MATH: the science of pattern and order

TIMSS (Trends in International Mathematics and Science Study) About the study:

o Largest study on math and science education ever conducted (41 nations) o mid-1990s, study repeats in 2007o Data gathered from 4th, 8th, and 12th grades

What we learned from the study: o US 4th and 8th graders are above average, but significantly outperformed by 8

countrieso US 12th graders are significantly below international averageo Top countries start math lessons with new problems/tasks; in 99% of US

classrooms, math class starts with review of hw.o US Math Curriculum is “a mile wide and an inch deep”

NCTM (National Council of Teachers of Mathematics) World’s largest mathematics education organization 6 Principals Fundamental to high-quality mathematics education

1. Equity- high expectations and strong support for all students2. Curriculum- must be coherent, focused on important mathematics, and well

articulated across the grades3. Teaching- requires understanding what students know and need to learn and then

challenging and supporting them to learn it well4. Learning- actively building new knowledge from experience and prior knowledge5. Assessment- done for students, to guide and enhance their learning6. Technology- essential to teaching and learning

5 Content Standards (in 4 grade bands: preK-2, 3-5, 6-8, 9-12) Number and Operations Algebra Geometry Measurement Data Analysis and Probability

5 Process Standards Problem Solving- learning and doing math as you solve problems Reasoning and Proof- justifying ideas through logical argument Communication- being able to talk about, write about, describe, and explain

mathematical ideas Connections- connections between ideas and to real world Representation- using charts, symbols, graphs, manipulative, etc to express ideas and

relationships

Characteristic of a Great Math Teacher Knowledge of Mathematics- profound, flexible, and adaptive knowledge of content

Persistence- demonstrate persistence Positive Attitude- leads to more effective instruction Readiness for Change- be prepared to unlearn and relearn math concepts to fit current

methods and standards Reflective Disposition- professional development, etc

Educational Theories that Inform Math Teaching Constructivism (Piaget)- students are creators of their own knowledge Sociocultural Theory (Vygotsky)- learning through social interactions Best classrooms use ideas from BOTH

Classroom Environment & Teaching Strategies (informed by educational theory) Students

o share ideaso look for and discuss connectionso view errors or strategies that didn’t work are opportunities for learningo embrace struggleo build new knowledge from prior knowledge

Teachero makes mathematical relationships explicito engages students in productive struggleo Provide opportunities to talk about matho Provides opportunities for reflective thoughto Encourage multiple approacheso Treat errors as learning opportunitieso Scaffold new contento Honor diversity

What does it mean to “Do Math”? Means generating strategies for solving problems, applying those approached, seeing if

they lead to solutions, and checking to whether the answer makes sense Basic facts and skills are important in enabling students to do math, but rote exercises

alone do not prepare students for real world math

What does it meant to understand Math? Understanding: a measure of the quality and quantity of connections that an idea has with

existing ideaso varies person to persono not an all-or-nothing propositiono exists on a continuum from relational to instrumental understanding

Instrumental Understanding: doing something without understanding Relational Understanding: knowing what to do and why (GOAL)

Different Ways to Represent Mathematical Ideas1. Oral Language2. Written Symbols3. Real-world Situations4. Pictures5. Manipulative Models- physical objects used to illustrate and discover concepts

Tips for Effectively Representing Math Concepts Teach and encourage the use of multiple representations; students develop deeper

understanding and stronger retention when they have more than one way to think about and test emerging ideas

Help students connect models to the concept but avoid parroting (“do as I do”) Variety of tools should be accessible for students to select and use freely

5 Strands of Mathematical Proficiency (foundations for CCSS Standards of Math Practice)1. Conceptual Understanding- comprehension of concepts, operations, and relations2. Procedural Fluency- skill in carrying out procedures flexibly, accurately, efficiently, and

appropriately3. Strategic Competence- ability to formulate, represent, and solve mathematic problems4. Adaptive Reasoning- capacity for logical thought, reflection, explanation, and

justification5. Productive Disposition- habitual inclination to see mathematics as sensible, useful, and

worthwhile, coupled with a belief in diligence and one’s own efficacy

Teaching through Problem Solving (not for problem solving) Students learn through real contexts, problems, situations, and models Problems presented at the beginning of a lesson and skills emerging from working with

the problems Don’t tell students how to do a problem

o choose a problem that lends itself to your targeted strategyo if no one uses your strategy, ask “could we have tried ____? What would that

look like? Let’s give it a try.”

What is a problem? Any task or activity for which the students have no prescribed or memorized rules or

methods (no specific “correct” solution) Routine problem- students can right away which operations to use (ex: 2 + 4 = __) Non-routine problem- students don’t initially know how to solve it (ex: 2 + _ = _ + 9)

Features of a problem: Must begin where the students are Problematic or engaging aspect of the problem must be due to the mathematics that the

students are to learn (content overrides context) Must require justifications and explanations for answers and methods

Worthwhile Tasks are: Cognitively demanding (problem-based; involve higher-level thinking) Accessible to every student (multiple entry & exit points)

Contextually relevant (reflect cultures and interests of students; connect to other disciplines)

Found in standards-based textbooks

Problem-Solving Strategies Draw a picture, act it out, use a model Look for a pattern Guess and check Make a table or chart Try a simpler form of the problem Make an organized list Write an equation

4-Step Problem-Solving Process (for students)1. Understanding the problem2. Devising a plan3. Carrying out the plan4. Looking back

Questioning: Tips for Asking Productive Questions Ask higher-level questions Good questions target both concepts and procedures Consider the pattern of questioning; 3 major types

1. Initiation-response-feedback (IRF)- teacher asks, student answers, teacher confirms or challenges

2. Funneling- teacher continues to probe student until they reach a particular answer3. Focusing- uses probing questions to negotiate class discussions and facilitate

understanding; THE BEST ONE Use strategies that ensure every student is accountable to think of the answer (ex: turn &

talk) Don’t automatically confirm a correct answer; instead, engage other students by asking

follow-up questions (brainstorm alternative strategies, check work, etc)

3 Things that Teachers Should Tell Students1. Conventions (symbols, labels, terminology)- should be introduced after concepts2. Alternative Methods- if important strategies don’t emerge naturally, teach the strategy as

“another way” of solving the problem3. Clarification or formalization of students’ methods

Calculators should: NOT be used to practice computational skills be used to explore patterns, conduct investigations, test conjectures, solve problems, and

visualize solutions

Drill vs Practice:

drill- repetitive, non-problem-based exercises designed to improve skills or procedures already acquired

practice- refers to different problem-based tasks or experience, spread over numerous class periods, each addressing the same basic ideas