycle to tie students’ “comfort zone” of learned skills and concepts into a students’ “discomfort zone” where learning becomes risky to them
Helping students become “comfortable with discomfort” in learning by providing bridges from what they know to what they need to know.
Learn it and leave it behind.
Check it off the list of course objectives.
Learning that never arises again during the course to be used as the “stairstep” to the next level of knowledge or skill.
Prior Knowledge theories rest on the concepts that learning happens only when there is conceptual change on the part of the learner.
IT MATTERS… Students learn by refining previously learned skills and concepts. This means that new student learning must begin at the current level of learned skills and concepts before being able to “refine” them in any way.
Lev Vygotsky’s Zone of Proximal Development refers to the difference between what a learner can do without help and what s/he can do with support.
IT MATTERS… Learners who work within the Zone of Proximal Development learn new skills and concepts without anger, frustration, or apathy. Learners who work outside the ZPD experience high frustration and shut down, masking their fear and confusion with anger and apathy.
Jerome Bruner’s concept of Spiral Curriculum rests on the belief that students can learn complex concepts at any age as long as the information is structured effectively.
IT MATTERS…Teachers who structure learning so that complex ideas are taught at the student’s level of understanding first, and then revisited at gradually increasing levels of difficulty lead students to learn how to learn, so to speak.
Piaget’s theories of “assimilation” and “accommodation” rest on the concept that people learn either by fitting a new outside idea into their existing internal conceptual categories OR by re-categorizing their internal conceptual world to accept a brand-new concept or skill.
THIS MATTERS… Either way, students can learn ONLY if the new material and the old material come in contact with each other and relate to each other.
Ne
w S
kill/
Info
rmat
ion
The “new stuff” that students must learn—and often feel a discomfort in trying because it is so far out of their comfort zone.
Skills and concepts that are within “shooting range” of their previous learning.
Skills and new concepts that are the logical “next step” of knowledge beyond the students’ current understanding and capability.
New knowledge out of context with prior learning.
New knowledge that require too much of a cognitive leap at one time for students
New knowledge or skills that are so far beyond students’ current learning that they shut down or give up because of an internal fear of failing or unsolvable level of confusion
Sim
ilari
ties
an
d D
iffe
ren
ces
Student generated
Higher order thinking: requires analysis of features, definition of what something is and is not, application to an “other”, evaluation, and synthesis skills
Enhances student ability to use facts/knowledge and transfer concepts
May be done in linguistic (words) or non-linguistic (charts/graphs) formats
Includes but not limited to: Comparing, including Venn diagrams
Classifying
Creating metaphors (abstract connections between unlike things)
Creating analogies that demonstrate different relationships o A:B::C:D (no linquistic
representation of an analogy) o EX: ____is to humans as carbon
monoxide is to plants o EX: Shakespeare: sonnet::____:
ode
Teacher generated
Pre-made materials
If you make students do this in a variety of ways, students will learn better than if you use any other strategy. Marzano’s High-Yield Strategy 1. This is the Super Strategy with a 1.61 effect size (very, very, very high yield on student learning). Students who experience this learning strategy have a 45% percentile gain simply because of this strategy.
Sum
mar
izin
g /
No
te-T
akin
g
Student created, not teacher created
Higher-order thinking involving categorizing (analyzing), prediction of future use, and evaluation of importance
Systematic and structured training of students to recognize and isolate important information to know, understand, and be able to do
Training in a variety of formats for notes and annotations
Ongoing revisiting of “tricks of the trade” to help students refine their summarizing and note-taking skills Summary frames for different kinds
of text
Paragraph Shrinks
Annotating (vs. highlighting)
Interactive note-taking strategies
Multi-modal note-taking strategies
Synthesis of information reviews or study guides
The more notes taken by the student, the better, says Marzano.
Verbatim note-taking
Copying from the board
Teacher-created
outlines or lists of information(unless used as a model)
A one-time “done deal”
Marzano’s High-Yield Strategy 2. This strategy has a very high yield on student learning, with a 1.00 effect size. Students who experience this learning strategy have a 34% percentile gain simply because of this strategy.
Re
info
rcin
g Ef
fort
/ P
rovi
din
g R
eco
gnit
ion
Systematic and structured training of students to recognize the role of their effort in their success
Includes but not limited to: Student self-assessment of his/her
effort, using a descriptive rubric
Student-created charts that dually track effort on tasks and resulting success on assessments
Regular discussion about student efforts on a task and student-shared ideas for anticipating/overcoming obstacles
Symbolic rewards and praise that are dependent on a standard of performance
Teacher practice of Pause-Prompt-Praise when students are facing difficulty: Pause and think what’s hard, Prompt with a specific suggestion, then come back and Praise for the specific improvement
“Bless your heart” responses
Accusatory responses (Why didn’t you do better????)
Assumption that all students believe that there’s a relationship between effort and achievement
Vague praise (“Good job!”)
Marzano’s High-Yield Strategy 3. This strategy has a high yield on student learning, with a .80 effect size. Students who experience this learning strategy have a 29% percentile gain simply because of this strategy.
Ho
me
wo
rk a
nd
Pra
ctic
e
Beyond-the-class learning
Purposeful—and that purpose is explained out loud
Extension of knowledge
Cycles of practice needed to refine proficiency and mastery of a skill
Used to springboard new learning
Student attempts at mastery that require comment/feedback
Feedback may be given in a variety of ways, from teacher and from peers
Feedback on homework is not the same as grading homework
Designed for an “independent” level of learning, not at a “frustration” level
Parent involvement in student homework
Irrelevant (or seemingly so) to what is being done during class
Unclear—easy to confuse when done independently
High-stakes
A one-shot experience with little connection to continued cycles of learning
Marzano’s High-Yield Strategy 4. This strategy has a high yield on student learning, with a .77 effect size. Students who experience this learning strategy have a 28% percentile gain simply because of this strategy. NOTE: Homework given at the high-school level results in higher percentile gains than homework given in elementary school or middle school.
No
nlin
guis
tic
Re
pre
sen
tati
on
s
Student-created
“Coding” of information in formats other than words, phrases, sentences, paragraphs
Transfer of information given in paragraph form into the same information arranged in other forms
Includes but is not limited to:
Graphic organizers
Charts, Tables, Graphs
Maps
Images, Pictures
Physical models
Movements
Mental pictures
Equations (turning situations and paragraphs into equations)
Teacher-created
Pre-made materials
Templates to fill in
Closed-ended products
Lower-order thinking tasks
Marzano’s High-Yield Strategy 5. This strategy has a high yield on student learning, with a .75 effect size. Students who experience this learning strategy have a 27%percentile gain simply because of this strategy.
Sett
ing
Ob
ject
ives
an
d P
rovi
din
g Fe
edb
ack
Student-generated goals for learning and habits of mind, adapted from and aligned with the standard course goals--in student’s own words. May be short-term (today, this unit) or longer-term.
Descriptive responses to student attempts at demonstrating learning that help students understand their current level and how to move forward.
Clear exemplars or descriptions of mastery , such as on a descriptive rubric
Teacher to student
Student to student
Student to self—how can I improve?
Ongoing and recursive --like the wheels of a bicycle that go round and round in order to propel you further and further
Bicycle wheels: Learning goals
Formative assessment Refined
goals next Formative
assessmentand further and further we learn
Student must do the “pedaling” -- setting his/her own goals, assessing his/her own growth, and being allowed to make some strategic choices along the way
Teachers “pave the road” and provide “the map”: rubrics, modeling, coaching when needed, etc.
Peers are interdependent on each other’s “rides”—helping to planning the journey, riding their own bikes, encouraging endurance on the uphill of learning, providing feedback (Try this gear!), troubleshooting the “flat tires” of learning, and celebrating the accomplishments
CCSS objectives
Standard Course of Study objectives
NC Essential Standards for a course
Teacher goals for students
Formative assessment conducted only by teacher to student
Marzano’s High-Yield Strategy 7. This strategy has a moderately high yield on student learning, with a .61 effect size. Students who experience this learning strategy have a 23% percentile gain simply because of this strategy.
Gen
erat
ing
and
Tes
tin
g H
ypo
thes
es
A student-generated prediction, hypothesis, guess, or estimate made prior to practicum and finding evidence
Requires students to explain their hypothesis and their conclusion using the evidence they found
Includes: System or Process Analysis:
Hypothetical What if’s for changes or glitches (circulatory or transportation or whatever)
Problem solving of “ill-structured” or messy situations
Historical investigation , plausible scenarios
Invention, new forms of [whatever]
Decision making, including moral/ethical dilemmas, economic/societal dilemmas
The teacher presenting his/her hypothesis and conclusions to the class, even as an explanation of how the course concept transfers to the real world
Marzano’s High-Yield Strategy 8. This strategy has a moderately high yield on student learning, with a .61 effect size. Students who experience this learning strategy have a 23% percentile gain simply because of this strategy.
Cu
es,
Qu
est
ion
s, A
dva
nce
Org
aniz
ers
Information that helps students remember ideas they already know or have heard of
Open-ended questions that prompt students to fill in missing information or critique points of view.
“Before-learning” materials that prompt student to scaffold ideas or concepts
Closed-ended prompts or materials, easily predictable or mundane and obvious. These do little to spark interest or wake up students’ minds.
Marzano’s High-Yield Strategy 9. This strategy has a high yield on student learning, with a .59 effect size. Students who experience this learning strategy have a 22% percentile gain simply because of this strategy.
Standards for Mathematical Practice
What It Is What It Isn’t Why It Matters #1
Mak
e se
nse
of
pro
ble
m a
nd
per
seve
re in
so
lvin
g th
em.
Students are reasoning, thinking, and/or proving their answers. They:
search for patterns and trends.
analyze information (givens, constraints, relationships, goals).
can explain and justify their solution with mathematical evidence.
Students are deomonstrating a structured approach to problem solving. They:
understand and can explain the meaning of the problem.
look for entry points to solving the problem.
plan a solution pathway.
monitor and evaluate their progress and change course as necessary.
Students check the reasonableness of their answers. They:
predict or estimate the solution before solving.
ask themselves if their final answer makes sense.
check that they have answered the original question(s) completely.
Students collaborate to understand the approaches of others. They:
compare solution approaches, analyzing usefulness in the context of the problem, efficiency and/or elegance.
can explain the approach taken by another student and use it in a similar problem.
add what they learn about the approaches of others to their toolkit of methods.
Teachers telling students step-by-step how to solve a problem and having them replicate.
Solving contrived “real-life” problems.
Requiring that students always use a certain method to solve a particular type of problem.
Requiring students to use the most efficient method.
Worksheets filled with “naked math” problems.
Adding It Up: Helping Children Learn Mathematics (2001) Although in school, students are often presented with clearly specified problems to solve, outside of school they encounter situations in which part of the difficulty is to figure out exactly what the problem is. Then they need to formulate the problem so that they can use mathematics to solve it. Consequently, they are likely to need experience and practice in problem formulating as well as in problem solving. They should know a variety of solution strategies as well as which strategies might be useful for solving a specific problem. With a formulated problem in hand, the student’s first step in solving it is to represent it mathematically in some fashion, whether numerically, symbolically, verbally, or graphically. Representing a problem situation requires, first, that the student build a mental image of its essential components. To represent a problem accurately, students must first understand the situation, including its key features. They then need to generate a mathematical representation of the problem that captures the core mathematical elements and ignores the irrelevant features. This step may be facilitated by making a drawing, writing an equation, or creating some other tangible representation. In building a problem model, students need to be alert to the quantities in the problem. Not only do students need to be able to build representations of individual situations, but they also need to see that some representations share common mathematical structures. Novice problem solvers are inclined to notice similarities in surface features of problems, such as the characters or scenarios described in the problem. More expert problem solvers focus more on the structural relationships within problems, relationships that provide the clues for how problems might be solved. In becoming proficient problem solvers, students learn how to form mental representations of problems, detect mathematical relationships, and devise novel solution methods when needed. A fundamental characteristic needed throughout the problem-solving process is flexibility. Flexibility develops through the broadening of knowledge required for solving nonroutine problems rather than just routine problems. Routine problems are problems that the learner knows how to solve based on past experience. In contrast, nonroutine problems are problems for which the learner does not immediately know a usable solution method. Nonroutine problems require productive thinking because the learner needs to invent a way to understand and solve the problem. Part of developing strategic competence involves learning to replace by more concise and efficient procedures those cumbersome procedures that might at first have been helpful in understanding the operation.
#3 C
on
stru
ct v
iab
le a
rgu
men
ts a
nd
cri
tiq
ue
the
reas
on
ing
of
oth
ers.
Students are engaged in mathematical discourse. They:
paraphrase mathematical ideas of others in their own words.
explain their reasoning.
verify
agree/disagree
add on
question each other Students are making and testing mathematical conjectures. They:
write conjectures using precise mathematical language.
take a systematic approach to testing conjectures.
use examples and/or counterexamples to test conjectures.
use simulations to test conjectures.
anylze situations by breaking them into different cases.
Students are making arguments to defend their reasoning. They:
understand and use defintions, stated assumptions, postulates, theorems, and previous results.
reason both deductively and inductively.
can build a logical progression of statements to support their reasoning.
communicate and defend mathematical reasoning using objects, drawings, diagrams, and/or actions.
determine the domains to which an argument applies.
take into account the context of the problem. Students understand and evaluate arguments of others. They:
listen to or read the arguments of others.
determine if the argument makes sense.
ask useful questions to clarify or improve the argument.
distinguish correct logic or reasoning from that which is flawed.
can explain the error of flawed reasoning.
Discourse occurring only between teacher and student.
Low-level, one or two word response questions.
Simply correcting a student’s mistake without exploring the thought process behind the mistake.
Requiring a certain format for formal proof.
Adding It Up: Helping Children Learn Mathematics (2001) An important part of classroom instruction is to manage the discourse around the mathematical tasks in which teachers and students engage. Teachers must make judgments about when to tell, when to question, and when to correct. They must decide when to guide with prompting and when to let students grapple with a mathematical issue. Their decisions do not simply rest with the mathematical task at issue. They also need to decide who should get the floor in whole-group discussions and how turns should be allocated. Teachers have responsibility for moving the mathematics along while affording students opportunities to offer solutions, make claims, answer questions, and provide explanations to their colleagues. The point of classroom discourse is to develop students’ understanding of key ideas. But it also provides opportunities to emphasize and model mathematical reasoning and problem solving and to enhance students’ disposition toward mathematics. Therefore, discourse needs to be planned with these goals in mind, not merely as a “checking for understanding” form of recitation. Discourse also shapes both the task and students’ opportunities to learn from it. Managing the discourse is one of the most complex tasks of teaching. Adding It Up: Helping Children Learn Mathematics (2001)
Creating classrooms that function as communities of learners has been the focus of much recent research and scholarship in mathematics education. In the research on teaching and learning mathematics with understanding, four features of the social culture of the classroom have been identified. The first is that ideas and methods are valued. Ideas expressed by any student warrant respect and response and have the potential to contribute to everyone’s learning. A second feature of a classroom community of learners is that students have autonomy in choosing and sharing their methods of solving problems. Students recognize that many strategies are likely to exist for solving a problem, they respect the methods used by others and that others need to understand their own methods, and they are given the freedom to explore alternatives and to share their thinking with the rest of the class. A third feature is an appreciation of the value of mistakes as sites of learning for everyone. Mistakes are not covered up; rather, they are used as opportunities to examine reasoning and to deepen everyone’s analysis. The appreciation of mistakes is a fundamental aspect of mathematical work outside the classroom; inside, it helps build the community. Finally, a core feature of these classrooms is the recognition that the authority for whether something is both correct and sensible lies in the logic and structure of the subject rather than the status of the teacher or the popularity of the person making the argument. The resolution of disagreements resides in mathematical argument. Adding It Up: Helping Children Learn Mathematics (2001) Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations. Such reasoning is
#4 M
od
el w
th m
ath
emat
ics.
Students are using mathematical models as evidence to support problem solutions. Models may include:
Drawings
Manipulatives
Symbols
Graphs/Tables Students are using and/or sharing multiple representations:
Verbal
Graphical
Tabular
Algebraic Students:
apply mathematics to solve authentic problems that arise in everyday life, society, and the workplace.
distinguish between important and extraneous information for a given context.
simplify a complicated situation with a mathematical model.
make appropriate assumptions and approximations when needed.
revise the model as needed.
use the model to describe relationships, draw conlcusions, and/or make predictions.
interpret aspects of the model in context.
determine the degree of usefulness of a model.
Using manipulatives as a novel way of performing algorithms without understanding.
Solving contrived “real-life” problems.
Using formulaic or fill-in-the-blank interpretations.
Adding It Up: Helping Children Learn Mathematics (2001)
Manipulatives should always be seen as a means and not an end in themselves. They require careful use over sufficient time to allow students to build meaning and make connections. Recent research has explored how students interact with manipulatives. Students may not look at these objects the same way adults do, and it can be a challenge for students to see mathematical ideas in them. When students use a manipulative, they need to be helped to see its relevant aspects and to link those aspects to appropriate symbolism and mathematical concepts and operations. Observational studies have documented cases in which students were taught to use manipulatives in a prescribed way to perform “wooden algorithms.” If students do not see the connections among object, symbol, language, and idea, using a manipulative becomes just one more thing to learn rather than a process leading to a larger mathematical learning goal. When used well, manipulatives can enhance student understanding. They can, for example, enable teachers and students to have a conversation that is grounded in a common referential medium, and they can provide material on which students can act productively provided they reflect on their actions in relation to the mathematics being taught. Manipulatives also help students correct their own errors.