There are a number of factors in the way that people manage debt payments that the article seems to question as illogical.
Just take a standard setup of a mortgage where we'll give a generous 7% interest rate, along with a couple of credit cards with a 15% rate and let's put some decent sized balances here for sake of comparison:
$300,000 mortgage @ 7% over 30 years
$10,000 credit card @ 15%
$20,000 credit card @ 15%
This is a random hypothetical with numbers made up out of thin air and without factoring in tax deductions. From a sheer cash flow perspective, say you have $4,000 to apply to your payments and the minimum payments are something like:
$2,500 for the mortgage
$300 for the first credit card
$500 for the second credit card
So you've got a total of $3,300 in payments with an extra $700 to use to accelerate payments that you've got to decide how to allocate between them.
If I apply it to the mortgage, this would be how the payment structure pans out over time (using Debt Repayment calculator)
$300,000 at 7% with payment $2,500
= 208 months or 11.4 years to payoff with $212,241.36 in interest (total paid $512,241.36)
$10,000 at 15% with payment $300
= 44 months or 3.5 years to payoff with $2,983.59 in interest (total paid $12,983.59)
$20,000 at 15% with payment $500
= 57 months or 4.75 years to payoff with $7,802.46 in interest (total paid $27,802.46)
Now, let's look at two strategies to applying the extra $700 cash with the mortgage vs the small debts.
First the mortgage way:
$300,000 at $3,200
= 137 months or 11.4 years to payoff with $184,138.98 in interest (total paid $484,138.98)
$10,000 at $300
= 44 months or 3.5 years to payoff with $2,983.59 in interest (total paid $12,983.59)
$20,000 at $500
= 57 months or 4.75 years to payoff with $7,802.46 in interest (total paid $27,802.46)
Now the smallest debt way:
$10,000 at $1000
= 12 months years to payoff with $1,354.08 in interest (total paid $11,354.08)
But now our formula changes because after 12 months, I now have $1,000 to apply to the next smallest:
$20,000 at $500 for 1 year (total interest $1,642.62, principle paid $4,357.38)
$15,642.62 remaining principle then at $1,500 after one year
= 12 more months to payoff (24 months total) with $2,091.00 in interest (total paid $23,733.62)
And now after 24 months I have $1,500 to apply to the mortgage.
$300,000 at $2,500 for 24 months (total interest $24,489.39, principle paid $35,510.61)
and then at $4,000 after that on the remaining $264489.39
= 86 more months to payoff with $107,086.21 in interest (total paid $442,597)
When you factor in cash flow into a total debt payment allowance you end up with an increase in applied payment if you payoff the small stuff first. The result here is a total net payment of $477,684.70 and no debt after 110 months or 9.16 years vs a total net payment of $524,925.03 and continual debt payments for 11.4 years.
EDIT: I need to go back in and adjust the mortgage way to accelerate payments more when the 2 credit cards are paid off. My bad.