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Here is a new appraisal of the Khan Academy, the accountable of a broadly apprehend column I arise Monday about the advertising and absoluteness of the academy. You can acquisition that column here. And you can acquisition a acknowledgment to that column from the architect of the Khan Academy, Sal Khan, by beat here.

The afterward was accounting by Christopher Danielson and Michael Paul Goldenberg. Danielson holds a Ph.D. in mathematics apprenticeship from Michigan State University. He teaches algebraic at Normandale Community College in Bloomington, MN. He maintains the blog “Overthinking My Teaching” and has accounting for Connected Mathematics. As of this writing, he has three badges and 11,041 activity credibility on Khan Academy. Goldenberg holds a master’s amount in mathematics apprenticeship from the University of Michigan, as able-bodied as master’s degrees in English and cerebral foundations of apprenticeship from the University of Florida. He writes the blog “Rational Mathematics Education” and was a co-founder of the accumulation Mathematically Sane. He currently coaches aerial academy mathematics agents in Detroit.

By Christopher Danielson and Michael Paul Goldenberg

Nearly anybody believes that K-12 mathematics apprenticeship in the United States is in atrocious charge of improvement. One being whom abounding feel has congenital a definitively bigger algebraic teaching allure is Salman Khan, whose chargeless on-line library of abbreviate advisory videos has had millions of hits (170,000,000 as of this writing) and fatigued abundance of acclaim and basic from such luminaries as Bill Gates. Gates has alleged Khan, “the best abecedary I’ve anytime seen.” But we argue that, rather than revolutionizing mathematics teaching and learning, Khan’s assignment adds a abstruse brighten to a corrupt angle of teaching and acquirements mathematics. What is more, his videos acknowledge an benightedness of how we apperceive acceptance apprentice mathematics.

So accepted and abject is the accepted media abode surrounding Khan Academy that back Newsweek ran a awning adventure afresh about the top 100 agenda innovators, the catechism was not whether Sal Khan would be on the list, but whether advocate would be acclimated to call him. Abiding abundant Newsweek gushed, “Khan Academy offers an avant-garde aperture that could accommodate the American educational system.”

We accept abundant account for the declared ambition of Khan Academy — “A free, world-class apprenticeship for anyone, anywhere.” Yet, we accept some austere apropos about the affection of the apprenticeship accouterment this education. Indeed, if either of us were authoritative Khan’s instruction, we would point to some authentic and important gaps in his practice.

What do you charge to apperceive to teach?

Our appearance is that agreeable ability abandoned is bare for affection instruction. While alive mathematics is of advance all-important and contributes to authoritative acceptable decisions in a algebraic lesson, it is not sufficient. Abounding mathematics educators accent addition affectionate of ability all-important to architecture and bear affection instruction: pedagogical agreeable ability (PCK). PCK refers to ability of agreeable as it relates to teaching.

A abecedary with able PCK knows from both educational analysis and able acquaintance which algebraic account and tasks abandoned acceptance are acceptable to acquisition easy, what they are acceptable to acquisition challenging, which examples, models and metaphors will best acceptable be accessible for acknowledgment a authentic idea, and what misconceptions acceptance will accompany to tasks that will advance them to accomplish added errors aural accustomed algebraic topics.

Pedagogical agreeable ability is an important foundation for planning acquaint and for accommodation authoritative during a lesson. Contempo analysis has accent the aberration amid agreeable ability and pedagogical agreeable ability in the area of elementary mathematics. In particular, advisers are award affirmation that authentic kinds of PCK are associated with greater assets in apprentice acquirements in elementary mathematics.

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Writing equations in all forms | Standard form | Khan … | standard form khan academy

Does Khan apperceive what he needs to know?

We argue that the videos at Khan Academy are alarmingly bare of PCK. We affirmation added that (1) the examples Khan chooses arise alleged at accidental and appropriately are, conceivably unsurprisingly, generally actually poor. They are decumbent either to actualize added confusion, or to abort to abode axiological questions acceptance are acceptable to have; and (2) Khan’s explanations are frequently off ambition in acclamation acceptable apprentice questions that accomplished agents would ahead and elicit.

We action two examples. It is important to agenda that we did not accept these examples afterwards watching hundreds of Khan’s videos. Instead, we came beyond them in the action of evaluating the account of Khan’s video apprenticeship for courses we teach, and for agreeable in which we are interested. We argue that these examples are archetypal and that our appraisal applies broadly.

Comparing decimals. Decimal fractions (decimals for abbreviate — the numbers to the appropriate of the decimal point) are a awfully arduous affair in the elementary algebraic curriculum. This adversity is due at atomic in allotment to the actuality that abounding rules that administer to accomplished numbers do not administer to decimals. Conceivably the best acclaimed of these involves comparing numbers. Back two accomplished numbers accept altered numbers of digits (e.g., 435 and 76), the one with added digits is greater (and usually by a lot!) This is not authentic of decimals: 0.435 is beneath than 0.76, admitting accepting added digits.

Therefore, any attack to admonition acceptance advance their compassionate of comparing decimals needs to accord with the case of decimals accepting altered numbers of digits to the appropriate of the decimal point. The accepted American curricular treatment, in which acceptance are instructed to adjoin zeroes to the beneath decimal to adjust the cardinal of digits helps acceptance to accomplish this assignment correctly, but fosters its own misconceptions.

Khan Academy avoids this accepted curricular approach. Instead, Khan has two videos involving comparing decimals. One is blue-blooded Comparing decimals; the added is titled, Decimals on a Cardinal Line. In the first, Khan uses actually one example, comparing 45.675 to 45.645. In the second, there is additionally actually one example, comparing 11.5 to 11.7. In both cases, a apprentice can get a actual acknowledgment by applying an incorrect aphorism (such as avoid the decimal point and amusement what charcoal as a accomplished number). What is more, the accompanying account does annihilation to abode accepted misconceptions.

An important affection of Khan Academy is its online exercises, and those who avert Khan Academy adjoin critiques of its videos generally point to the contest as authoritative up for any deficiencies in the videos.

The aboriginal of the decimal allegory videos we altercate has no exercises. The additional has contest that ask acceptance to locate numbers on a cardinal line, but these numbers all accept one decimal abode and are to be placed on a cardinal band that is already subdivided into tenths. In short, the contest action no bookish accuracy and do not abode our axial concern.

A apprentice who thinks that 0.435 > 0.76 is offered annihilation in the way of alteration on Khan Academy. In fact, one of the top questions on the folio for this video (as of July 18, 2012) is “So is .02009 greater than .0207?” This is actually the array of catechism that a competent abecedary of addition needs to ahead and to answer. Khan fails to affectation it.

Equality. The according assurance (=) is acclimated in addition and algebra to announce that two things accept the aforementioned value. Acceptance generally alter the according assurance to beggarly and now acquisition the answer. In adjustment to apprentice algebra meaningfully, acceptance charge to adapt the according assurance correctly. Back fourth graders were asked to address a cardinal in the box to accomplish the cardinal book beneath true, the best accepted acknowledgment was 12.

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8 4 = ☐ 5

This represents an incorrect appearance of the according sign. A apprentice thinks, 8 additional 4 is 12, address that, again additional 5… This additionally explains addition accepted answer, 17, in which the apprentice adds all numbers in the announcement and writes the acknowledgment in the abandoned box.

Ideally, a high-quality algebraic abecedary would: (1) altercate the acceptation of the according assurance frequently and explicitly, and (2) archetypal actual use of the according sign.

Yet back Khan is teaching acceptance to accumulate in the Multiplication 4: 2-digit times 1-digit cardinal video, he multiplies 16 x 9, which leads to the ciphering 9 x 1 5 (the 5 is the ‘carry’ from 6 x 9). He notates that as follows:

9 x 1 = 9 5

In a afterwards video, Khan notates article agnate as follows:

3 x 3 1 = 10

In both cases, the characters goes unexplained. He does not actual or altercate the antecedent notation; he aloof starts autograph it differently. This seems to be because he does not apperceive that the aboriginal characters perpetuates important apprentice misconceptions. He seems to abridgement the ability of how acceptance apprentice (and abort to learn) the abstraction of equality. He knows that the aboriginal characters is incorrect, but he doesn’t apperceive (or abroad doesn’t care) that it will reinforce students’ amiss ideas. And his alertness methods or abridgement thereof don’t advance him to ahead such apprentice cerebration and/or misunderstanding.

Views of research. Some readers will actually apperceive our appraisal as animadversion Khan’s errors. All agents accomplish mistakes, afterwards all, and Khan should not, the acumen goes, be alleged out for authoritative the aforementioned mistakes we all make. There is accuracy in this altercation to be sure. But competent agents are absorbed in acclimation those mistakes and in compassionate their importance.

In particular, competent agents accept that acquirements analysis offers insights for their own teaching. Sal Khan is on almanac as absolution this research. In a contempo account he did with Harvard’s EdCast, he said the following:

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Writing linear equations in all forms | standard form khan academy

I anticipate frankly, the best way to do it is you put being out there and you see how bodies acknowledge to it; and we accept contest on our armpit too, so we see whether they’re able to see how they acknowledge to it anecdotally.

Khan will put the video out there and see how bodies acknowledge to it. He perceives this to be a bigger access than accumulation after-effects of affection analysis projects into his advisory decisions. In the age of No Child Left Behind and its authorization for “scientifically based research” as the foundation for classroom instruction, this seems lazy.

Furthermore, as acclaimed above, abounding of Khan Academy’s contest do not acknowledge the misconceptions that abounding acceptance accept about the affair at hand, e.g. decimals.


Khan has abounding fans. We are acquainted that our critiques will not amuse those fans. And at the aforementioned time, it seems bright that we are not abandoned in our concerns. Back we apprehend apprentice comments on the site, we get the all-embracing consequence that abounding acceptance — decidedly those who may accept grasped the action in chic but ambition to see the teacher’s address bargain to a short, reviewable anatomy — anticipate that Khan is a dream appear true.

But we additionally see from comments that the apprentice who comes to these videos actually absent is acceptable to leave them aloof as absent if not worse off, decidedly if there’s annihilation conceptually or procedurally arduous that they hadn’t ahead baffled or understood.

If Khan’s videos occasionally popped up in a Google search, we would be agreeable to accept him backpack on. There is lots of worse admonition accessible on the web. But Khan is hailed as “unbelievable” ( Bill Gates) and his assignment as “sparking a anarchy in education” ( USA Today) In the description of his accessible book, he is declared as accepting “established himself as an outsider, with no teaching accomplishments to tie him to burst models.”There actually are burst models in education, but there is actually no affirmation that competent ability of apprentice acquirements and cerebration is one that agents can allow to jettison.

We do not advance that Khan should accept in a abecedary apprenticeship affairs (although conceivably that is not such a bad idea). Instead, we advance that Khan Academy badly needs choir of teaching experience. Khan could tap into any cardinal of absolute networks of admirable teaching, conceivably from recipients of the Presidential Award for Excellence in Mathematics and Science Teaching or the National Board of Able Teaching Standards. He could seek these teachers’ admonition and accord on issues ample and small. A bashful but advantageous angle would be to accept accomplished agents adapt the examples he uses in his videos. In this way, he would not charge to lose his brand unrehearsed style.

Whether baby accomplish or large, we appetite Sal Khan and his funders to put their time, accomplishment and dollars to the best accessible ends, decidedly back it comes to authoritative decisions ashore in accurate, anxiously advised pedagogical agreeable ability . The Khan Academy videos as they anon abide abatement far abbreviate of this goal.


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