Functions: Mathematical Functions: Mathematical Tools for ScientistsTools for Scientists

Project Pathways: Course 1, Cohort 2Pilot for revised course 1Pressure Volume LessonFebruary 13, 2006

CRESMET

Pathways Objectives for Teachers

• Deepen teachers’ understanding of foundational mathematics & science concepts and their connections (function, rate-of-change, covariation, force, pressure)

• Improve teachers’ reasoning abilities and STEM habits of mind (as defined in the problem solving framework)

• Support teachers in adopting “expert” beliefs about STEM learning, STEM teaching, and STEM methods (problem solving, scientific inquiry, engineering design)

• Improve ability to monitor, reflect on, & modify classroom instruction (K-12 and ASU)

Explorations of Pressure and Variations of

Pressure But first, what is pressure?

Can you think of examples of pressure (e.g., tire pressure)?

How is pressure measured--What units are used to describe pressure? Tire pressure is measured how?

What does lbs/in2 mean? (Force (in pounds)/square inches

exerted on a surface Pressure of a diver under water is

measured how?

Covariation questions (finger tool) As the area changes, how does the pressure

change, assuming the force stays constant.

As the force changes, how does pressure change, assuming the area is held constant?

Covariation question (finger tool) As the depth under water changes, how does

the pressure change?

As the pressure on your ears change, how does your depth under water change

How is pressure in a volume measured?

kPa (Kilopascals)

What is a Pascal? pascal [Pa]The pascal is the SI unit of pressure. One

pascal is the pressure generated by a force of 1 ﾊ newton acting on an area of 1 square metre. It is a rather small unit as defined and is more often used as a kilopascal [kPa]. It is named after the French mathematician, physicist and philosopher Blaise Pascal (1623-62). Pascal--What is a pascal?

1 Pa is the pressure generated by 1 N/m2

What does N/m2 mean? (Force (in newtons)/square meters exerted on a

surface

So, we’ll be working in kPa

What is a newton (N)?

1 N = the force exerted at the surface of the earth by a 1 Kg mass

newton [N]The newton is the SI unit of force. One newton is the force required to give a mass of 1 ﾊ kilogram an acceleration of 1 metre per second per second. It is named after the English mathematician and physicist Sir Isaac Newton (1642-1727).

What is a kilogram? 1 kg = ? lbs 1 lb = ? kgs.

Finger tool--as pounds vary how do kilograms varyFinger tool--as kilograms vary how do pounds vary

Building a Formula (Function)

• Write a formula (function) for converting lbs. to kg.

• K = 2.2. L

• K(L) = 2.2 L

• How are these two functions related?

• Now, write a formula (function) for converting kg. to lbs.

• L = .45 K

• L(K) = .45 K

How are these two functions related?

ConversionsForce: Measured in Newtons Pressure: Measured in kPa and psi

What is the meaning of atmosphere (atm)? air pressure at sea level 1 atm = 101.325 kPa 1 atm = 14.696 psi

Write a formula or function for converting psi to kPa. psi = ? kPa

What is the air pressure at sea level? The air pressure at sea level is: 98.37 KPa

Pressure-Volume Experiment You’ll need your calculators and computers

Demonstration of what we’re going to do

Project Pathways

Partnership of ASU, four school districts & Intel (ASU: Mathematicians, scientists, engineers, math and science educators, professional development experts)

Primary Goal: • To produce a research-developed, refined & tested

model of inservice professional development for secondary mathematics and science teachers.

Core Strategies: • Four integrated math/science graduate courses + linked

teacher professional learning communities (lesson study approach)

Pathways Objectives for Students

• Increase challenging course-taking

• Improve problem solving, scientific inquiry, engineering design skills

• Improve understanding of foundation concepts and reasoning abilities of Freshman STEM courses at the University

• Improve confidence and interest in math & science

Instruments for Assessing Pathways Progress and Effectiveness Function Concept Inventory

Developed--instrument validation continues

Beliefs about STEM habits and STEM teaching Developed--early in the validation process

PLC Observation Protocol Still collecting qualitative data--not yet developed

RTOP Already validated and published

Developmental Research on the Notion of Rate-of-

Change Reasoning about and with the concept of rate-of-change requires specific ways of thinking

Covariational reasoning, the simultaneous coordination of two varying quantities and attending to the ways in which they change in relation to each other, is non-trivial for students (Thompson, 1994; Carlson et al., 2002)

-Is needed for interpreting and representing models of dynamically changing events

-Is foundational for understanding major concepts of calculus (limit, derivative, accumulation, the FTC)

Understanding these ways of thinking is useful for determining learning trajectories for instruction

Taxonomy

Reasoning Abilities R1: Apply proportional reasoning R2: View a function as a process that accepts

input and produces output R3: Apply covariational reasoning

Coordinate two varying quantities while attending to how the quantities change in relation to each other

Conceptual Abilities C1: Evaluate and interpret function information C2: Represent contextual function situations using

function notation C3: Understand and perform function operations C4: Understand how to reverse the function

process C5: Interpret and represent function behaviors C6: Interpret and represent rate of change

information for a function

Planning

PhaseBehavior

Resources Heuristics Affect Monitoring

OrientingSense makingOrganizingConstructing

Mathematical concepts, facts and algorithms were accessed when attempting to make sense of the problem. The solver also scanned her/his knowledge base to categorize the problem.

The solver often drew pictures, labeled unknowns, and classified the problem. (Solvers were sometimes observed saying, “this is an X kind of problem.”)

Motivation to make sense of the problem was influenced by their strong curiosity and high interest. High confidence was consistently exhibited, as was strong mathematical integrity.

Self-talk and reflective behaviors helped to keep their minds engaged. The solvers were observed asking “What does this mean?”; “How should I represent this?”; “What does that look like?”

ConjecturingImaginingEvaluating

Conceptual knowledge and facts were accessed to construct conjectures and make informed decisions about strategies and approaches.

Specific computational heuristics and geometric relationships were accessed and considered when determining a solution approach.

Beliefs about the methods of mathematics and one’s abilities influenced the conjectures and decisions. Signs of intimacy, anxiety, and frustration were also displayed.

Solvers reflected on the effectiveness of their strategies and plans. They frequently asked themselves questions such as, “Will this take me where I want to go?” and “How efficient will Approach X be?”

ExecutingComputingConstructing

Conceptual knowledge, facts, and algorithms were accessed when executing, computing, and constructing. Without conceptual knowledge, monitoring of constructions was misguided.

Fluency with a wide repertoire of heuristics, algorithms, and computational approaches were needed for the efficient execution of a solution.

Intimacy with the problem, integrity in constructions, frustration, joy, defense mechanisms, and concern for aesthetic solutions emerged in the context of constructing and computing.

Conceptual understandings and numerical intuitions were employed to reflect on the reasonableness of the solution progress and products when constructing solution statements.

Checking Verifying Decision making

Resources, including well-connected conceptual knowledge, informed the solver as to the reasonableness or correctness of the solution attained.

Computational and algorithmic shortcuts were used to verify the correctness of the answers and to ascertain the reasonableness of the computations.

As with the other phases, many affective behaviors were displayed. It is at this phase that frustration sometimes overwhelmed the solver.

Reflections on the efficiency, correctness, and aesthetic quality of the solution provided useful feedback to the solver.

The Multidimensional Problem Solving Framework: A The Multidimensional Problem Solving Framework: A Characterization of Effective Problem Solving PracticesCharacterization of Effective Problem Solving Practices

Beliefs about STEM Learning and Teaching Taxonomy

Confidence (a) Teachers Confidence in their mathematical ability (b) Teachers confidence in their scientific reasoning ability (c) Teachers confidence in their pedagogical ability

Methods of Mathematics and Science(a) Process of learning mathematics and science (b) Learnability of mathematics and science (c) Authority in learning mathematics and science

III. Methods of Teaching Mathematics and Science (a) Instructional goals (b) The role of homework and exams (c) Attention to the learning process (d) Teaching Constraints (Factors of Resistance) (e) Instructional Approaches/Teacher Classroom Practices

Teachers confidence in their pedagogical ability

I feel prepared to create learning opportunities for my students that promote connections between mathematics and science.

I feel threatened when students ask questions for which I do not have an answer.

I have a clear understanding of how the concept of mathematical function develops in students.

When a student asks a question I feel confident in posing a good question to help him construct his own meaning and understanding.

Methods of Mathematics and Science

Learnability of mathematics and science Truly understanding mathematics in the mathematics classroom

requires special abilities that only some people have.

Process of learning mathematics and science Making unsuccessful attempts when working on a mathematics

problem is an indication of one’s weakness in mathematics.

Authority in learning mathematics and science I try to get my students to rely on their own reasoning and logic

when attempting solutions to problems. It is important that students look to their teacher to verify the

correctness of their answers and explanations.

Instructional goals A primary goal of my instruction is to teach facts, skills/procedures that my

students may need in more advanced courses.

The role of homework and exams The primary goal of my exams is to assess if my students can memorize facts and

carry out procedures like ones required for completing the homework.

Attention to the learning process It is important to understand what a student is thinking when s/he asks a question.

Teaching Constraints (Factors of Resistance) The pace at which I must cover material does not allow me to teach ideas deeply My school administrators value my efforts to get students to understand ideas

deeply. I am able to adapt my pacing so I can spend instructional time on central concepts

of my courses

The Role of Concept Assessment Instruments in Evaluation

Provides coherent assessment of central ideas and what is involved in understanding those ideas

Provides valid and reliable assessment of student understanding

Easy to administer to large populations Can assess course and instructional effectiveness May serve as a tool to assess readiness for a course

Limitations of Concept Assessment Instruments Do not reveal thinking of individual students Do not reveal insights into the process of

learning Are not tied to specific instruction or course

materials Limited in assessing creative abilities

Closing Remarks

Use of individual cognitive models of learning or knowing, such as the Covariation Framework and the Multidimensional Problem Solving Framework increase the purposefulness of instructional materials, instructional actions, and serve as guiding frameworks for instrument development.