Communicating Use mathematical language while solving and arguing for the correct solution Make conjectures to build logical progression and explore ideas Defend reasoning using representations Listen and make sense to other arguments or explanations and ask probing questions
Indicator 4: Connecting Apply prior knowledge to solve real world problems Make connections with science and other disciplines Look for patterns and recognize that quantities can be represented in different ways Recognize the concepts and model by using a structure for a similar problem
Draw a pictures or a model Make an organized list Look for a pattern Guess and check Work a simpler problem Make a table Guess and check Work Backward Act it out
Structures are made up of parts that go together and help one another. A structure is only as strong as its parts. Structures can be both natural and man-made. Structures can combine to form other structures. Structures can facilitate or hinder change.
We cannot “see” a student’s understanding, we make inference about what it may be. The theory of constructivism suggests that telling does not help us understand the ideas, the ideas must be constructed to make meaning.
Ball et al. (2009) suggest a set of criteria for high leveraging practices: 1.Supports work that is central to mathematics. 2.Helps to improve the learning and achievement of all students. 3.Is done frequently when teaching mathematics. 4. Applies across different approaches to teaching mathematics.
5.Can be articulated and taught. 6.Is accessible to learners of teaching. 7.Can be revisited in increasingly sophisticated and integrated acts of teaching. 8.Is able to be practiced by beginners in the field-based settings. (p.461)
Actively engaged minds in thoughts equal effective learning. No ideas exist in isolation. Existing ideas give meaning to the new idea, new connections are formed between the new idea and the existing ones. Ideas…Understanding…Learning
Models are not the same as concepts Models are objects, pictures, drawings or manipulatives that represent the concept and its relationship Models and other Tools for Learning Tools to help learn concepts Procedure Knowledge as a Tool Learn hand and hand with concepts
Involve s directing student attention toward specific learning Highly structured environment Content/concepts broken down into parts Involves modeking skills, behaviors and thinking Engage and Hook Explain and model Explore and apply Evaluate and close
… it is the teachers using particular teaching methods, teachers with high expectations for all students, and teachers who have created positive student-teacher relationships that are more likely to have above average effects on student achievement.
High cognitive demand(Stein et. Al. 1996: Boaler & Staples, 2008 Significant content(Hiebert et. al, 1997) Require justification or explanation (Bolaer & Staples, in press) Make connections between two or more representations (Lesh, Post, & Behr, 1988) Open-ended (Lotan, 2003; Borasi & Fonzi, 2002) Allow entry to students with a range of skills and abilities Multiple ways to show competence (Lotan, 2003)