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| Didax "Class Ideas" Newsletter Archive |
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Welcome to another issue of Class Ideas. Let's think mathematically! Mental Math is an important skill that is getting more and more attention as standardized testing increases in the U.S. This issue focuses on developing students' Mental Math skills with an informative article from math educator Paul Swan, lots of great downloadable activity pages and some useful Internet links. And if you're an email subscriber, don't forget to check out this month's special offer.
I love to hear from you, so feel free to email me if you have any comments on this newsletter or suggestions for future ones!
Anna Mullen, Editor |
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| Mental Math Strategies, A Practical Explanation |
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by Paul Swan
What are mental strategies or techniques?
There are several explanations, but I like the definition offered by McIntosh, Reys and Reys (1997) who described the purpose of mental or thinking strategy as turning a "calculation we cannot do into a calculation we can do by employing relationships between numbers and operations." (p. 323)
Subitising
Before discussing the development of mental strategies, it is appropriate to comment on children?s ability to subitise. Subitising is a technical term that comes from the Latin root subito, meaning suddenly or immediately. Basically, it means you have the ability to glance at a small group of objects and know how many are in that group. For example, you might look at the dots on one face of a die and realize that three dots are showing without needing to count the individual dots.
Dominoes and dot dice may be used to enhance this ability. Dominoes have the advantage of coming in double nine, double twelve and even double fifteen configurations. A simple game for developing this instant recognition involves rolling two dot dice and then matching them to the equivalent domino piece. For example, if a 5 and a 3 are rolled, the child would need to look for the corresponding 5, 3 domino piece.
Later, a numeral die may be substituted to encourage the matching of numerals with dots. Eventually, two numeral dice may be used.
Counting Strategies
Young children primarily rely on counting strategies. For example, when adding 3 and 5, some children will count 1, 2, 3, 4, 5, 6, 7, 8 using their fingers to keep track. Other children will use a more sophisticated strategy, counting 3, 4, 5, 6, 7, 8. An even more sophisticated strategy would involve counting up from the larger number 5, 6, 7, 8. This strategy relies on the child understanding that the result is the same even if you change the order of the addition.
Some children may become overly reliant on counting strategies, which can cause them difficulties later. For example, a student using a counting up strategy to add 28 and 27 would likely lose count or miscount. Another problem is that children can over-generalize a particular strategy. In the first addition example, the commutative property of addition was applied; that is 3 + 5 will produce the same result as 5 + 3. However, this property does not apply in the case of subtraction, where 5 ? 3 is not the same as 3 - 5. Examples of children employing this incorrect thinking may be seen when children start performing written calculations. When confronted with a question like 243 ? 125, rather than rename the 243 to account for the fact that five ones can not be subtracted from three ones, the child simply swaps the numbers, effectively saying, "I can't do three take away five, but I can do five take away three." This is a mistake that teachers commonly note.
As students? thinking develops they start to adopt more sophisticated strategies, such as doubling, or the use of near doubles. Bridging a ten is another important strategy that children develop to ease the burden of mental calculation. For example, when adding nine and six it is simpler to add one to nine to make ten and then add ten and five. this strategy relies upon the ability of a child to partition or split numbers up. This strategy is extremely important as it is the basis behind many other strategies. For example, in later years when required to add 25 and 27 you may choose to partition 27 into 25 + 2 so 25 and 25 and then 2 to reach an answer of 52. Alternatively, you could add 25 and 5 and then 22. Of course, you may have used an alternative strategy such as doubling to reach the same answer.
Eventually, the basic addition and subtraction facts become known and may be used to derive other facts. A child might know that 8 and 4 make 12 and use this piece of knowledge to work out the result of adding 8 and 5. The strategy of using a known fact to derive the answer to an unknown question relies on developing a bank of unknown facts. Mike Askew (1998) uses the following diagram to explain how this relationship works.
Eventually, derived number facts become known number facts and, in turn, as the range of known facts expands, so do the opportunities children have for developing facts.
Over time, children develop fluency with the basic addition and related subtraction facts. Basic addition facts run to 9 + 9 and the associated subtraction facts from 18 ? 9. It is not until much later that children are ready to learn the multiplication facts, or tables as they are commonly referred to.
What can I do to build up students? mental strategies?
Basically, give children opportunities to discuss and share their mental methods in a supportive environment. A rapid-fire twenty questions at the start of a lesson does not help a child to develop mental strategies. Likewise, there is evidence to suggest that teaching children standard written algorithms too early may cause them to abandon sound mental strategies and adopt mental versions of the written algorithm. In effect, the children become worse at mental computation.
Paul Swan is a senior lecturer in early childhood, elementary and middle school mathematics. He was awarded his PhD for his research on children?s computation choices and methods.
References:
Askew, M. (1998) Teaching Primary Mathematics: A Guide for Newly Qualified and Student Teachers. Great Britain: Hodder & Stoughton.
McIntosh, A. Reys, R. & Reys, B. (1997) Mental Computation in the Middle Grades: The Importance of Thinking Strategies. Mathematics in the Middle School, 2(5), 322-327. |
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| Downloadable Mental Math Activity Pages |
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 This month's downloadable activity pages come from our two World Teachers Press Mental Math series, Daily Mental Math and Mental Math Workouts. These popular books offer activities in a range of grades, from 2 to 9. Follow the link and I'm sure you'll find some useful pages for helping your students develop stronger Mental Math skills. |
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- Download pages from Daily Mental Math, Grade 2
- Download pages from Daily Mental Math, Grade 3
- Download pages from Daily Mental Math, Grade 4
- Download pages from Daily Mental Math, Grade 5
- Download pages from Daily Mental Math, Grade 6
- Download pages from Daily Mental Math, Grade 7
- Download pages from Mental Math Workouts, Grades 4-6
- Download pages from Mental Math Workouts, Grades 5-7
- Download pages from Mental Math Workouts, Grades 6-8
- Download pages from Mental Math Workouts, Grades 7-9
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| Mental Math Internet Links |
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 The Internet is a great resource for both lesson plans and background planning information. Here are a few websites to get you started on learning more about Mental Math and developing your students' skills. From a position paper from the NCTM and lessons to teach computation to a great article about Japanese students developing incredible mental calculation skills, there's information just a click away. |
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| December Newsletter Theme: Winter |
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 Winter is coming! Next month, Class Ideas will give you great ideas and resources to bring this fun, chilly season indoors. |
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