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Overview of use of MAB |
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How to convince students that a large cube can represent 1 |
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Activity: Column Overflow |
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Why we prefer LAB to MAB |
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Overview of use of MAB
Base 10 MAB are found in many primary schools and are commonly used in teaching the representation of and operations with whole numbers. To use them to represent decimals, students need to be convinced that blocks do not always have the same value. For example, the large cube may represent 1, rather than 1000. (See below for how this might be done). MAB can then be used in teaching the representation of and operations with decimal numbers. Activities such as making and naming numbers, ordering by size, estimation, addition, subtraction, multiplication and division (as repeated addition and subtraction) in the new decimal number realm can be given a concrete embodiment. This assists in avoiding just giving lists of rules for dealing with decimals.
Some students may think it is babyish to use MAB for decimals because they have used them in younger years. In this case, consider using Linear Arithmetic Blocks (LAB) or Area Cards instead. They look different but can do the same things.
Instructions on using MAB correctly in teaching whole number operations (addition, subtraction, multiplication and division) can be found in standard primary mathematics education textbooks (Booker et al, 1997; English and Halford, 1995).
Concrete materials such as MAB are of great assistance to demonstrate WHY addition and subtraction algorithms work and the meaning of notation. However, very careful attention must be given to making the link between the concrete and the symbolic. Teachers need to be very careful that children see the parallels between moving the blocks and carrying out operations on numbers.
It is appropriate to discuss (before or after starting to introduce decimals) that the blocks can be grouped together to make new and larger units. For example, 10 large cubes can be put together to make a superlong, 10 of these would make a superflat and 10 of these would make a supercube, and that the process need never stop!
Building up these base ten links between columns is a crucial conceptual underpinning for whole numbers and for decimals.
How to convince students that the large cube can be used to represent one
Display one of each piece of MAB in a row (largest on left) with labels on cards (block, flat, long, mini or whatever other names are familiar). Start from the large block and discuss the shape that results when it is cut into ten pieces (i.e. flats) and then the shape that results when the flat is cut into ten pieces (i.e. longs) and the result of cutting a long into ten pieces (i.e. minis). Explain that you want to continue this process, and wish to cut the mini into ten pieces. Using a cube of clay the size of the mini, start to cut the clay with a knife, at the same time asking for predictions for the shape of the result (i.e. flat); suggest miniflat as an appropriate name. Use tweezers to hold the clay and ask what shape will result when the miniflat is cut into ten pieces (i.e. long); suggest minilong. Finally, try to illustrate that the result of cutting the minilong into ten pieces will be a very small cube; suggest minicube (or tinimini). Explain to the students that as this is hard to visualize with such small pieces, they need to imagine that they have become the size of a Borrower (anyone seen the movie?). Ask them to close their eyes as you remove the clay and then display the original MAB pieces but with their new labels - miniflat, minilong and tinimini.
Activity: Column Overflow
This activity addresses common misconceptions about the place value of decimal numbers.
Use the large block to represent the number one. Ask students to make 14 hundredths from blocks. Show them the following 7 numbers (possibly written on individual cards) and ask them to choose decide which is 14 hundredths: 1400, 140, 14, 1.4, 0.14, 0.014, 0.014, 0.0014. Repeat with 14 thousandths and the same numbers. Then encourage discussion about common mistakes. Someone may have chosen 0.014 (for 14 hundredths) and 0.0014 (for 14 thousandths). Most classes will contain at least one or two of these column overflow thinkers who use zeros to indicate the empty columns and then squash the number 14 into the relevant column. A few students may pick 1400 for 14 hundredths as would the reverse thinking student in the Case Studies.
Why we prefer LAB to MAB
Our research has demonstrated that LAB has a number of advantages over MAB.
Our published research paper (Stacey et al, 2001) compares the two materials on the basis of epistemic fidelity (how true the model is to the mathematical principles involved) and accessibility for students. Two teaching experiments involving 30 matched students indicated that LAB is considerably more accessible for students. There are three reasons for this:
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students get confused with MAB simply because it has been used before with the "mini" representing one (see above); |
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LAB models number with length whereas MAB models number with volume and many students in upper primary do not yet have a strong grasp of volume; |
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the various pieces of MAB seem to be of different dimensions (1-D, 2-D, 3-D ) and this makes generalizing to more place value columns difficult. |
Use of LAB was associated with more active engagement by students and deeper discussion. Epistemic fidelity is critical to facilitate teaching with the models, but we attribute the enhanced classroom environment to the greater accessibility of the LAB material.
Both models have excellent epistemic fidelity, so that they both show how the size of numbers depends on the digits and the place value columns and they both can be used to demonstrate the operations. However, a significant difference between the two models is that LAB, with pieces laid end to end, has structural similarity to the number line, and is thus better able than MAB to model number density (the property that between any two decimals, a third decimal can always be inserted). LAB is therefore better able to demonstrate the principles of rounding. The limitations of MAB in regard to the continuous properties of decimals were noted by Hiebert, Wearne and Taber (1991). Following instruction with MAB, Year 4 students' performance on discrete-context tasks (e.g. writing the number represented by a picture of MAB; choosing the larger of two decimals) improved. However their performance on continuous-representation tasks (e.g. shading 2.6 of a continuous quantity; finding a number between two decimals) did not. Thus MAB appeared to support understandings of decimals as discrete quantities but not as continuous quantities.