Mathematical Understanding: An Introduction Fuson, Kalchman, and Bransford

Why the negative association with math? Instead of connecting with and building on student’s math understanding and intuitions, math instruction often overrides student’s reasoning and replaces it with a set of rules that disconnects problem solving from meaning making.

Why the negative association with math? Instead of organizing the skills required to do mathematics around a set of core concepts, the skills become the center (and possibly the whole) of the instruction.

Why the negative association with math? Since procedural knowledge is separated from meaning making, students are not using meta-cognitive strategies when they solve math problems.

An Observation by John Holt One rather good students was working on the problem “If you have 6 jugs and you want to put 2/3 of a pint of lemonade into each jug, how much lemonade is needed?” The student responded with “18 pints” The teacher followed this answer up with some questions.

An Observation by John Holt T- How much is in each jug? S- Two-thirds of a pint T- Is that more or less than a pint? S- Less T- How many jugs are there? S- Six T- Then 18 doesn’t make sense if the 6 jugs each have less than 1 pint

An Observation by John Holt The student shrugs and responds “Well that is the way it worked out” Holt argues that the student has long since quit expecting school to make sense. They tell you these facts and rules and your job is to put them down on paper the way they tell you. Never mind whether they mean anything or not.

Adding it Up- National Research Council report Their report argues for an instructional goal of mathematical proficiency (far beyond mastery of procedures) by detailing five intertwining strands. Conceptual Understanding Procedural Fluency Strategic Competence Adaptive Reasoning Productive disposition

These 5 strands map directly to the principles of How People Learn

PRINCIPLE #1: Teachers must engage students’ preconceptions People posses informal strategy development & mathematical reasoning that can serve as a foundation for learning more abstract math but the link is not automatic. Teachers must build on existing knowledge and engage student’s preconceptions

Common Preconceptions about Mathematics Math is about learning computations Math is about following rules to guarantee the correct answer Some have the ability to do math and some do not. (widespread belief in US but not elsewhere. Other countries attribute effort to math success. And teaching reflects this)

Instruction therefore must: Allow students to use their own informal problem solving strategies and then guide their math thinking toward more effective strategies Encourage math talk- clarify their strategies and compare benefits/limits of alternate approaches Bridge common conceptions & math understandings

PRINCIPLE #2: Understanding requires factual knowledge and conceptual frameworks Teachers must help students build & consolidate prerequisite competencies, understand new concepts in depth, and organize both concepts and competencies into a network of knowledge Math proficiency requires students master concepts AND procedures to reason & solve problems effectively.

PRINCIPLE #2: Understanding requires factual knowledge and conceptual frameworks Emphasizes importance of both conceptual understanding and procedural fluency. The concepts underlying an area of mathematics must be clear to the teacher if they are to develop mathematical proficiency within their students Balance must be maintained between learner-centered and knowledge-centered needs.

PRINCIPLE #2: Understanding requires factual knowledge and conceptual frameworks Viable alternative methods for solving a problem, and discussions of the advantages and disadvantages of each that facilitates flexibility and deep understanding of the math involved.

PRINCIPLE #3: A Metacognitive Approach Enables Student Self- Monitoring Learning about oneself as a learner, thinker, and problem solver is an important aspect of metacognition. A teacher can help a student become a better mathematical thinker by moving the student through the following levels of the “math-talk learning community”:

1. Traditional, Teacher Directed Format -where the teacher asks short answer questions and the student provides answers directed at the teacher 2. “Getting Started” Level -where the teacher begins to pursue and assess students’ mathematical thinking 3. “Building” Level -where teacher elicits more student responses with more descriptions of their thinking. Also multiple methods to solving a problem are discussed 4. “Math-Talk” Level -where students justify their own ideas, ask questions of other students and help other students

Instruction That Supports Metacognition “Debugging” – shift focus from right and wrong answers to “debugging” answers by finding the error, figuring out why it is an error, and correcting the error. “Dialoguing” – teachers need to emphasize the importance of communicating mathematics. Students need to reflect on and communicate their mathematical thinking. “Seeking and Giving Help” – students need enough confidence to engage in problem solving and try to solve them but also to seek help when they get stuck.

The Framework of How People Learn: Seeking a Balanced Classroom Environment Learner-centered – draws out and builds on student learning. Knowledge-centered – focuses simultaneously on the conceptual understanding and the procedural knowledge of a topic. Assessment-centered - there are frequent opportunities for students to reveal their thinking on a topic so that teachers can shape their instruction in response to student learning. Community-centered – the members of the classroom value student ideas, encourage productive interchange, and promote collaborative thinking.

Next Steps: Create teacher-learning communities (our PLCs) Learn to use student thinking in video clubs (upcoming teacher observations) Have lesson studies on how to build on student knowledge (possibly in the future) Use teachers as curriculum designers to determine learning paths (currently working on with PROM/SE)