Workshops page

Introduction (2 pages) 2Herget, Wilfried: Good estimating and hardly calculating – one problem, but many solutions (3 pages)

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Higginson, William/Colgan, Lynda: The joy of X: Helping prospective teachers see the creative potential of mathematics: Contextual remarks for a workshop (4 pages)

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Mason, John: Student-constructed-examples (1 page) 11Meissner, Hartwig: Creative use of calculators (6 pages) 12Sheffield, Linda Jensen: When the problem is solved the creativity has just begun (6 pages)

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Wollring, Bernd: Examples and working environments for the geometry of paper folding in the primary grades (6 pages)

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Please turn over and select the pages you are interested in.

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Creativity and Mathematics EducationProceedings of the International Conference

WORKSHOPS

Introduction

Again, mathematics looks different through the eyes of a child. Specially in

the workshops we will concentrate on these views. All participants will be in-

volved in creative activities. We will concentrate on practical issues. Each

participant will get the chance to experience new ideas through "learning by

doing", they must "begreifen". This is a German word with a double meaning,

to touch or to feel, but also to understand or to comprehend. "Begreifen" is the

one goal of our workshops. Communication or discussion about that process

of "Begreifen" is the other goal.

Much emphasis is given to the use of technology. The use of calculators and

computers reduces a lot of the burden of traditional mathematics teaching. We

must rethink our approach to teaching geometry and algebra and of course,

equally important, to paper and pencil skills. There will be no longer a strong

future for traditional procedures. The potential of a computer or of the TI 92,

but even of simple calculators in primary grades, will cause a dramatic change

of the curricula.

We will get a new discussion about the value of procedural knowledge.

Which skills will our children need for their daily life and professional life

situations? Pressing keys along a given sequence of symbols only is a syn-

tactical use of technology. What about the semantical aspects? Guess and test

procedures will become more important.

We need creative ideas to facilitate learning and understanding. The use of

technology is one possibility. But to enrich the horizon of the students, other

topics might help also. Nonmathematical problems may be fascinating. Math-

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ematics and Art is such a favourite topic. Paper folding is another attractive

activity for the children.

And we will need manipulatives for a creative teaching. Specially Wolfgang

Reitberger will give us a whole bunch of new ideas how to produce easily a

broad variety of simple "bricks" for creative applications with low-cost mater-

ials.

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Wilfried HERGET, Halle-Wittenberg, Germany

GOOD ESTIMATING AND HARDLY CALCULATING –

ONE PROBLEM, BUT MANY SOLUTIONS

The newspaper article on the giant shoe (see fig.) uses to serve as the starting

point: "Which shoe size has this giant shoe?" Such a question is very unusual,

both in mathematics lesson at school and in preservice and inservice teacher

training. I am always very strained, how many different ways can be found to

give an answer. A standard version is to start with the eyeglasses or with the

apron width: Their size can be measured easily both in the picture and in real-

ity. Simple calculating gives the length of the shoe. However, the question of

the shoe size remains still open. Did you ever think about it, how shoe length

and the different scales of shoe size are related? A small, interesting research

project!

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Fig.: Which shoe size has this giant shoe?

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Another idea comes from a colleague, who teaches art and mathematics: "A

shoe is approximately as long as a face!" On the basis of a standard shoe size,

say 7 or 8, we can calculate easily the shoe size of the giant shoe - presup-

posed, the relation between shoe length and shoe size is linear. Is it really

that? How could we realise it?

"Maths means calculating!"

"Maths means calculating!" - but not only and not always! For the problem

above, the focus does not lie on calculating, but rather on mathematizing and

on finding an appropriate solving strategy: "Here is a situation. Think about

it!" (Henry Pollak). And at the end there is no perfect result of the type "The

shoe size is 127½" or "The shoe length is 197.386 cm”. Giving and solving

such a problem in a maths lesson is very unusual, both for the pupils and for

the teachers. But nevertheless fundamental mathematical considerations en-

able at least an approximate statement for the shoe size - however there is no

unique, perfect result, and the situation is not divided yet into small "mouth-

fair" bits, to be eaten easily in the submitted order.

Fermi problems

Enrico Fermi (1901-1954), Nobel prize 1938, often preferred direct, rather

provisionally seeming solving methods instead of the "more elegant", fine-in-

timate and complex methods. In order to teach this to his students, he de-

veloped a certain sort of questions, which since that time were called "Fermi

problems" (von Baeyer).

"How many piano tuners are there in Chicago? " - this is probably the most

well-known Fermi problem. First one does not have the quietest notion, how

the response could read, and one is rather sure that too little of information is

given, in order to be able to find any solution. However, If one splits the ques-

tion up into subproblems and courageously makes some plausible assump-

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tions, then one arrives rather fast at an approximate answer: Chicago has

about three million inhabitants, an average family consists of four persons,

and a third of all families possess a piano. Thus there are 250,000 pianos in

this city. If each piano is tuned every ten years, then this means 25,000 tasks

per year. If each piano tuner is able to tune about four pianos per day, then

during 250 working days in a year he can tune about 1000 pianos, thus there

must be about 25 piano tuners in this city.

Of course, the response is not particularly exact, it could just as well be only

ten or even 100 – the point is that one arrives on very different ways at ap-

proximate solutions, which are nevertheless situated all "in the correct area".

"How many hairs do humans have on the heading?" - this Fermi question

already gave cheerful hours to many pupils (Wilfried Rohm): Usually the

haired part of the heading is regarded as an hemisphere, and the pupils de-

termine with the ruler at the heading of their neighbours an estimation for the

number of hair per cm2. Here still some similar problems: How much fixed

garbage is produced by the households of our country per year? How many

pigs are slaughtered daily? What are the "middle hourly wages" of humans,

related to their lifetime?

The value of these problems - or more correctly: the value of the solving pro-

cesses - lays in the pleasure of making oneself creatively and courageously on

the way towards an answer and in the experience of having arrived independ-

ently at an (admittedly approximate) solution instead of looking up reverently

to a solution found by somebody else.

Many similar problems and situations shall be presented - and as many of

them as possible shall be experienced actively, too.

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William HIGGINSON, Lynda COLGAN, Kingston, Canada

THE JOY OF X: HELPING PROSPECTIVE TEACHERS SEE THE CREATIVE POTENTIAL OF MATHEMATICS: CONTEXTUAL REMARKS FOR A WORKSHOP

Stimulated by a set of powerful factors such as the increasing availability of ever more powerful and affordable computing devices and discussions stimulated by results of international comparisons of achievement, such as TIMSS, many jurisdictions have been engaged in the generation of new curricula for school mathematics. Three factors which have come close to being universal constants in this process are: concerns about the competence levels and attitudes of the teachers who will be called upon to implement these new curricula, debates about the role of technology, and a very strong ‘utilitarian’ rationale for the place of mathematics in education. A group of mathematics educators at Queen’s University in Kingston, Ontario, concerned about the excessively narrow conception of their subject, has over the past few years been involved in a number of projects which have attempted to show the benefits of a more broadly defined conception of mathematics. Three of these initiatives have been: Tomorrow’s Mathematics Classroom: A Vision of Mathematics Education for Canada (Higginson and Flewelling, 1997), Creative Mathematics: Exploring Children’s Understanding (Upitis, Phillips and Higginson, 1997) and The Joy of X (Colgan and Higginson, 1999).

In the three by three matrix which provides the framework for the Vision statements mathematics is simultaneously presented as a tool, a language, and an art. This triadic conception implies triple roles for the teacher [informer, facilitator and artist], and the student [complier, cognizer and creator]. Creative Mathematics is an account of the results of introducing a “constructive-aesthetic” component into a Grade Three classroom in Vancouver. The Joy of X was a supplementary enrichment series for elementary teacher education candidates enrolled in the Primary/Junior stream of the Bachelor of Education program at Queen’s during the 1998 - 1999 academic year. This initiative brought together some 60 volunteers interested in broadening and improving their understanding of mathematics and how it might be taught. Participants attended up to twelve sessions (most of these were 90 minutes long; because of timetable constraints a few had to be fitted into 60 minute slots) presented by carefully chosen (enthusiastic and knowledgeable) speakers that focussed on envisioning new practices through narratives and resources. The aim of the sessions was to support the reconstruction of these beginning teachers’ professional identities as they attempted to extend their mathematical knowledge and to improve their

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disposition toward the subject by hearing about experienced teachers’ personal reflections/transforming experiences, participating in hands-on activities and discussing salient curricular and instructional issues.

The Joy of X was predicated on two hypotheses, namely, that: (i) rich, carefully-chosen, professional development experiences could strongly impact the mathematical knowledge and disposition of beginning teachers; and, (ii) by making available detailed exemplars of classroom scenarios the pedagogic repertoires of beginning teachers could be significantly increased.

The teacher candidates were given opportunities to participate in mathematical learning opportunities that represented the presenters’ efforts to share a qualitatively different type of school mathematics – one that has the potential to transform mathematics instruction and/or one’s personal mathematical identity. The forum provided a safe setting in which alternative approaches to mathematics teaching and learning were explicated, discussed, tried and assessed and where such efforts were encouraged and supported. Included in the sessions were hands-on workshops that focussed on themes such as:

Nimble Numbering; Mathemagic; Multicultural Mathematics; Origami and Mathematics; Mathematics through Children’s Literature; Spiders and Flies; Mathematics and Music; Mathematics on the Web.

Early analysis of the extensive data collected prior to, during, and after the series revealed that: (i) despite the fact that these individuals were, arguably, among the most able members of their national peer group, there was widespread unease and dissatisfaction about their earlier encounters with mathematics:

•Math makes me nervous.

•I wouldn’t hesitate to say that I fear math.

•My background is very limited - I only got the 2 math credits I needed to graduate high school and both credits took me 2 tries.

•I took math courses out of necessity - not out of interest.

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•My teachers managed to take all the joy out of the subject - I don’t remember ever having a teacher that was excited or enthusiastic about teaching math.

•In my math class we were never told or shown why something works in math, just that this is the way it works and we should memorize it.

(ii) the two identified hypotheses were confirmed for a high percentage of the participants

•I really felt like I was catching on to the beauty of mathematics for the first time in my life.

•I viewed math as a rigidly defined field with right and wrong answers. I thought that math and creativity were diametrically opposed. I was wrong! Math and creativity are inextricably linked.

•The creative ways that can be used to teach math made me feel more comfortable and less afraid.

•I want to take my students out of the classroom (literally and figuratively) to discover the math around us and to let them experiment and play.

•I realized that I learned the concepts much easier when we could touch and manipulate to get the answer to a problem. I want to use the same types of openers you used to get my class on track and focused.

•I wasn’t aware that there were mathematical links to music. I know now that I can use music in the classroom to introduce new mathematical concepts. I am eager and impatient to begin teach-ing how to compose music in the classroom!

•Math is fun ... finally! My new enthusiastic attitude will definitely help me teach and to instill an interest in - not a fear of math.

•I will try to frame math as beautiful ... not just numbers and dead facts.

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(iii) a feature of the series which was particularly effective in bringing about these views was the use of a technique we called “openers”.

A typical ‘Joy of X’ class began with 2 or 3 activities that could be set out with little or no explanation as an ‘opener’ for people to do as they arrived. The purpose of the ‘openers’ was to help our candidates to make the transition from their last class by being challenged to focus and concentrate on an interesting problem. The ‘openers’ were designed to help the students concentrate immediately on the pleasure of doing mathematics rather than getting the right answer, in other words, to establish a nonthreatening, comfortable environment by encouraging question-asking and discussion. The activities represented the full gamut of mathematics curriculum: Data Management & Probability, Patterning and Algebra, Number Sense and Numeration, Spatial Sense and Geometry and Measurement. All of the activities were chosen because they promote problem solving and mathematical reasoning, they involve the use of concrete materials and they can be enjoyed again and again. The activities were selected to help the students to develop the confidence to take some risks, try something new, and make mistakes.

Participants in the Creativity and Mathematics Education conference will have the opportunity to engage in an abbreviated The Joy of X session. In this session participants will experience an instructional sequence through which we develop opportunities for students to examine the effects of changing the dimensions of the area of two-dimensional objects on the volume of three-dimensional objects using origami.

References

Colgan, L. and W. Higginson, The Joy of X: An MSTE Group Project Kingston, ON: MSTE Group, Queen’s University, 1999

Higginson, W. and G. Flewelling, Eds. Tomorrow’s Mathematics Classroom: A Vision of Mathematics Education for Canada Kingston, ON: MSTE Group, Queen’s University, 1997 [The Primary, Junior, Intermediate and Senior versions of these documents are available at http://tilp.educ.queensu.ca/faculty/tmc/ ]

Upitis, R., E. Phillips and W. Higginson Creative Mathematics: Exploring Children’s Understanding Routledge: London and New York, 1997

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John MASON, Milton Keynes, Great Britain Abstract

STUDENT-CONSTRUCTED-EXAMPLES

One of the principal ways in which students of mathematics at every age make

sense of mathematics is through the construction of their own examples:

examples of objects and examples of problem-exercises. The examples can be

routine or novel, but example-construction constitutes a domain for the

exploiting of creativity.

In this workshop participants will have an opportunity to experience example

construction as a domain of creativity. They will be able to experience both

the effect on their own appreciation of a topic and the release of energy

available when

expressing their own generality, specialising for themselves,

constructing mathematical objects which satisfy certain properties and

not others,

characterising types of problems, and trying to locate the most general

problem they can of a given type.

Styles of mathematical prompts will be offered which are suitable to students

of every age.

Curricular Goals

When students take control of constructing mathematical objects, and control of problem creation, they appreciate the mathematics in a much fuller way, akin to the lecturer who finally understands a topic when called upon to teach it. Furthermore, when students powers are invoked and when creative energies are provoked, students experience pleasure, satisfaction, and increased personal confidence, providing

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them with the requisite energy to pursue mathematics a little bit further.

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Hartwig MEISSNER, Münster, Germany

CREATIVE USE OF CALCULATORS

What is a creative use of a tool, of calculators, computers, machines, ...?

We cannot and we will not define what creativity may be. We associate

with “creativity” some individual abilities like being able to invent new

(important) ideas, to discover new relationships, to try or vary old tech-

niques in new modes, to connect (till then unconnected) fields of experi-

ences, etc. But we also associate social aspects like being able to commu-

nicate, to cooperate in team work, to convince by arguments, to

motivate, ...

Reading this list of different abilities does not give us the feeling that we

are talking about existing mathematics education. Nevertheless, these abil-

ities will be the background for our activities in this workshop.

Often calculators are forbidden in primary grades: “Our students first must learn the basics”. And when calculators are used later on they often only serve as a tool for quicker and more complex calculations with safer results. In this workshop we will concentrate on a more creative use of calculators. A workshop needs activities. We will present different types.

Part I. Workshop Activities

1. Skill training. There are some creative ideas to reduce monotony and

boredom by the use of technology:

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1.1 Training camp: We need worksheets with sets of problems for mental arithmetic and a calculator. We do “parallel processing”, i. a. pressing the keys while computing mentally. = will be pressed only after we got the result mentally. The display then will give an immediate feed back (similar to vocabulary learning).

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1.2 Competition I (mental arithmetic): Some students are allowed to (later on: must)

use calculators, the other group does not use calculators. The teacher calls the problem,

which group is first with a correct answer?

1.3 Competition II (worksheets for mental arithmetic, calculators allowed): Use the calculator or not, who is first to get all answers correct?

1.4 Competition III, similar to 1.3 for the following types of paper and pencil al-gorithms: 3461 7 (more-digit times one-digit),

2301 addition of several 8294 simple subtraction of one+ 4682 more-digit-numbers - 4637 more-digit-number+ 7059

1.5 Results after some training: “For many problems I do not need the calculator. I am as safe as the calculator, but I am quicker in my head than in pressing all the keys.”

2.3 Mathematics with Arrows

What is the name of the last dot? Write down all names of the dots. Which is the quickest way to find the name of the last dot? Use decimals/fractions instead of the integers.

2.4 Bingo: The given numbers are products of two factors each. Find for the products the appropriate factors out of 7, 13, 16, 19, 23, 28, 43.

3. Discovery learning3.1 Find the number z with z z = 2 (3, 4, 5, 6, ...). Write a protocol of your guesses (table of inputs/outputs).

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361 817 133 529 448247 688 304 989 644784 299 364 91 256169 532 196 368 437301 112 161 208 559

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3.2 Multiplication: Select a path from A to B. Change the direction at each corner. Multiply the numbers of each step you go. Find the path with the smallest product. You have 4 trials.

3.3 Programming operators: Press n n = ( for +, -, or ) and 5 = .Try on a simple and cheap calculator. You should get the result of 5 n, i. e. thecalculator is programmed now as a n -machine (operator n ). Cheap calculators often have this property for all 4 basic operations. Experience various operators.

3.4 Partner work: One student “hides” an operator, the other student has to find out the operator by guess and test (writing a protocol (table of inputs/outputs) and arguing).

4. Calculator games. For the following 4 games you must guess and test.

You always should write a protocol of your inputs/outputs.

4.1 Big ZERO, partner work: Student A hides the operator -a , student B must find the input, which leads to the output zero.

4.2 Big ONE, partner work: Similar to Big ZERO, but you hide the operator a and the partner must find the input for the output one.

4.3 Hit the Target: You get an operator a ( for +, -, or ) and interval [c, d]. Find a number z with z a [c, d]. Try for all 4 basic operations. Most challenging is a multiplication operator, why?

4.4 Multiplication chain: You get a target interval [c, d] and a starting number s. Find a first factor f1 with s f1 = f2 [c, d]. If you fail f2 is your next starting number for

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finding factor f3 with f2 f3 = f4 [c, d]. If you fail again f4 will be the next starting, etc. Analyse the sequence of the fi in your protocols.

Part II: Reflecting the Activities

We distinguish between the syntactical use and the semantical use of

calculators (and also computers). The syntactical use is dominated by

pressing buttons along a given sequence of symbols (of a formula or of a

computational task). In our skill training activities we have used the

calculator syntactically, also in 2. and 3. To “program” a calculator as an

“operator” also is a syntactical activity, also the use of these “operator-

machines” later on.

The semantical use centres around “what?” and “why?”. A given problem

leads to questions where the calculator is a tool to find quick answers or

“narrow” answers or hints for answers, or ... The calculator basically is the

tool for different types of guess and test approaches to a given problem.

A semantical use is given when we must reflect the output for getting hints

for the new input. We then reflect the relationship input-operator-output or

later on with computers the relationship input-function-output or still more

complex, we study “Simulations”.

When the input and the operator (or function) is given and we ask for the

output, of course we have a syntactical use. But when we ask for an

appropriate starting number to get a given output usually problems arise.

Traditional mathematics uses reverse functions or algebraic transformations

to solve the problem. But by the help of quick calculators we also can make

a guess of the starting number and the output will give us a feed back about

the quality of our guess. With this hint our next guess might be better (or

worse, if we did not understand the hidden relations). A next guess may

help or another guess etc.

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Having a quick calculating machine the guess and test approach to find a

solution of a problem often is much easier than the use of algebraic

transformations. Thus this type of problem solving gets more and more

important in mathematics education. We have this approach in activity

2.4: Look at the last digits and at the magnitude of the factors and make

your guess (and test). Or in 3.2 students often at their first time get the

conscious experience, that multiplication with decimals can lead to

smaller numbers, and that repeated multiplications with factors between 0

and 1 can bring the product down to zero. Or they get a feeling about

approximation in 3.1 or about the relationships of the 4 basic operations in

3.4, 4.1 and 4.2. The activities in 3.1 and 4.4 even need an unconscious

“percentage-feeling”.

Our calculator games in 4. develop number sense for the four basic

operations, each “wrong” output gives a better hint for a better input.

Specially 4.3 develops a feeling for the magnitude of the two factors

necessary for a product of a given magnitude. We often “play” this game

with third graders. They develop an excellent feeling for finding a good

starting number and with one or two more guesses they have a correct

answer.

Easy problems in mathematics education are characterised by having a

rule, an algorithm or a formula for computing syntactically. Difficulties

arise when there are too many formulae (which one do I need?) or when

we need reverse functions or the algebraic transformations of formulae.

To overcome these difficulties our project group TIM has introduced a

new teaching method, which we call ONEWAY-Principle. According to

this method we integrate between the syntactical procedures and the

algebraic transformations systematically guess and test procedures. In this

teaching part the already known syntactical use of calculators is

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maintained and will be used as a tool to discover semantically the

relationships between the input, the “operator” and the output: Stay on the

ONEWAY of this easy syntactical procedure, make your guess and test.

We will describe the ONEWAY-Principle along the teaching unit “percent-

ages”. 6 3 5 + 6 % = is the syntactical use. If you look, before

pressing = , at the display (of a certain kind of simple calculators) you

will see 38,1, i. e. 6% of 635. After pressing then = you will get the sum

673,1. With that sequence of key strokes you can by guess and test solve

also all other percentage problems like: With tax you pay 280,60 $, without

tax 244 $, tax rate in %? Or: How big was the loan with an interest rate of

6% to pay back after one year with 3551 $? Or: You bought shoes in a sale

with 30% reduction for 68,60 $, how much did they cost before? Or: How

many years do you need to double your 1000 $ by getting a compound in-

terest rate of 5%?

We also use the ONEWAY-Principle for introducing the topics “Growth and

Decay”, “Trigonometric Functions”, “Which is the algebraic expression for

that graph of a function?”, etc. The main idea is, keep the ONEWAY

“Syntactical Calculations” as long as necessary and use your brain for guess

and test:

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Guess and test with the ONEWAY-principle

INPUTOUTPUT creativity feed back

Input Outputsyntacticalcalculation

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Linda Jensen SHEFFIELD, Fort Thomas, USA

WHEN THE PROBLEM IS SOLVED, THE CREATIVITY HAS JUST BEGUN

Too often, students are satisfied with getting an answer to a problem and not looking at it any further. In this way, they miss the excitement of thinking deeply about mathematical ideas and discovering new concepts. Students need to learn to explore problems to find that the fun has only just begun when the original problem has been solved.

Using the following open-ended heuristic is one way to get students to become more creative in the exploration of problems.

Relate

Investigate

CommunicateEvaluate

Create

Students might start anywhere on this model and proceed in a non-linear fashion to creatively investigate the problem. For example, a student might relate ideas about solving this problem to previous problems that have been solved, investigate those ideas, create new problems to work on, evaluate solutions, communicate the results, and think of other related problems to work on.

Questions to Encourage Thinking and ReasoningIn order to encourage students to investigate mathematical concepts on a deeper, more creative level, you should use rich, interesting problems that can be explored on a variety of levels. Expect problems to be solved in a variety of ways and give students a chance to explain their reasoning to each other. Use one problem as a springboard for several others. Work on these problems with colleagues before trying them with children and see how many solutions, patterns, generalizations, and related problems you can find. Ask questions that help students explore the big ideas, such as:

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Comparisons and Relationships1. How is this like other mathematical problems or patterns that I have seen? How does it differ?2. How does this relate to "real-life" situations or models?3. How are two factors or variables related?Structure, Organization and Representation1. How can I represent, simulate, model, or visualize these ideas in various ways?2. How might I sort, organize, and present this information?3. What are the essential elements of this problem?Rules and Procedures1. What steps might I follow to solve that? Are they reversible? Is there an easier or better way?2. Do I have enough information? too much information? conflicting information?3. What if I change one or more parts of the problem? How does that affect the outcomes?Patterns and Generalizations1. What patterns do I see in this data?2. Can I generalize these patterns?Reasoning and Verification1. Why does that work? If it does not work, why not?2. Will that always work? Will that ever work?3. Is that reasonable? Can you prove that? Are you sure?Optimization and Measurement1. How big is it? What is the largest possible answer? The smallest? 2. How many solutions are possible? Which is the best?3. What are the chances? What is the best chance?

Assessment CriteriaIf you wish students to develop deeper understanding of mathematical concepts, you should use criteria for assessment that encourage depth and creativity such as:Depth of understanding - the extent to which core concepts are explored and developedFluency - the number of different correct answers, methods of solution, or new questions formulated

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Flexibility - the number of different categories of answers, methods, or questions.Originality - solutions, methods or questions that are unique and show insightElaboration or elegance - quality of expression of thinking, including charts, graphs, drawings, models, and wordsGeneralizations - patterns that are noted, hypothesized, and verified for larger categoriesExtensions - related questions that are asked and explored, especially those involving why and what if

Help students learn to move from skill and drill problems with one right answer to puzzles that require reasoning and justification and to open-ended problems and explorations that have several solutions or related problems that will deepen and extend the mathematics being learned. Remember that the real mathematics frequently begins after the original problem has been solved.

Number ProblemsI. Drill and Skill: Study the following sequence.1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5 ...List the next ten numbers in the pattern.Puzzle: Study the following sequence.1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5 ...What is the 1,000th number in the pattern?Explain your reasoning.Explorations: Study the following sequence.1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5 ...Continue the sequence and search for interesting patterns. List as many patterns as you can find and predict the consequences of continuing the pattern to the "nth" term. Justify your reasoning. Compare this to similar problems.

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II. Skill and Drill: Fill in the next three lines on the chart below and explain your method.

135

79 1113 15 17 19

Puzzle: Without completing the chart, tell where the number 289 will appear. Explain your reasoning.Explorations: List all the patterns you can find on the chart above. Make and validate generalizations about your patterns.

Geometry and Measurement ProblemsI. Skill and Drill: Find the area and perimeter of a rectangular plot of grass measuring 5 meters by 4 meters.Puzzle: I have a rectangular plot of grass with an area of 24 square meters and a perimeter of 20 meters. What are the dimensions of the plot?Exploration: I have a rectangular plot of grass with an area of 24 square meters. What is the smallest possible perimeter of the plot? What is the largest possible perimeter? What if the shape of the plot of grass is not a rectangle? If the area of the plot doubles, will the answers to these questions also double?

II. Skill and Drill: Find the area and perimeter of a given polygon on the geoboard.Puzzles: Make a polygon on the geoboard with an area of two square units and the smallest possible perimeter. Make a polygon on the geoboard with an area of two square units and the largest possible perimeter. Exploration: Make as many polygons as possible on the geoboard with an area of two square units. Develop a classification scheme to determine if you have found all the possible polygons.

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Algebra Problem

I. Skill and Drill: Dan has decided to buy tops, toy planes, and/or stuffed animals for the school fair, and he must spend according to the following prices:

Tops $ .50 apieceToy planes $ 1.00 apieceStuffed Animals $10.00 apiece

If he buys 15 tops, 35 planes, and 50 stuffed animals, how much will he spend?

Puzzle: Latisha has exactly $100 to buy toys for the local children's charity and wants to buy exactly 100 toys. She has decided to buy tops, toy planes, and/or stuffed animals, and she must spend according to the following prices:

Tops $ .50 apieceToy planes $ 1.00 apieceStuffed Animals $10.00 apiece

What combinations of toys might she buy?

Exploration: Jerome has exactly $100 to buy toys for the local children's charity and wants to buy at least 60 toys. He has decided to buy tops, toy planes, and/or stuffed animals, and he must spend according to the following prices:

Tops $ .50 apieceToy planes $ 1.00 apieceStuffed Animals $10.00 apiece

What combinations of toys might he buy? Make a chart to display your findings. What patterns do you notice?

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Probability and Statistics Problem

I. Skill and Drill: Study the following triangle. Continue the triangle for 4 more lines.

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

What is the pattern in the sum of the lines of the triangle?

Puzzle: How could you use the triangle to predict the probability of getting exactly 3 questions correct if you guessed on every question on a 6 item true-false test?

Exploration: Study the following triangle. Continue the triangle for 4 more lines. List as many patterns as you can that can be found in the triangle.

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

After exploring some of these problems, encourage the students to create other puzzles and explorations of their own. Guide them to investigations of significant mathematics that allow them to develop their power as creative mathematicians. Use some of the questions suggested to get them started, and teach them to evaluate their own and their peers' work using some of the suggested assessment criteria.

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Bernd WOLLRING, Kassel, Germany

EXAMPLES AND WORKING ENVIRONMENTS

FOR THE GEOMETRY OF PAPER FOLDING IN THE PRIMARY

GRADES

Mathematics teaching for beginners in the primary grades still needs

impulses with respect to geometry, social learning, real invention situations

and active-constructing learning. We propose to realize working

environments on paper folding geometry in the primary grades and related to

this we describe design elements for teaching units and first findings from

empirical research related to this.

Many children are aquainted with elements of paper folding geometry from

the kindergarten and from their playing. Experiences on folding activities

may be assumed, more efficient ones than to other geometrical working

environments. As an essential advantage of paper folding geometry we see

that experience and knowledge on activities can be enriched before the

objects and facts are grasped by the language. This coincides with general

findings on learning proposing that competence related to actions precedes

competence related to language. We characterize products, tools and

documents related to paper folding geometry. As material to start with we

take quadratic origami-paper or paper in DIN-A-format, which nearly

everywhere is easier available than other materials for the teaching of

geometry.

The choice of the products to be folded is guided on the one hand by the fact

that they should have a material sense ("Werksinn") out of the range of

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mathematics and on the other hand should have a sense within the field of

mathematics.

The first is often neglected too much within mathematics teaching. What we

mean is that the folded pieces have a material sense as an object of art or as a

gift or as a model or as a part of another object with a sense which is accepted

in the everyday life of children. It is especially favourable, if such a material

sense can be attributed to a larger ensemble of those parts. We call this

ensemble sense, for instance > stars or > blossoms have it. Many of the paper

folding objects dedicated to children in the literature are chosen by an

artistical point of view.

So the teacher has to choose the paper folding objects also specially from the

second point of view, which means that they constitute substantial working

environments also with respect to mathematics. This is the case for instance

when symmetry is a central principle of generating the folding construction

or when spatial objects are constructed or spatial reasoning is an essential

presupposition for the constructing of the objects. Objects with mathematical

sense are among others those which arise by systematic variation of basic

patterns like > blossoms or which only make sense as symmetric objects like

> bugs or which show different types of symmetry like > modular stars or

which get their spatial shape by being assembled from different pieces named

modules like > the modular Denecke-cube.

Both of these product senses can be integrated by a product format we call >

folding picture which means a painting or picture that is completed by a

glued in folded motive. Several folding pictures can be collected to a >

folding picture book with or without the inegrating structure of a tale. A

product format that allows differentiating and working with respect to several

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ranges and additionally induces a rhythm of folding activities through the

year is the cumulative assembling of > calendars consisting of paper folding

pictures with motives fitting to the years seasons. Like the cube this calendar

arises from modules. This last mentionend aspect is of special importance:

Objects of modular origami, that means assembled folding products, are

ideally suited for the learning of partners or the learning in smaller or larger

groups, especially if it is possible to construct different objects from identical

or similar modules. This for instance is the case for the modules by which the

> Sonobe-cube or the > Kasahara-cube are constructed. Especially the

working with inner differentiation is supported by working envronments on

modular origami.

So far we marked senseful products in paper folding geometry, now we look

on > tools. By this we do not mean scissors or glue, which are refused in

traditional origami. What we mean are tools for measuring and comparing

consisting again of folded paper, which are used like circles, rulers, models

or patterns during the folding of other objects.

Often an object of folded paper arises by folding first some lines for

measuring an then making the shaping foldings which yield the final gestalt.

A tool now may arise when after having gained some routine one exports the

lines for measuring to an auxiliary paper and then folds the object, which

now does no longer carry these subsidiary lines, as it is the case for instance

in folding a > division of a straight line or folding > modular Denecke-cubes

with completely smooth surface. Such tools have a special purpose and

additionally have the didactic profit to induce a reasoning about the principles

and techniques by which they are created.

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But products and tools for their own do not give sufficient reasons to us for

intesive working on paper folding geometry in mathematics teaching. A

special accent should lie on the reflection of the construction process. With

respect to this it turns out that an essential advantage of paper folding

geometry lies in the fact that children are able to articulate their folding

constructions in an elementary and child appropriate way, so > folding

documents arise.

There are basically three ways to start activities when it is intended that

children reconstruct a given paper folding object. The first is using a printed

document from an origami book. It is encoded according to international

rules: Valley folds appear marked with little lines, mountain folds appear

marked with dots etc. The start with this type of document is often not

successful for children even if the objects are not complicated. As far more

successful we found in our investigations the analytic access, where the

children get a folded object as a pattern which they have to reconstruct or to

vary. Having completed their own folded object they give the pattern back. In

the analytic access the children at first work folding backwards then

completing their own object they work folding forward. The third way is

following a document which is created by other children.

As extremely fruitful however our investigations showed the task for the

children to create a purpose-guided but not standardised document about

folding an object dedicated to other persons, an instruction by children for

children. A written text with drawings in general is not successful in the way,

that the acceptor succeeds in folding the object. Documents in form of a >

folding book or a > folding map turned out to be especially efficient. A

folding book is a document for individual use, on its pages succeeding

folding states are fixed by glue. Additional texts may arise but are not

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necessary. A folding map is a document to be used as well individually as by

a group, for all the fixed folding states are to bee seen simultaneously. Here

also additional texts are possible but not necessary. In the task to construct

such a document arising from the reflection of the own constructing

dedicated to other people we see the proper demand on creativity in the

geometry of paper folding. Generating a folding document is a substantial

working envirionment for the learning of mathematics, which fulfills nearly

all the demands of an action-oriented cooperatively discovering and

constructing form of working. One way to generate such a document cosists

in the procedure to materialize the folding states at first and then together

discuss decisions on choosing and arranging them on a map with limited size,

before they are fixed.

At the moment we focus the interest of our qualitative empirical

investigations on the process of creating such folding maps by children and

on the effects which arise from the feedback given by other children who try

to construct the described object using such a map. The folding maps turn out

to be child appropriate construction descriptions, an efficient form of

nonverbal texts, which can be succeeded by formulating the process by

language and the introducing of special mathematical concepts. A special

form of folding maps arises whe children get the opportunity of articulation,

to represent the generation of spatial objects like the > Sonobe-cube on a

map consisting of a sequence of threedimensional spatial parts of the

complete object. These > 3D-maps generated by children belong to the most

efficient spatial construction descriptions, which we know at all among the

eigenproductions of children.

Folding documents especially in form of folding maps show similar to a

concept map but more consistent in the arrangement of the steps, whether

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geometrical construction principles like symmetry or congruence or

similarity guide the folding constructions. We observe > symmetry concepts

already in the maps of second graders, and third and fourth graders show by

their maps how they try to > support spatial reasoning of the acceptors of

their maps.

(The objects and facts marked by > are introduced during the workshop.)

References

Fusé, Tomoko (1992): Unit Origami, Multidimensional Transformations.-

Tokyo, New York: Japan Publications INC. 1992.2, 1990.1, ISBN 0-87040-

852-6

Gibbs, W. (1994): Polyhedra from A-sized Paper.- Mathematics in School,

September 1994

Macchi, Pietro; Scaburri, Paola (1997): Nuovi Origami.- Milano: Giovanni

De Vecchi Editore 1997, ISBN 88-412-4635-9

Momotani, Yoshihide (1990): Origami Architectures.- printed in Japan:

1990, ISBN 4-416-39014-9

Müller, Gerhard; Wittmann, Erich (1995): Das Zahlenbuch 3.- Stuttgart:

Klett 1995

Wollring, Bernd (1997): "Man darf nicht immer gleich aufgeben, wenn's mal

nicht klappt." Mädchen und Jungen bauen gemeinsam Würfel aus gefaltetem

Papier.- Sache-Wort-Zahl 25 (1997), Heft 7: "Junge und Mädchen", S. 25 -

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