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chapter11 manipulatives

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    Page 11.1 1990 Center for Innovation in Education, Saratoga, California

    Mathematics Their Way Summary Newsletter

    CHAPTER 11:

    PLACE VALUE

    A firm u nd erstand ing of place value is a pr erequisite for all work

    in arithmetic. Stud ents who do n ot und erstand the concept of

    place value can not p rogress through the four basic operations

    without difficulty. They learn the operations by memorizing an

    increasing n um ber of seemingly u nrelated facts and procedures.

    Even the most capable students become confused w hen the load

    becomes too great, usually at the point w hen they mu st learn

    mu ltiplication and division. Mathem atics becomes increasingly

    mysterious for these students, wh o have little hope of und erstand-

    ing the conten t of second ary-level mathem atics courses. It is

    essential that p lace value be given ma jor emp hasis in the primar y

    grades and that students have frequent experiences with m anipu-

    lative materials that demonstrate place value.

    1985 California Mathematics Framework, p.23

    In order to effectively implem ent the place value objectives outlined in

    the 1985 California Mathem atics Framew ork (above) in the p rimary

    grades, the m athematics curriculum mu st include place value experi-ences that:

    t use d evelopm entally app ropriate manipu latives.

    t use a var iety of bases to teach the place value system.

    t explore place value patterns (linear and m atrix).

    t provide regroup ing and non -regroup ing multi-digit addition and

    subtraction experiences simultaneously.

    INTRODUCTION

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    Chapter 11: Place Value

    1990 Center for Innovation in Education, Saratoga, CaliforniaPage 11.2

    PLACE VALUE MANI PULATIV ES

    Place value manipulatives fall within two distinct developmental

    levels: concrete and repr esentational. Concrete place value experiences

    mu st come first. Represen tational activities are appr opr iate later in the

    elementary grades after children h ave engaged in a v ariety ofexperiences with concrete mater ials.

    Concrete MaterialsConcrete materials beans and cup s and U nifix cubes allow th e

    children to construct grou pings. Experiences with th ese materials

    stress one-to-one correspondence between the n um ber and the mate-

    rial it represents.

    Kathy Richardson addresses the issue of one-to-one correspondence at

    the p lace value level in her book,Developing Number Concepts Using

    Unifix Cubes (Chapter 5, p. 133):

    Coun ting grou ps [of objects] requ ires a different kind of thinking

    from counting single objects. Childr ens first counting experiences

    require an un derstanding of one-to-one correspondence. They

    learn that one nu mber w ord goes with one object. But, wh en

    dealing with nu mbers above ten, children are required to count

    group s as thou gh th ey w ere individual objects. The question

    How m any tens in thirty-four? assum es the child can conceive of

    ten objects as one entity. The qu estion H ow many hun dred s in

    346? assum es the child can conceive of one hun dr ed ob jects as

    one group (all the while remembering that each h und red is also

    ten group s of ten).

    Representational MaterialsBean sticks and mu lti-base blocks are fixed mater ials. The mater ials

    impose a ready-mad e structure on the stud ent. Although children

    man ipu late materials e.g., chips, m ulti-base blocks and bean sticks

    they are requ ired to think abstractly. A child with insu fficient

    experiences using concrete materials may be confused w hen a nu mber

    is represented by m ulti-base blocks. They must already un derstand

    that one block in base ten represents either ten flats, a hu nd red longs,

    or a thousand un its. Rather than u nderstand ing that a num ber like 362

    is represented by three flats, six longs, and two u nits, a child m ay only

    see eleven separate blocks of varying sizes.

    Chip trading is also representational but at a m ore abstract level.

    Children mu st be able to und erstand the abstraction that one chip

    could represent a group of ten or a h und red chips (chips can also

    repr esent grou pings in other bases). The necessity for children to be

    surroun ded with an abund ance of concrete experiences before they

    move on to representational materials cannot be emp hasized strongly

    enough.

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    Page 11.3 1990 Center for Innovation in Education, Saratoga, California

    Mathematics Their Way Summary Newsletter

    Choose AppropriateManipulatives

    Because mathematics is made by human beings and exists only in

    their minds, it must be made and remade in the mind of each

    person w ho learns it . In this sense mat hemat ics can only be

    learned by being created.

    Author Unknow n

    Teachers often overlook the limitations set by representa tional materi-

    als. One Math Their Way teacher told how she mad e bean sticks with

    her second g rad e class one year. The class had experienced p lace valueactivities earlier in the school year an d in the first grade, using concrete

    materials. Once the sticks and rafts were constru cted, the children u sed

    them to m ake add ition and subtraction p roblems. The children loved

    the bean sticks. They d idnt app ear to have an y p roblems w ith the

    fixed arrangement of the sticks and rafts.

    The teacher kept the bean sticks and used them w ith classes after that

    year. The new second gr aders didn t seem to relate to the materials in

    the same m ann er as the first group. In retrospect, the teacher felt the

    reason the bean sticks were more su ccessful the first year was because

    the children participated in the construction of the sticks and rafts.

    They und erstood wh at the beans on the sticks and rafts represented.

    Even though they had similar classroom experiences to prepare themfor the activity, the children who used the pre-assembled materials the

    following years d idnt experience the creation of the sticks and rafts.

    The pre-assembled (representational) materials imposed a structure

    that subsequent classes of second grade children w ere not ready to

    accept.

    Be careful not to move p rematu rely to the representationa l level.

    Allow children time to construct their ow n u nd erstanding of place

    value by engaging in experiences with an assortm ent of concrete place

    value materials.

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    Chapter 11: Place Value

    1990 Center for Innovation in Education, Saratoga, CaliforniaPage 11.4

    USE A VARIETY OF BASESPlace value is merely a system of organ izing large numbers. Teachers

    tend to think th at the base ten system is the only base we use in the

    real world . For this reason, they often question wh y other bases are

    taught in the lower grades. Its true that ou r society frequently uses

    base ten, but we also use other bases every day without th inking about

    it. Every time w e bake, sew, measu re with a r uler, or tell time, we areworking w ith other bases. The comp uter op erates on a binary system

    or base tw o. Yet, textbooks tend to teach isolated p lace value skills

    (e.g., borrow ing and carrying) in base ten only. Children w ill develop

    a better und erstanding of place value if they are allowed opp ortunities

    to compare and contrast relationships (using real materials and real-

    life experiences) in other bases.

    TEACH REGROUPING AND N ON-REGROUPINGADDITION AND SUBTRACTION TOGETHER

    The 1985 California Mathematics Framework, p. 22, emph asizes the

    importance of teaching m ulti-digit add ition an d subtraction in a w aythat stud ents und erstand the p rocess, rather than viewing each algo-

    rithm as a sepa rate, isolated function:

    The p ractice of a skill in isolation is seldom effective in developing

    the u nd erstanding required to make the skill useful. Instruction in

    computational algorithms should emph asize und erstanding the

    procedures that are being u sed.

    For examp le, it is now com mon to teach tw o-place add ition and

    subtraction w ithout regrouping before introducing regrouping.

    This app roach leads stud ents to focus on separate procedures and

    hinders their un derstanding of the basic operations. Most students

    view a problem such as as two one-place problems pu shed

    together ( and ) and do not think of add ing 40 and 20.

    Because stud ents get a great d eal of practice in this kind of prob-

    lem, they assume that can be calculated in the same way and

    see nothing w rong w ith the answ er of 612. After learning a new

    procedur e in which 1 is carried or p ut on top of the tens colum n

    every time, students are confused w hen p resented w ith a mixture

    of problems that requ ire regroup ing and problems that do not. The

    child w ho asks, Do I regroup on this problem? has almost no

    und erstanding of the concept of two-digit ad dition or subtraction.

    This d ilemma can be avoided by teaching mu lti-digit ad dition and

    subtraction w ith and w ithout regrouping simultaneously, using

    manipulative place value materials, and relating the process torealistic situations. When manipu lative materials are used, it is

    easy to demonstrate when regroup ing is needed; and th ere is no

    need to teach regrou ping as a separate algorithm.

    4+2

    3+5

    43+25

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