Cognitive Demand
Match the following definitions to their level of cognitive demand
“Are focused on producing correct answers rather than developing mathematical understanding”
“involve either reproducing previously learned facts, rules, formulae, or definitions, OR committing facts, rules, formulae or definitions to memory”
Procedures without connections (1)
Memorization (2)
Describe what you feel makes a student “high status” and what you feel makes a student “low status”.
High status: over-participator (always volunteers), immediately gathers resources, dominates group discussions, etc...
Low status: Under-participator, quiet, avoids eye contact, rarely raises hand, prefers not to speak in group, etc...
What are three aspects of student responses that are important to anticipate before teaching a lesson?
1.The errors they might make
2. The different representations they might create
3.Their responses to your questions
Determine if the following are advancing or assessing questions?
1. Can you tell me what you did?
2. Where do you see your equation in the diagram?
3. Is this the only solution?
1. Assessing
2. Advancing
3. Advancing
What is one thing that you have observed your mentor teacher do that you want to incorporate into your future classroom?
Erika: One thing that my mentor teacher does well that I want to do is create a positive relationship with all of the students. Most of her students really like her because she takes time getting to know them personally and she truly wants all of her students to succeed.
Ryan: My mentor teacher displayed great use and implementation of classroom norms. For example, each and everyday, the students walk into the room and know to begin their warm-up and take out their homework. She explained to me that these norms are developed by holding the students accountable from day one and not letting disruptive/ destructive behavior go unaddressed.
Match the following definitions to their level of cognitive demand
“Require some degree of cognitive effort. Although general procedures may be followed, they cannot be followed mindlessly”
“Require students to explore and understand the nature of mathematical concepts, processes, or relationships”
Procedures with connections
Doing Mathematics
Name two equitable practices that you would like to implement in your classroom.
1: Group roles (facilitator, resource manager, includer, reporter, etc..)
2: Assigning competence - drawing public attention to a particular student’s mathematical thinking/ idea and ensuring that this student receives credit for that idea.
Name two benefits of using a monitoring chart in your classroom?
1. Organize student responses in anticipation of sequencing during whole group discussion
2. To keep track of the progress of each group during the task (pacing)
In your own words, what is the difference between assessing and advancing questions?
Assessing: Used to clarify what the student has done and what the student understands about what he or she has done.
Advancing: Move students beyond their current thinking by pressing students to extend what they know to a new situation.
What is one thing that you have observed your mentor teacher do that you maybe do not want to incorporate into your future classroom?
Erika: I have seen my mentor teacher give timed math facts quizzes during class that count for a grade and do not relate to the current material of the class. I do not think that this promotes a positive math identity within the students or has a productive purpose for the class in general.
Ryan: One strategy that my mentor teacher used to promote studying for exams was to display a gold star with the name of each student who gets an A+ on an exam. Although I can see that this may serve as positive reinforcement to those students who excel at test taking, it can certainly be destructive to a student’s confidence who may not do well when taking exams.
Categorize the following task:
Graph the following equation. y = 2x + 3
Procedures without connections
How can you reframe this sentence to make it more equitable?
“My low students did not understand today’s lesson at all.”
"There are some students in my class that would benefit from more time on this topic."
Why can it be beneficial to select student work to share rather than to ask for volunteers?
When asking for volunteers, you will often get the same small group of students to participate. When selecting student work, you have the opportunity to raise the status of certain students, and involve more members of the class in whole group discussion.
“Possibly the greatest skill within the art of questioning is knowing how to build in a ______ at the appropriate time. This means both students and teacher—everyone needs to learn the value of a _____” (Johnson, 1994, pg. 25-26).
Pause, Pause
Explain one time you have observed your mentor teacher practice good classroom management.
Erika: One thing that I have realized that my mentor teacher does is she strategically places students at certain tables based on their behavior and participation patterns. She tries to mix up the groups frequently so that they are interacting with more of their peers and so that they can all focus and succeed during class.
Ryan: My mentor teacher had been lecturing for a good amount of time during a block period (90 min) and she could see that the students were losing interest and getting antsy. She had the students stand up and walk around the room, forming groups with other students based on specific criteria (eye color, number of siblings, etc.). The students had fun with the mini activity and were able to focus up for the remainder of the lesson.
Categorize the following task:
Find 1/6 of 1/2 using pattern blocks.
Procedures with Connections
When working with small groups, how can you, as a teacher, ensure that all students’ ideas are heard and valued?
Assign group roles (everyone has a job).
Ask other members of the group to restate their peer’s thinking to validate the statement made by the student.
The teacher’s positioning when speaking to small groups
Explain one method that a teacher might use to sequence student work to facilitate a meaningful class discussion?
One method that a teacher might use to begin with groups that used concrete models (numbers, examples, diagrams) and work towards a more abstract model (equations, general formulas,etc.)
Match the question type to the description:
Gathering Information and Probing Thinking
1.Students explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or the completion of a task.
2. Students recall facts, justifications, or procedures
Probing Thinking (1)
Gathering Information (2)
If there is one thing that you could have done differently while implementing your signature lesson plan, what would it be?
Erika: If I could re-do my lessons with the class, I would have tried to have students share more of their thinking. Most days I ran out of time and I wish that I could have used the students’ thinking more than I did.
Ryan: For the second task that i implemented during my signature lessons, I would have provided the students with a bit more scaffolding in writing their system of inequalities. The goal of the lesson was for the students to be able to solve systems of linear equations through graphing. However, the students were so caught up in writing the equation that I felt that this goal of the lesson was missed.
How could you change this task to make it a higher level of cognitive demand?
A football team has an away game, and the bus breaks down. The coaches decide to drive the players to the game in cars and vans. Four players can ride in each car. Six players can ride in each van. There are 48 players on the team. The equation 4x + 6y = 48 models this situation, where x is the number of cars and y is the number of vans.
a.) Graph the equation. Interpret the intercepts.
b.) Find four possible solutions in the context of the problem.
"A football team has an away game, and the bus breaks down. The coaches decide to drive the players to the game in cars and vans. Four players can ride in each car. Six players can ride in each van. There are 48 total players on the team.
a) With what combinations of cars and vans can the coaches drive the players to the game? Explain how you determined these combinations using visual representations on your poster board.
b) Can you determine which combination will require the least number of vehicles? If so, how?"
What does equity mean to you?
To us, equity does not mean treating each and every student the same. Equity means that you, as a teacher, are making the necessary, individual accommodations that foster a successful and comfortable learning environment for ALL students.
Explain the importance of connecting student thinking as opposed to having students share their work in a “show and tell” style.
When students share their work in a show and tell fashion, oftentimes connections are missed not only between different students’ work, but connections with the mathematics are missed as well. In addition, students may pay less attention to their peers. When we connect student thinking, students are learning from one and another and all of their ideas build up to create one overall mathematical idea.
Match the question type to the description:
Making the mathematics visible and Encouraging Reflection and Justification
1. Students discuss mathematical structures and make connections among mathematical ideas and relationships.
2. Students reveal deeper understanding of their reasoning and actions, including making an argument for the validity of their work.
Making the mathematics visible (1)
Encouraging Reflection and Justification (2)
What were your two biggest takeaways from your field practicum experience?
Erika: My biggest takeaway from my field practicum experience was that students really do feed off of the energy and attitude of the teacher. I notice that some days when my mentor teacher was tired or quiet, the students were not motivated to learn. When she had a more upbeat attitude, I could tell the students were substantially more engaged.
Ryan: 1: My biggest takeaway from my field practicum experience this semester is that true classroom management and student effort boils down to the relationship that the teacher builds with his/her students. I found that the students that I interacted with more frequently seemed to put forth more effort and were far less hesitant to ask questions.