#1 IDS/SAGE summer math courses

.

A Blog for Teachers in our Courses

At the heart of the work of IDS/SAGE are the Growing Minds courses designed to help teachers see and work with the developing mechanisms of intelligence students must use as they attempt to understand foundational ideas of the math curriculum. These courses address this difficult task of helping teachers look deeper behind students’ learned answers and performances to see students’ thinking at work that generate the answers and procedures they produce. Our research using student 'Mind Maps' vividly shows underachieving math students lack the mental systems necessary to understand basic number operations of the whole number system, the beginning forms of mathematical representation, place value, simple Euclidean geometry, beginning coordinate geometry. As a result, their learning remains weak and tied to memorization rather than understanding. Our success in helping teachers see and work with the developing mental machinery of students depends on videotapes, activities, explanatory texts, analysis activities, and, perhaps most powerfully, the diagnostic assessment tasks teachers learn to use. They develop their assessment abilities by practicing on students recruited for the class. With these concept by concept diagnostic tasks, we then help teachers see how to fit specific instructional activities to the point of each student’s development.

Being able to see and work with cognitive systems enables teachers to go beyond an exclusive instructional focus on correct answers and procedures to look deeper into the causes of poor math learning. Learning depends on having the ability to learn. In specific terms, a math ability is the completed mental system that controls what students’ see, understand, and learn. Learning without understanding tends to be trivial and temporary in students. The research gives teachers the technical descriptions and grade levels for each of these completed cognitive systems that afre needed for specific content in math learning. Without the internal development of a cognitive system, students cannot understand or benefit from the learning activities. They cannot generalize or apply since they must rely on memorization. Every learning activity depends upon a system which makes that learning possible. And a systems approach helps teachers see and work with the development of math ability in students.

A systems approach clarifies exactly what it is that comprises math "ability" in a learning activity so that teachers can see its specific and necessary components. This professional research-based knowledge of students’ math development gives teachers a look at how the deeper understandings produced by students’ cognitive systems produce learning in the math programs teachers are using. Our goal is to make teachers more powerful in their ability to produce student growth by helping them directly promote the development of these underlying intellectual math abilities in their students. This kind of instructional power depends on teachers’ ability to 'see' and work with the cognitive systems that organize and regulate students’ activity and learning, and we believe this teacher ability to see cogntive systems is the primary indicator of teacher professional development. To help teachers with this challenging goal, we divide the courses each session into theory and instructional application. We want teachers to see the powerful and very practical value that understanding the development of students’ mental systems can provide them.


Growing Minds in Math

At the K-3 level (the grade levels that are the focus of our first two courses) the essential logical/mathematical systems students must construct are those of the concept of number, those of classes used in naming things according to their properties, and describing differences to produced ordering among objects according to an attribute. Students also begin the construction of all the math relationships of basic math facts and place value numeration. Growing Minds in Mathematics looks at how students use these number, inclusion, order, and numeration systems to represent all kinds of situations and objects such as understanding number as a concept separate from counting or any spatial arrangements, one-to-one correspondence of number, simple classification of animals, multiple classification systems involving a matrix of classification of things, awareness of simple number operations, then students' conceptions of equations and representation of math facts, and finally, beginning placed value concepts.


Growing Minds in Geometry and Measurement

The second course, Growing Minds in Geometry and Measurement, tackles the essential systems for representing the spatial properties of objects such as the concept of length, perimeter, and area. These lead to various forms of geometry knowledge. To represent the spatial properties of objects producing geometry depends on students first putting together topological concepts. This beginning form of primary geometry, here again operating as cognitive systems in parallel with the mathematical systems, reconstruct the spatial form of objects by using proximity, boundaries, inclusions, separations, to take apart and put together the spatial parts and wholes of objects. Intermediate students then construct the concept of a straight line out of the simpler topological line students previously used to define boundaries of objects. Students also develop a simple concept of the order of spatial parts they use in forming their representations. Later, out of the formation of a point of view and a line of sight, students develop projective concepts and relationships such as masking, the straight line, and perspective. At the same time, students are developing the concept of a coordinate system that organizes right left, front back, and up down directions. They are simultaneously developing Euclidean concepts of length, area, distance, order of placement, and perimeter. The Euclidean concepts lead to developing units for each of the properties and measurement using these units. Combined with concepts of time, students at the middle school level can begin to look at causal relationships of change and beginning notions of graphing these relationships using the coordinate plane. These come to be represented and used mathematically in analytic geometry and science courses.

Understanding and working with these cognitive systems in students’ thinking is challenging but essential in tackling the math achievement gap. We organize the content of the courses around the ‘big ideas.’


The Three Course Objectives

The three big ideas of the courses are
1) the cognitive systems that make up the concepts;
2) the concept-specific interview methods that teachers use to see the systems at work in students' thinking; and
3) instructional activities that are directly fitted to these cognitive systems.

Being able to understand and work with cognitive systems is essential and useful if an instructional program is to be successful with all students. Being able to diagnose how far each student has achieve in constructing each of the math systems enables teachers to identify the next step of construction in the formation of conceptual understanding. The assessment methods of the tasks are powerful criterion referenced diagnostic assessments with their Rasch statistics derived from research. This task method of developmental assessment reaches down to the pre-school level to assess "readiness" among pre-school, kindergarten, and primary students. The courses open new directions, offer much to investigate, and create exciting potential for instructional development. We always follow the courses and teachers’ development with great interest.