Concrete-Representational-Abstract CRA in Review: –Describe each component of CRA –What are the benefits of using CRA? –What are some examples of how it can be used to deepen student understanding of content? With a partner, discuss: –How did you use CRA principles in your classroom since the last inservice? –What worked? –What would you do differently?
Connections to CRA Multiple knowledge types to be addressed in mathematics instruction: –Conceptual – representation of major concepts, relationships, & connections –Declarative – knowing “that”; e.g. facts –Procedural – knowing “how”; knowledge directly applied to doing a task, problem- solving steps/process/procedure –Conditional – understanding when/why a strategy is used
Types of Knowledge: Application Are knowledge types discrete categories in your daily lessons or intertwined? How can CRA be used to bridge connections between knowledge types? Take out the lesson you taught last week… –What is the focus of the lesson (CRA)? –What other lessons would be needed to support student knowledge and understanding? –How would you sequence instruction on the topic?
Connections to Research Wu, Milgram, & Leping Ma: –Wu reflected on a commonly held opinion regarding American Teachers’ reluctance to introduce, teach, and discuss mathematical concepts from the abstract or symbolic reference point (Ma, 2006) –Practices what he preaches: Wu lead discussion on fractions building from concrete, representative to abstract (Wu, 2008, p 5-7)
Connections to Wu From elementary to high school: –“The narrow concern with the speciﬁc value of each computation, gives way to the exclusive concern with applying the basic laws of operations correctly and judiciously” (Wu, 2009, p. 10)
Connections to Wu
Connections to Wu Preparing students for Algebra requires an increase in the power of instruction at the middle grades –Shift from focus on arithmetic (computations with speciﬁc numbers) to algebra (concepts of generality and abstraction)requires instruction to be precise
Connections to Wu Students need to understand the concepts and procedures behind each algorithm to: –Reason through abstractions –Make generalizations –Reduces the amount of time students spend memorizing Teachers who attended… –Other comments or connections?
Today’s Focus: Questioning
Instructional Survey Results Instructional Element Importance Preparation
Rationale for Questioning Maximizes student engagement Increases opportunities for students to respond Correlated to achievement Redirects students to task Supports monitoring of students and teacher decision-making
Benefits of Questioning How good questioning impacts math achievement: –Meaningful responses to students help them think and let you see what they are thinking –Helps students build different types of knowledge –Can stimulate student discourse that leads to increase student understanding
Questioning Strategies Types of questions: –Closed: Questions that require answers with limited scope Example: What unit should be used to measure this room? –Open: Probing questions to encourage students to think about several related ideas Examples: How could we measure the length of this room? What choices of units do we have? Why would some units seem more appropriate than others? (http://www.utdanacenter.org/mathtoolkit/support/questioning.php)
Questioning Strategies Response techniques involve: –Waiting –Requesting a rationale for answers and or solutions –Eliciting alternative ideas and approaches What techniques have you used?
Linking Questioning to Student Discourse According to NCTM's Professional Standards, math teachers should orchestrate discourse by: –Posing questions and tasks that elicit, engage, and challenge each student's thinking –Asking students to justify their ideas orally and in writing Through modeling of investigative questioning, the teacher should help students learn to conjecture, invent, and solve problems
Linking Questioning to Student Discourse The Professional Standards propose five categories of questions that teachers should ask:
Your Use of Questioning Strategies With a group of teachers who share your curriculum: Take out the lesson you taught last week: –What categories of questions do you typically ask? –What’s an example of a question you asked your students? –Could you have gone in greater depth? –Should the questions have been more basic to scaffold learning? As a whole group: Which categories of questions lead to deeper understanding? Which categories promote earlier questions that support later questions?
Application to Your Classroom With a partner, answer the questions: –How will you take the information re: questioning back to your classroom? –How will you use questioning strategies to supplement the questions already posed in your curriculum? With the lesson you will teach next week: –Think about the enduring understandings you want your students to have and the responses that would demonstrate this understanding –What questions will you ask students in the lesson to get the results you want?
Making the Most of Your Data
Incorporate a 10 minute computational strand into your daily instruction Routine –Day 1: AIMSweb Probe administration –Day 2: Students review and graph probe results from previous week. Teacher can see graphs as students construct and enter data –Day 3: Teacher/class selects one type of problem to focus for the week - presents solution strategies –Days 4-5: Students practice 3-5 of these problems as an entry task Practical Probe Strategy
Formative assessment and feedback to students has positive effects Students are more motivated by timely and specific feedback Meets the instructional design feature of regular practice and review Can be modified to teach prerequisite computational skills needed for upcoming investigations How Strategy Links to Research
Take out intervention tracker Share with a partner: –How are you using the intervention tracker to document interventions for students? –Thinking about the last time you looked at your data, will you make any adaptations to the interventions you are using? What will you do differently? We will look at the intervention alongside your data in April’s meeting –Meanwhile, keep tracking! Intervention Tracker
Closing Activities Questions? Spring meetings on Wednesday evenings –4/8, 5/6, 6/3 –4-6 p.m. Next meeting: April 8 –Review questioning in lessons –Examine data with intervention tracker –Close-out instruction module Mathematicians Workshop Series –Dan Erman, April –Trish Koontz, May –Richard Sharr, May –Media Services, Rm. 41 for all future sessions Evaluation