Highlights & summary pages from the ILS Math Institute

-- supports, resources, and materials to help you in your work

-- these materials must be considered within the context of the ILS Mathematics Institute to be meaningful; this does not represent the entire

content of the Institute

-- created by E. O’Connor, Ph.D. © 2004; photocopy permission for participants in ILS Mathematics Institute

Now, zooming in on the people . . . :Chatham population data

Perspectives on math

relevance… & inspiration

Multiple representations / novel integration

• Population statistics overlaid on the geographic map of the united states

http://www.census.gov/geo/www/cenpop/meanctr.pdf

Perspectives on math

relevance… & inspiration

Guiding Student Understanding / (developing constructs for the “Do” of FDEP process)

Relevance of constructIntroduced --

real-world if possible

Inductive exposure toMath construct / patterns Illustrated in the process

(hands-on if possible)

If relevant, introduction of Formulas or Algorithms

(simplifies math)

Practice with Algorithms in Many context

General math Model becomes

Evident to student

Once the model, formula, & algorithms are understood, the student can apply this model to understand & calculate other situations (deduction)

Students should return to the model to check reasonableness of answers and when need to test their problem-solve with a simpler problem – model must be firmly understood & assimilated for return checking to be possible

Different situation

Different situation

Different situation

How math models the real world and vice versa /the importance of the various components both for the

classroom and for mathematics

• What are the cluster of performance indicators from among the various key ideas that relate to real world problem to be solved?

• What are the various mathematical models and underlying understandings that the student must grasp to be able to understand these problems in a mathematical formats? How can an understanding of the model be applied to other problems or real-world application?

• What are the algorithms that eventually support the mathematical models and simplify the process of applying the mathematics?

Algorithms

Mathematics model(s) Mathematics model(s)

“Real World” &

Complex problems

Reduces problems to simpler arithmetic

(this is not a one-to-one correspondence)

Students must understand

there are limitations to

any model

Considering multiple representations of fraction - whole-part model, sets, lengths, areas

• Students can have problems because they do not understand what is the “whole” or the set / sometimes the model used sends the wrong message to the students

0 1/4 1/4 1/4 1

Sets of different objects / the shaded areas is compared to the whole set

Length (no. line) – fractional components are represented on the number line

Fractional areas are shaded & equal

Physical models / patterns generating a “need” for a formula

Physical models / patterns demonstrating a formula development

thru concrete experience

Data tables / exercises with powers pattern recognition & model (interactive in Excel)

What questions can you raise as students examined different patterns? What other representations can be shown for this data? In what other areas of math might this information serve useful?

Data tables / expressions / graphs pattern recognition & model understanding (Excel)

What questions can you raise as students examined the effect of changing coefficients? What are the unique characteristics of this type of data, expressions & graphs? How could this graph lead to a discussion of finding roots? How could the students interact w/ the data and the graph?

Data tables / exercises with powers patterns & graphs (interactive in Excel)

What questions can you raise as students examined different patterns in combination with graphic representations? How does the graph help & hinder understanding? How can the students articulate answers to these questions?

How can interactive changes be used to discuss scale and graphing?

Local school data / multiple representations meaningful

patterns & formulas

Multiple representations patterns & formulas

Physical models / patterns multiple representations (2 & 3d) & manipulations

Physical models / patterns demonstrating use of grids, coordinates

& more thru concrete experience

How might you integrate coordinate pairs and tabular-data-organization in these lessons (multiple representations)? What about measurement, scaling, proportions?

Works well with numbers / operations / geometry / measurement / probability

Works well with stats, data analysis, graphing

Meaningful ways of presenting math at the intermediate level

Guiding thru inductive reasoning(learning from a pattern)

Demonstrating & doing (within a meaningful

Experience)

Show relationship betweenData & representations

(spreadsheet interactivity)

Deductive (applying a modelOr formula to situations)

How can reasoning, multiple solutions, “no right way”, etc. be used as the learning environment?

Developing math understanding thru multiple models and representations

Math Reasoning / problem solving

ensure tenacity & search for

truth

test all models & representations against all "limits" of the associated

problems

use multiple models & representations

have students explain to others (oral, reports, presentations, teach

someone youngerprovide adequate time & require rigorous thinking

in multiple spaces

ensure differentiated lessons for both

high & lowillustrate misleading

nature of models in some contexts (pros/cons)

have students create their own models for different problems

model yourself the value of multiple

representations and complex thinking

guide students into situations that test models in many ways

(ie. division of whole #, whole # & fraction, fraction & fraction)

Integrated Standards Approach –math in context / requiring student ownership

Numeration Operations Measurement

Reasoning

&

ConjectureModeling

Uncertainty

Patterns

Assessment / test prep throughout

Individual: for assimilation, Deeper reflection,

Extended applications; Individualized projects

Individual (differentiation): Additional support or

challenge, as needed

What are the considerations when creating class “arrangements”?

Math learningTask

Complex learning And reasoning

Practice and Drill

Group: for complexSolutions, multiple

Perspectives, explanations & discussion; projects

Overarching framework to guide to unit development

ProcessProcess

MathMath

Students Students

Assessment - Testing

Assessment - Testing

Ways That Math Represents the World

• Hands on, measurement, scaling, physical – of space & relationships

MathMath• Patterns to models (inductive) / models to world (deductive);

formulas, graphs, equations, symbols, algorithms

• Interconnections between constructs, problem solving w/ multiple models & representations

ILS Math Approach - Student Arrangements

• Working together; preferably solving & discussing complex problems

StudentsStudents

• Work independently – for assimilation, reflection, personal integration & for math practice

• Customized instruction for special needs (low / high)

Task definition

is a critical issue

The Process of Standards-Based Mathematics Teaching (involving students)

• Locate construct &/or its use in everyday life

ProcessProcess

• Construct / Solve / Explain thru patterns, models, formulas, & algorithms (based on relevant Math)

• Extend/connect to other math constructs, real world & other disciplines

• Students practice techniques & problems; teacher assesses thru all

FDEP Matrix – overview MATH ITEM Find Do Extend Practice

OBJECTIVE OF EACH

COMPONENT

- to situate the math concept in the physical or social world- to ground the do/practice components in a meaningful context- to connect new construct (Do) to past experience

to have students:  - derive & develop math  knowledge & formulas (from  experience when feasible) - perform the basic analyses, computations and manipulations

to challenge students to:- value the construct in their lives- to see & solve problems w/ this construct- to connect w/ other  processes &  constructs

to have students:- develop test questions & strategies - determine rehearsal areas in vocabulary & formulas

TASK DESCRIPTION

- locate, identify, observe, and/or  quantify the math  construct in physical / social world - bring to class and/or brainstorm- conduct sessions for observations

describe, test, derive, use the math constructs and formulas, ensuring have experiences (present or prior) on concrete, representational, calculation levels

- extend the construct learned (Do) into student life - solve problems w/ construct; combine with & apply to other constructs & disciplines

- encourage internalization & learning of language & formulas- prepare for tests by analyzing construct in test scenario

DIRECTING ACTIVITIES

- teacher requires & directs info to be gathered but does not provide answers; teacher steers students towards the questions to be answered in Do

- teacher sets-up, structures and explains scenarios but lets students conduct Do with little direct instructing; teacher directs a summarizing final discussion

- teacher guides students towards extending ideas & connected math constructs; the activity works best when  students determine the connections

- teacher mandates practice question development, however, students originate the practice problems

CONTEXT / SETTING

can be varied for interest, renewed attention, or classroom needs; present activities with humor/silliness, drama, useful and/or real life applications, treasure hunts, holiday accoutrements, relevance to other areas, or direct student interest; consider a school year or major math-construct theme - EXTEND is particularly well suited to creative variations

FDEP Matrix – details, part 1  SPECIFIC COMPONENTS TO CONSIDER: (use as appropriate to your class)

EXPLAIN

students:

FIND

describe, list, enumerate the place, use, application and/or location of the construct in world

DO/DISCUSS

speak / write about their thinking & its derivation, teach others the construct

EXTEND (CONNECT)

tell / write about problems solved w/ construct; apply to life; connect to other constructs

PRACTICE

continue using constructs; create, design, administer & assess test questions

VISUALIZE

 students:

draw / outline in photos the construct in physical -social contexts

represent, sketch, measure, count, diagram, photograph, graph

design / illustrate new applications; draw with other constructs

create visual for test questions 

MANIPULATIVE/ MEASUREMENT

apply as appropriate; use purchased ones if available; develop personal ones

develop your own or  use purchased if available; apply to construct; save manipulatives

use to compare math construct in other applications

allow students to practice with measurements & materials

RECORD(combine w/ technology)

students:

save findings, lists, totals - use journals, worksheets, spreadsheets

save drawings, representations, graphs; take digital photos; make binders

present thru reports, charts, tables; make PowerPoint; incorporate other constructs

store vocabulary, formulas & student-made test questions; list in journal areas for practice

INTERDISC* / INTEGRATED

students:

find construct within other disciplines & classes; consider role of math in world

measure, count, manipulate, study, or graph info related to other disciplines & school needs

help other disciplines w/ math; solve or explain other problems w/ math; connect

reinforce  constructs thru  connections to other contexts; use in Flashbacks

FDEP Matrix – details, part 2

DIFFERENTIATE

 teacher:

FINDsaves class created materials, projects & lists for students needing more work

DOarranges group tasks on different levels and  complexities so all may participate

EXTENDdesignates different follow-up tasks; may require continued DO

PRACTICEtailors practice to the needs of the individuals; has more able student teach 

TECHNOLOGY

 students:

store, compile,  save, arrange (sort/filter); capture & save relevant data, images, photos,  drawings

create statistics, graphs, formula tests, & calculations; use online interactive technologies w/ key math constructs

create authentic studies,  reports, projects & surveys that use, solve problems with, & apply construct

review saved materials, test questions, teacher notes;  interactively work w/ spreadsheet & online resources

TECHNOLOGY teacher:

although downloadable templates are available, teachers should invest the time in learning spreadsheet & graphing programs (such as, Excel) to be able to develop their own materials, measurements, and displays to meet exactly the needs of their students 

PROJECTS

teacher:

engage students in an extended written or reported activity that integrates the various activities within the exploration of the math construct -- to explore the construct, to situate the construct within the larger context of application, history, problems solved, and importance of the particular math understanding; for ideas on choosing projects., have students find math constructs in news & events

ASSESSMENT

teacher:

observes, listens to, assembles, reads, annotates, and evaluates the evidence presented in the components above (verbal, written, drawn, reports, projects) for evidence of learning; uses "practice" time when test question are being developed to listen to student thinking; creates additional F/D/E experiences if understanding is missing; use standard classroom games for rehearsing & memorization only after understanding on a conceptual level has been demonstrated by the majority of students

Saving student work

Save student works:  important aspects of learning and understanding mathematics relate to connecting new information with prior knowledge and delving more deeply into constructs.  At the end of concept introduction lesson (such as the ones that follow in this unit), persevere student best works - by placing products within plastic sheets and storing in a binder, by taking digital photographs of products and saving on the computer, by saving PowerPoint and Word files that were used during the lessons.  These products can be used continuously in later lessons thereby reinforcing the important awareness that mathematics continuously "talks" about the same world, reducing the amount of learning time that is related to understanding the context of a mathematics problem, and encouraging more pride and quality in the original work (if students understand that their work may be selected as a product to be saved).

The reflective process /requirement for thinking in learning

• “An educative experience has two criteria: It must involve interaction between the individual and his or her environment, and it must have the element of continuity. The relationships and connections that an individual perceives and makes within and between experiences are what give meaning to experiences that would otherwise be meaningless. Such connections subsequently allow the individual to move into future experiences with greater awareness, understanding, and purpose, providing direction and therefore assuring growth.”

• “the process of reflection requires teachers to confront the complexity of teaching and learning. Any action the teacher takes will therefore be a considered one rather than an impulsive one. In like fashion, once teachers learn to think reflectively, they can teach their students to do the same, for teachers teach best what they understand deeply from their own experience. From there they can encourage their students to confront thoughtfully the phenomena of their world.”

• C. Rogers, 2002, Defining Reflection: Another Look at John Dewey and Reflective Thinking; State University of New York at Albany

• http://www.tcrecord.org/ExecSummary.asp?ContentID=10890

The reflective process /requirement for thinking in learning

• “Second, reflection is a systematic, rigorous, disciplined way of thinking, with its roots in scientific inquiry. Under this criterion, which forms the heart of the article, the steps in the reflective process as conceived by Dewey and outlined in How We Think are summarized. As Dewey defines it, reflection is a particular way of thinking and cannot be equated with mulling something over. Such thinking, in contrast to reflection, is, in a word, undisciplined. Six phases of reflection are identified, and extensive explanations and examples of each are offered. The six phases include the following:

• an experience; • spontaneous interpretation of the experience; • naming the problem(s) or the question(s) that arises out of the experience; • generating possible explanations for the problem(s) or question(s) posed; • ramifying explanations into full blown hypotheses; and • experimenting or testing the selected hypothesis • At the end of the process one feels that the meaning one has ascribed to an experience

fits, makes sense, and can be relied on in future experiences. More often than not, of course, once one has tested one's theories in action, more questions, more problems, and more ideas arise. In this sense, reflection comes full circle, part of a dialectical process: The testing becomes the next experience, and experiment and experience become synonymous. If one takes the process of reflection seriously, it is impossible for it not to change how one acts in the world.

• C. Rogers, 2002, Defining Reflection: Another Look at John Dewey and Reflective Thinking; State University of New York at Albany

• http://www.tcrecord.org/ExecSummary.asp?ContentID=10890

The reflective process /requirement for thinking in learning

• “Third, reflection needs to happen in community, in interaction with others, and it requires attitudes that value the personal and intellectual growth of oneself and of others. The community also serves as a testing ground for an individual's understanding as it moves from the realm of the personal to the public. It also provides a forum wherein the individual can put form to what it is he or she was thinking-or feeling-in the first place. One of the interesting by-products of working in a supportive community is that it allows teachers to acknowledge their interdependence in a world that scorns asking for advice and values, above all, independence for both students and teachers. Dewey, always leery of dualisms, recognized that teachers and students need both the support of the community and the ability to act independently within the larger world.

• Finally, reflection requires attitudes that value the personal and intellectual growth of oneself and of others. Dewey believed that the attitudes that the individual brought to bear on the act of reflection could either open the way to learning or block it. Awareness of our attitudes and emotions, and the discipline to harness them and use them to our advantage, is part of the work of a good thinker, he argues. He recognized the tendency in all human beings to see what we wish were true, or what we fear might be true, rather than to accept what evidence tells us is so. Reflection that is guided by attitudes of directness, whole-heartedness, open-mindedness, and responsibility stands a much better chance of broadening one's field of knowledge and awareness and serving the communities within which one lives.”

• C. Rogers, 2002, Defining Reflection: Another Look at John Dewey and Reflective Thinking; State University of New York at Albany

• http://www.tcrecord.org/ExecSummary.asp?ContentID=10890