A colleague asked me to have a look at mathematics in the Khan Academy. I clicked at a random exercise; it happened to be not an ““average word problem“, but a “word problem about averages”:

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 ofon the 7th test, then the average score on all 7 tests will be :x

(522+x)/7

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:

Hint 4:

x=24

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?

Answer:

6*87=522Question 2. How many points in total Gulnar needs to get in 7 tests?

Answer:

7*78 = 546Question 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”).

I would like to use your analysis to illustrate the point I have been trying to make about Khan’s materials.

Imagine this scenario:

STEP 1. A student experiences the richness of the Socratic questioning you describe with a good mentor, understands this type of problems, makes up a few, solves a few and otherwise LEARNS the two key mental actions in their application to averages.

STEP 2. The student goes about her life, which does not happen to include problems about averages for the next couple of months. When she next sees such a problem, she remembers there was a nice way of going about it, which she understood well, but not its steps.

STEP 3. The student searches Khan site like you did, arrives at the problem page, and either watches the video (the most likely scenario, by the way) or goes through hints. Maybe she goes, “Wait, what?” at Hint 2, or has to pause the video at 4:30 when it comes up, but overall, the resource jogs her memory of the wonderful past learning experience enough that the average problems (sic) make sense again. This is empowering to the student, and cheaper than involving her wonderful mentor again.

This is one scenario where Khan materials are quite useful. Another scenario is called “flip lessons” (and it’s not about rude gestures). The meta-assignment is, “Go watch a video lecture and prepare three questions about it.” Then students discuss their questions with one another and their mentor. In this scenario, videos become advance organizers for learning. I can think of some other scenarios, but these are probably the most common.

– Oh, lieutenant, I hate cats!

– My lady, you just need to know how to cook them.

We can also compare Khan Academy with AoPS, which comes with a wonderful peer and mentor support forum where very different learning scenarios take place.

By:

Maria Droujkovaon May 28, 2011at 11:30 am

@Maria: As a revision tool for standard tests the Khan Academy could perhaps be VERY good — I do not question that. But you in effect compare KA with a microwave: not very convenient for serious cooking, but great for re-heating ready meals.

The most telling facet of your comment is the “wonderful tutor” that you are referring to: in my case she was a village school teacher in Siberia who received her training in the Kyakhta Pedagogical College, in Kyakhta (find it in Wiki!). “Socratic questioning” was simply part of standard curriculum: every teacher taught the same way.

By:

Alexandre Borovikon May 28, 2011at 12:50 pm

I like the microwave analogy a lot. Reheating ready meals or thawing a raw slab is what it does well.

I also think of Khan-style materials as a technical dictionary, with similar use cases.

Kyakhta had its own Pidgin, apparently. Interesting.

By:

Maria Droujkovaon May 28, 2011at 1:09 pm

I watched a few of the Khan tutorials, but got car-sick watching the cursor haphazardly move all about the screen.

By:

loopylooon June 4, 2011at 2:56 am

Your solution was more intuitive than Salman’s algebraic one. But he was doing it for an algebra student. What would have been better is for him to show the arithmetic solution before he does the more elegant one using algebra. Most kids I taught in Algebra 2 never learned in Algebra 1 that algebra was a generalized form of arithmetic. Showing (and reminding) them how that works is very useful in teaching algebra.

-Ihor

By:

Climeguyon August 9, 2011at 10:44 pm

See Parody Critiques Popular Khan Academy Videos and John Golden’s post.

By:

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please show the steps in solving this problem rationalize the denomenator of the following problem- the cube root of 7x divided by the cube root of 5 y squared.

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