Integration of the Internet (and Other Resources) into Mathematics
By
Mhairi (Vi) Maeers
At the
National Council of Teachers of Mathematics Regional Conference
October 17, 2002
10:30-12:00, Glenelm Room, Regina Inn
E-Mail: maeers@uregina.ca
Phone: 585-4601
INTRODUCTION
Today we are going to explore rational numbers, examining mainly the regional or area model of common fractions at mainly the elementary grades. The focus will be on fractional relationships, comparing amounts of fractions to determine equivalency, more than or less than, and some beginning addition and subtraction.
You will be exposed today to a variety of different resources that can be used in rational number classroom learning
stations. Children can rotate through these stations at regular intervals, or they can move "in their own
time" from station to station. These stations would normally occur at the end of a unit on fractions (as
a performance assessment opportunity) as the activities, materials, and mathematics vary from station to station.
As teacher, you can observe, ask questions, mediate learning, work with the children, encourage communication
of ideas, encourage sharing, take notes of how the children are "making sense" based on what you see
them do and what you hear them say--to you and to each other.
There are many websites that include fraction activities/ideas/lesson plans. Some sites are like workbooks where all a student would do is insert the one right anwser. Other sites offer a more open-ended environment--a place to create. I have pulled together a few sites that I think are useful for children to use in the classroom as environments for fraction learning.
During and/or after each activity discuss the following with your partner:
INTEGRATION
Over the last 10 years, mainly in the faculty of Education, I have explored and conducted action research on integration. With colleagues, we arrived at three major categories of integration: object-oriented; event-oriented; conceptual. I will explain each idea.
Another whole body of research addresses the evolution of one's technological/curriulum journey--ACOT and Moersch
The ACOT project revealed that technology integration is a developmental process marked by five stages of thought
and practice among teachers (Apple Computer, Inc. 2000)
1 Entry Teachers experiment with technology for their own personal and professional use.
2 Adoption Teachers have experienced enough success to introduce technology into their classrooms
3 Adaptation Teachers try to fit technology into what they already teach, turning technology into a solution
in search of a problem. As a result, many teachers remain at this stage3
4 Appropriation Teachers begin to harness technology’s capabilities for improving the learning process.
5 Innovation Teachers become comfortable enough with the use of technology to broaden their thinking about its
potential.
Research tells us that technology integration reflects the philosophy and underlying goals of instructional paradigms.
Thus, technology use in a traditional teaching environment will simply reflect that particular set of instructional
values.
In ELE, technology will also reflect instructional values. When those values are crafted with the end in mind,
technology will be more appropriate and supportive--in other words, more engaging.
Christopher Moersch more or less comes to the same outline as the ACOT model.
The above stages addressing one's use of technology could just
as easily be applied to stages in how we use any new resource (e.g., children's literature, manipulatives etc.).
We cannot suddenly decide we are going to use pattern blocks to teach fractions if (1) we have not played with
these blocks ourselves and understand their potential, and (2) if we have not provided time for children to play
without any interference of teacher direction (other than to play).
I find in my travels that some teachers desperately want to use, for example, manipulatives but have not had sufficient
or appropriate inservice in how they can be used effectively. On the other hand there are some teachers who really
would rather NOT use them because they prefer other resources, do not see the potential, have no idea how to use
them, or have a very rigid view of the potential of these materials.
This workshop is about fractions and it includes a variety of different resources that can be used to teach fractions.
If you like, for example, pattern blocks and can see how you would structure a learning environment using pattern
blocks to enable fraction learning then try it. Likewise with any other material you will work with today.
VARIABILITY
Zoltan Dienes, many years ago outlined two major ideas that have stuck with me: Perceptual Variability and Mathematical Variability. The following table illustrates a possible way to introduce and then build on concepts. We'll pick fractions.
|
Perceptual Variability |
Mathematical Variability |
Teaching Strategy |
| one variable (e.g., pattern blocks) | one concept--equivalence | teacher-directed/whole class--everyone doing the same thing/hands-on; experiential |
| two variables (e.g., pattern blocks; geoboards) | one concept--equivalence | teacher works with one group; other group works independently |
| multiple variables (e.g., pattern blocks, play-doh, paper coverings) | one concept--equivalence | interactive--indirect teaching; small groups; stations |
| one variable (e.g., paper coverings) | two or more concepts (e.g., equivalence; comparing fractions, addition of fractions) | teacher-directed/whole class--everyone using the same material--perhaps going tthrough each concept with teacher and/or working in groups with task cards to explore the different concepts |
| multiple variables (e.g., paper coverings, games, software, children's literature, play-doh, geoboards, cuisenaire, etc) | multiple concepts (e.g., equivalence, area concept, part of a set concept, measurement concept, comparing, addition and subtraction, etc) | small groups working at stations--rotating; this is a useful strategy at the end of a unit and can be effective as performance stations |
I advise my preservice teachers that they start with one perceptual variable and one mathematical variable and
then build each as they and the students they are teaching are comfortable with both the materials and the mathematics.
I have seen many teachers use stations in mathematics and I have seen children do very interesting mathematics,
but I have also observed that some teachers have difficulty assessing the mathematics that is occurring. They
are trying to do much math at the same time and there's simply too much distraction of variables. A multi-variable
approach is great for the end of a unit for assessment purposes. For concept-building purposes I feel it is best
to work with one or related concepts and gradually build the perceptual elements.
CONCEPTUAL INTEGRATION AND PERCEPTUAL AND MATHEMATICAL VARIBILITY
In this workshop we will work with rational numbers--common fractions and we will
concentrate on exploring fractional relationships with the intention of comparing fractional amounts to determining
equivalence, and to determine more than and less than. We will also engage in some exploration of addition and
subtraction of fractional amounts. All the resources, including the Internet resources, and all the activities,
have been selected to create learning environments in which the above concepts are most likely to be explored.
For the purpose of this workshop I will assume that you all know about fractions and this workshop will thus serve
as an opportunity for me to conduct performance assessment of what you know. I have therefore included both perceptual
and mathematical variability/
A-1 Fractions with Cuisenaire Rods
1. Find All the Fractions
2. Fractional Equivalents
If the brown rod is assigned the value of 1 find all sets of rods that equal 1. Designate a different rod to be
1--or maybe use two rods together to designate 1. Now find all the fractional equivalents for this new "1."
NOTES:
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A-2 Fractions with Cuisenaire Rods
Take a hundred flat. This will now become a "1."
Your orange rods will be a fraction of "1." How many orange rods cover your 1 board?
How many white rods will cover 1?
Use combinations of the orange and white rods to create the following fractions. Read them as decimal fractions,
rod combinations, and common fractions:
1. .3 = ______ orange + _____ white
= ______ tenths + ______ hundredths
2. .24
3. .05
4. .11
5. .76
6. .19
7. .99
NOTES:
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A-3 Fractions with Cuisenaire Rods (middle years)
Again begin with a hundred flat to designate one whole. Use the hundred flat as a template on which to construct
your fractional amount.
This time use whatever combinations of rods you like to create the following fractions. Record each fraction both
as a common fraction and as a decimal.
1. .35 (1/10 + 10/100 + 3/100 + 1/50 + 7/100 + 3/100) or .1 + .10 + .03 + .02 + .07 + .03)
Now you try!!!
2. .67
3. .51
4. .48
5. .83
NOTES:
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B. Color Tiles
Use color tiles to build rectangles with each of the fractional proportions given:
1. 1/2 red, 1/4 yellow, 1/4 green
2. 1/3 blue, 2/3 red
3. 1/6 red, 1/6 green, 1/3 blue, 1/3 yellow
4. 1/2 red, 1/4 green, 1/8 yellow, 1/8 blue
5. 1/8 red, 3/8 yellow, 1/2 blue
6. 3/8 blue, 1/4 red, 3/8 green
7. 1/2 red, 1/3 blue, 1/6 yellow
8. 1/4 yellow, 2/3 green 1/12 blue
For the following problems, determine what fraction is missing:
9. A rectangle is 1/2 red, 1/3 green, 1/10 blue, and the remainder is yellow. How much of the rectangle is yellow?
10. A rectangle is 3/5 red, the rest is blue and yellow, but not in equal amounts. What could the rectangle look
like?
Write a riddle similar to # 9 or # 10 for someone else to solve.
NOTES:
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C. Pattern Blocks
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D. Geoboards
On your geoboard make a 2-dimensional shape with an elastic. With a different colored elastic try to divide your
first shape in half. Experiment with different shapes and with different fractional amounts. Make shapes such that
you can find the fractions 1/2, 1/3, 1/4, and 1/5.
Work with a partner. Make half a shape. Your partner will make the other half--with a different colored elastic.
Now your partner will make a third of a shape and you will complete the whole shape. Try other amounts--such as
1/4 of a shape, 1/5, 1/10 etc.
Again work with a partner. This time make 2/3 of a shape and your partner will complete--with a different colored
elastic--the remaining 1/3. Try starting with 3/4 of a shape, 2/9, 3/5 etc.
NOTES:
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E. Rational Numbers Websites
The following are some interesting sites for learning about fractions. Visit these sites and determine for yourself
if you think they would be effective sites for your classroom teaching/learning environment.
The following site (Math Forum: Fractions: Elementary Lessons and Materials) found at http://mathforum.org/paths/fractions/e.fraclessons.html has great resources for creating fraction learning stations. This site is a list of many sites
that could have potential as learning tools for fractions.
Find one site at Math Forum that would be useful for you
in teaching fractions. Write a few comments about the usefulness of the site you selected.
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K-grade 8 math lessons at AOL can be found at http://members.aol.com/_ht_a/iongoal/mathlessons.htm There
seems to be lots to do at this site. It appears to be mostly interactive drill and practice, but probably a lot
more fun that workbook pages.
Do one review drill. What do you think? Would this be better than or at least as good as a worksheet or page from a textbook?
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At the Internet Learning Center you'll find a variety of different learning
centers, one being fractions. The activities are very basic and similar to worksheets, but they may have some value.
http://www.geocities.com/EnchantedForest/8112/learning.html
Go the fraction part of the above site. It's quite elementary, but it might have some value for you. What do you think?
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The Fraction Shape site uses the familiar pattern block shapes to build fractions. This is a fun site that uses
a java applet interactive environment.
http://math.rice.edu/~lanius/Patterns/
Find the java applet part of the above site and "play" with the interactive pattern blocks. If the yellow block = one whole, then make the following numbers with the blocks. Beside each rational number below make a small drawing of the blocks you used.
2 1/23 2/6 5/6
1 1/32 1/6 4 1/2
3 2/31/2 4 5/6
The Math Central Resource Room database has some fraction resources.
The following is an activity which can be worked on offline--following the on-line instructions.
http://mathcentral.uregina.ca/RR/database/RR.09.95/hanson4.html
Using pattern blocks (off-line), follow the directions at the above site.
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Fraction Hotlist (from one of my graduate students)
http://www.kn.pacbell.com/wired/fil/pages/listfractionj.html
Go through the hotlist above and examine the sites my student found. What do you think of them? Are they useful?
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Pre-K-Grade 2 Fraction Websites
General
http://mathforum.org/library/levels/elem1/
Fractions
http://mathforum.org/varnelle/knum.html
The above site has some very simple rational number activities for very young children. Explore the site and see what you think.
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F. ELEMENTARY COVERING ONE KIT ACTIVITIES
(the fraction kit can be home-made from cut out paper pieces or it could be a commercial kit such as Fraction Factory)
EXPLORING ONE-HALF --1
Make one-half with your fraction kit.
Make five-eighths with your kit.
Compare your one-half with your five-eighths and answer the following questions. Discuss the questions with a neighbor
and then record your answers in your scribbler.
EXPLORING ONE-HALF -- 2
Make one-half with your fraction kit.
Make three-fourths with your kit.
Compare your one-half with your three-fourths and answer the following questions. Discuss the questions with a
neighbor and then record your answers in your scribbler.
EXPLORING ONE-HALF -- 3
Make one-half with your fraction kit.
Make nine-sixteenths with your kit.
Compare your one-half with your nine-sixteenths and answer the following questions. Discuss the questions with
a neighbor and then record your answers in your scribbler.
MIDDLE LEVEL COVERING ONE KIT ACTIVITIES
EXPLORING ONE-HALF --1
Make one-half with your fraction kit.
Make five-twelfths with your kit.
Compare your one-half with your five-twelfths and answer the following questions. Discuss the questions with a
neighbor and then record your answers in your scribbler.
1. How many twelfths equal one-half?
2. How many twelfths make one-whole?
3. How many twelfths is one-twelfth less than one-half?
4. How many twelfths is two-twelfths more than one-half?
5. When we write 1/2 = 6/12, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 5/12 equal to?
EXPLORING ONE-HALF -- 2
Make one-half with your fraction kit.
Make five-eighths with your kit.
Compare your one-half with your five-eighths and answer the following questions. Discuss the questions with a neighbor
and then record your answers in your scribbler.
1. How many eighths equal one-half?
2. How many eighths make one-whole?
3. How many eighths is one-eighth less than one-half?
4. How many eighths is two-eighths more than one-half?
5. When we write 1/2 = 4/8, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 3/8 equal to?
EXPLORING ONE-HALF -- 3
Make one-half with your fraction kit.
Make three-fourths with your kit.
Compare your one-half with your three-fourths and answer the following questions. Discuss the questions with a
neighbor and then record your answers in your scribbler.
1. How many fourths equal one-half?
2. How many fourths make one-whole?
3. How many fourths is one-fourth less than one-half?
4. How many fourths is one-fourth more than one-half?
5. When we write 1/2 = 2/4, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 1/4 equal to?
EXPLORING ONE-HALF -- 4
Make one-half with your fraction kit.
Make four-sixths with your kit.
Compare your one-half with your four-sixths and answer the following questions. Discuss the questions with a neighbor
and then record your answers in your scribbler.
1. How many sixths equal one-half?
2. How many sixths make one-whole?
3. How many sixths is one-sixth less than one- half?
4. How many sixths is one-sixth more than one- half?
5. When we write 1/2 = 3/6, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 1/6 equal to?
EXPLORING ONE-HALF -- 5
Make one-half with your fraction kit.
Make nine twenty-fourths with your kit.
Compare your one-half with your nine twenty-fourths and answer the following questions. Discuss the questions with
a neighbor and then record your answers in your scribbler.
1. How many twenty-fourths equal one-half?
2. How many twenty-fourths make one-whole?
3. How many twenty-fourths is one twenty-fourth less than one-half?
4. How many twenty-fourths is one twenty-fourth more than one-half?
5. When we write 1/2 = 12/24, what kind of fractions are these? Can you record other fractions that equal 1/2?
6. What is 1/2 + 7/24 equal to?
EXPLORING ONE-HALF -- 6
Make one-half with your fraction kit.
Make one-third with your kit.
Compare your one-half with your one-third and answer the following questions. Discuss the questions with a neighbor
and then record your answers in your scribbler.
1. How many thirds equal two-halves?
2. How many thirds make one-whole?
3. How many thirds is one third less than one-whole?
4. How many thirds is one third more than one-whole?
5. What is 1/2 +1/3 equal to?