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14. Define overspecializing. List an example that you might see from a student. The process of deriving conclusions that are restricted inappropriately. This may happen because students focus on irrelevant attributes of a limited number of instances. Students who know that to add or subtract fractions you have to have a like denominator believe that the same has to be true for multiplication and division of fractions.
Chapter 2 Error Patterns
A-W-1
-When Gary is using his fingers to count, he is counting one of his fingers twice.
Conceptual
Number sense
One-to-one correspondence
Rational countingProcedural
Addition
One-to-one countingBasic Fact
Doesnt know addition
-To help Gary you could have him make sets of marks for each addend.
A-W-2
-Mike would add the numbers together and record the answers with no regard to place value and without regrouping.
Procedural
Double digit addition
Regrouping
Conceptual
Double digit addition
Number sense
Place valueBasic Fact
No problem
-To help Mike you could use bundles of 10 and single sticks and have him show both addends and then make a 10 to help him understand regrouping.
A-W-3
-Mary is adding from left to right and regrouping in that order also.
Conceptual
Multi-digit addition
Regrouping/reminder
Place value
Number senseProcedural
Multi-digit addition
Regrouping/reminderBase Fact
No problem
-You could have Mary use base ten blocks to get her answer. Have her show both addends and then show the answer with as few blocks as possible.
A-W-4
-When there is a multi-digit number on top and a single digit number on bottom Carol add them all together as if they were all units.
Conceptual
2 by 1 addition
Place value
Regrouping/reminder
Number senseProcedural
2 by 1 addition
Regrouping/reminder
Basic Fact
No problem
-You could have Carol draw a line to separate the tens and units that way she could see which number is being added to which.
A-W-5
-When Dorothy adds the numbers in the tens column she adds in the one-digit number again with it.
Conceptual
2 by 1 addition
Number sense
Place valueProcedural
2 by 1 additionBasic Fact
No problem
-To help Dorothy you could have her label units and tens and you could also have her use graph paper to write the problem in so that she can see what needs to be added together.
S-W-1
-Cheryl is not looking as the numbers as a whole subtraction problem but looking at each digits as its own problem so she is just taking the smaller number from the larger number.
Conceptual
Multi-digit subtraction
Place value
Number sense
Reminder/regroupingProcedural
Multi-digit subtraction
Reminder/regroupingBasic Fact
No problem
-Have Cheryl show you the number altogether with base ten blocks so the she understands that it is a whole number not separate numbers and then have her continue solving the problem using the base ten blocks.
S-W-2
-George seems to overgeneralize and renames in a subtraction problem even when it is not necessary.
Conceptual
Reminder/regrouping
Place value
Number sense
Multi-digit subtractionProcedural
Reminder/regroupingBasic Fact
No problem
-To help George you could use base tens blocks to show the minuend and then ask questions such as can I take away so many units without trading and how do I know when I need to trade and when I do not need to.
S-W-3
-Donna writes 0 for the missing digit whenever she is subtracting a number and zero rather than writing the number down.
Conceptual
Zero property
Reminder/regrouping (?)
MDS with identity
Number sense with zero(?)Procedural
MDS with identity
Reminder/regrouping(?)Basic Fact
No problem
-Give Donna a calculator and have her try several addition problems, multiplication problems, and subtraction problems with zero and then have her come up with a rule for each and then have her compare the rules to see how they are alike or different.
S-W-4
-Barbara is having a problem with regrouping when you have to regroup more than once in a problem.
Conceptual
MDS with regrouping
Reminder/regrouping
Number senseProcedural
MDS with regrouping
Reminder/regroupingBasic Fact
No problem
-Teach both situations, one column at a time so that Barbara is aware of when regrouping is necessary and when it is not.
M-W-1
-When Bob multiplies he is using his remainder from the ones when he multiplying the tens also.
Conceptual
Reminder/regrouping
Multi-digit-multiplication
Place valueProcedural
Reminder/regrouping
MDM
Place valueBasic Fact
No problem
-To help Bob have him make two problems with the ones and tens and have him work each of them. Have him remember the reminder in his head instead of writing it down.
M-W-2
-Joe is adding the reminding number to the tens before he is multiplying.
Conceptual
Reminder/regrouping
MDM
Number sense
Place value
Oder of operationsProcedural
MDM
Reminder/regrouping
Order of operationsBasic Fact
No problem
-Have Joe write the reminder below the bar as a small number to remind him to add before recording a product.
M-W-3
-Doug uses a mixture of multiplying and the addition algorithm.
Conceptual
MDM
Reminder/regrouping
Number sense
Procedural
MDM
Order of operations
Reminder/regrouping (overgeneralizing)
Distribution propertyBasic Fact
No problem
-You could have Doug use a calculator to compute each partial product and make sure the correct values are multiplied.
D-W-1
-Jim thinks as each digits as ones and just thinks they have to be divided and divides the greater of the two numbers by the lesser and records the result.
Conceptual
1 into 3 division
Place value
Number senseProcedural
Division algorithmBasic Fact
?
-To help Jim you could use manipulatives and teach the algorithm as a step by step record of activities.
D-W-2
-Gail records her answer from right to left instead of left to right.
Conceptual
Place value
Number sense
Multi digit division
Overgeneralizing multiplicationProcedural
Division
Basic Fact
no problem
-To help Gail have her estimate the quotient before computing the problem so that she can see if her answer she gets is close to what she estimated and if not she can go back and rethink what she did.
D-W-3
-If there is a zero in the tens place and John cannot divide he goes to the ones place but he does not record a zero to show that there are no tens.
Conceptual
MDD
Place value
Number sense
Procedural
DivisionBasic Fact
No problem
-Have John use graph paper when solving so that he can see when one digit is missing and hopefully will understand that there has to be a number in that spot.
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