I kinda lament the fact that kids have a hard time getting to grips with mathematical logic. However the following two examples of questions and answers, together with the corrections and explanation the teacher who is marking the answers is giving leave one even more sad for the fact that the wording of the question is confusing, and at worst demoralising when the student recieves the test back.
Exhibit A
The pupil's answer to us mere mortals is right and logical. However the marking which seems to have deducted a point says something different.
I think therefore, it all hinges on the word 'reasonable'. This word is code for a guess, hence the teacher's comment about 'estimate'.
Exhibit B
5 x 3 = 15 , which the student puts as 5 + 5 + 5, which is factually true. However, the teacher deducts a mark as it is wrong and writes instead 3 + 3 + 3 + 3 + 3 instead.
What? My only guess is in the way 5x3 is formed. If we substite 3 with a letter Q say, we have 5xQ=__ as the question. I common algebra, 5xQ is written as 5Q, so no matter what value the variable Q is, it must be multiplied five times. (Online discussions about this ventured into the realms of matrices, but you really don't need to go that far to see why)
So, 5xQ can be written Q + Q + Q + Q + Q. Inserting back the value 3, we have what the teacher wrote.
Ah, but what if the question were 5x0=__ this would still apply
0 + 0 + 0 + 0 + 0
= 5 x 0
= 0,
however, you couldn't write the same using as an addition of fives to sum up to zero in the same way.
This follows in question 2 where again, the pupil got it wrong. The pupil had to draw the array 4 x 6 and gets marked down. Using the 4 x Q =
Q + Q + Q + Q
each Q is now a row of six dashes, and forms 4 rows
' ' ' ' ' '
' ' ' ' ' '
' ' ' ' ' '
' ' ' ' ' '
Despite the fact that the commutative properties of 4x6 = 6x4, what is sought here is the rigid visual interpretation of the statement 4 x 6 rather than the answer which is 24.
It may seem incredulous, but there is a method in this madness, the way you set out a problem and how it is interpretated by the student will affect the outcome in later more advanced subjects like the afore mentioned matrices...
Exhibit A
The pupil's answer to us mere mortals is right and logical. However the marking which seems to have deducted a point says something different.
I think therefore, it all hinges on the word 'reasonable'. This word is code for a guess, hence the teacher's comment about 'estimate'.
Exhibit B
5 x 3 = 15 , which the student puts as 5 + 5 + 5, which is factually true. However, the teacher deducts a mark as it is wrong and writes instead 3 + 3 + 3 + 3 + 3 instead.
What? My only guess is in the way 5x3 is formed. If we substite 3 with a letter Q say, we have 5xQ=__ as the question. I common algebra, 5xQ is written as 5Q, so no matter what value the variable Q is, it must be multiplied five times. (Online discussions about this ventured into the realms of matrices, but you really don't need to go that far to see why)
So, 5xQ can be written Q + Q + Q + Q + Q. Inserting back the value 3, we have what the teacher wrote.
Ah, but what if the question were 5x0=__ this would still apply
0 + 0 + 0 + 0 + 0
= 5 x 0
= 0,
however, you couldn't write the same using as an addition of fives to sum up to zero in the same way.
This follows in question 2 where again, the pupil got it wrong. The pupil had to draw the array 4 x 6 and gets marked down. Using the 4 x Q =
Q + Q + Q + Q
each Q is now a row of six dashes, and forms 4 rows
' ' ' ' ' '
' ' ' ' ' '
' ' ' ' ' '
' ' ' ' ' '
Despite the fact that the commutative properties of 4x6 = 6x4, what is sought here is the rigid visual interpretation of the statement 4 x 6 rather than the answer which is 24.
It may seem incredulous, but there is a method in this madness, the way you set out a problem and how it is interpretated by the student will affect the outcome in later more advanced subjects like the afore mentioned matrices...