Gulnar has an average score of 87 after 6 tests. What does Gulnar need to get on the next test to finish with an average of 78 on all 7 tests?
The website provides an option of getting hints. This was the first thing I looked at.
Hint 1: Since the average scores of the first 6 tests is 87, the sum of the scores of the first 6 tests is 6*87=522
So far so good. But the next hint appears to be designed to force the student back on the “known” method of solution :
Hint 2: If Gulnar gets a score of x on the 7th test, then the average score on all 7 tests will be :
Followed by a more logical and timely
Hint 3: This average needs to be equal to 78 so:
(522+x)/7 = 78
and by the answer in the next hint:
And now I would like to make a few comments about these hints.
The Gulnar problem is a classical three steps problem of elementary school arithmetic (more precisely, two-and-half steps problem, because the two first steps are almost identical, as I will show you soon; in teaching, it is useful to emphasise the benefits of re-use of the same step). It is a pleasure for me to revisit the art of “questioning”, the key ingredient of solving word problems as it was taught in Russian elementary schools of my time:
Question 1. How many points in total Gulnar got in 6 tests?
Question 2. How many points in total Gulnar needs to get in 7 tests?
Answer: 7*78 = 546
Question 3. How many points Gulnar needs to get in the 7th test?
Answer: 546 – 522 = 24
In the encapsulation/de-encapsulation terminology, what is needed for solving the problem (and therefore what is interiorised in the student’s mind) is
- de-encapsulation of the concept of an average, and
- reversing the operation of forming an average.
Mastering this two two mental actions is needed not so much for further development of the theme which uses “an average” as a starting point, but for better understanding of the very concept of an average. In my opinion, this key methodological point appears to be missed by the writer of hints provided on the Khan Academy website.
Please notice that in the Khan Academy’s hints, Hint 2 is in itself a multi-step problem. Most likely, it is rooted in the material which is marked as prerequisite for “Average word problems”, namely “Systems of linear equations” preceded by “Linear equations 3″ preceded by “Linear equations” 2 and 1 preceded by “Adding and subtracting fractions” and “Dividing fractions” and so on up to “Addition 1″. In my opinion, the nature of hints restricted to recently learned material suggest a strict modular structure of material. As my solution shows, a synoptic approach with back references to much earlier material (basic subtraction and multiplication) could be didactically more useful.
And my last comment: when I was a child, I was taught to start solving problems like that by asking questions (which I would describe in my adult parlance of nowadays as meta-questions):
Question 0′. ”Gulnar has an average score of 87 after 6 tests. ” What questions can be asked about these data?
Question 0”. ”Gulnar needs to get an average of 78 on all 7 tests“. What questions can be asked about these data?
Basically, I and my peers were taught to do logical analysis of data (and of data structure). This explains the great didactic value of word problems.
Of course, I have to add a disclaimer: it is wrong to judge a systematically developed learning material after looking at the very first randomly chosen exercise. I would not write a similar post about a random exercise in a textbook (I mean, a real textbook, printed on paper). Almost by default, textbooks are linearly structured. On the contrary, the highly modular structure of the Academy’s website makes it open to assessment of randomly chosen nodes in its intricate dependency of topics flowchart (“Knowledge map”).