Every Monday for the past 7 weeks I’ve been attending a teaching credential class where I’m learning how to teach elementary students math. The class is focused on teaching students math through problem solving instead of introducing the algorithms first. This allows students to come up with inventive ways to solve a problem and find the answer. It’s a very interesting strategy that I’m trying to incorporate into my daily teaching, but sometimes I find it difficult since it’s not the way I learned math.

However, I’m learning some alternative ways to solve everyday addition and subtraction problems; new algorithms and methods that I never realized before. One of these is the subtraction algorithm called “borrow and pay back.” It’s the way many European and Spanish speaking countries solve their subtraction problems, and is similar to our borrow and regroup method of subtraction. The basis for how we learned math is ingrained in our behaviors, and this difference in a simple algorithm may be one cause of our problem with debt.

Before the 1930’s, the United States also solved subtraction problems using the borrow and pay back method and this was the method that was taught in school. It was changed due to a study that made students catch on to subtraction quicker. However, if you’ll note the names of both methods, there is a subtle difference that reflects our changing psychology of that era: borrow, borrow, borrow (also called the decomposition method). Below is a comparison of the two subtraction algorithms. They are quite similar, except in the borrow and pay back method, the numbers retain their place value. In our current borrow and regroup method, the numbers are isolated. Many students struggle with subtracting across zeros because of this.

Comparing Subtraction Algorithms
Comparing Subtraction Algorithms

Looking at the borrow and pay back method (also called the equal additions method or Austrian method), you may have been confused as I was. But once it is explained, it makes much more sense. The algorithm is basically “borrowing” a ten from one place value, and “paying it back” in the other number in the place value next to it. Our current algorithm we use today borrows and regroups. There isn’t any paying back going on. Does this sound familiar?

So was this change in a subtraction algorithm also a change in the way we viewed money and credit? After doing a little more research I found some information on the study that changed the way we subtract, but not a clear understanding of why it caught on so quickly. The Brownell study in 1937 found that students using the “crutch”, or current subtraction algorithm that uses the decomposition method with crossing out the numbers going across, were able to solve subtraction problems quicker and more accurately than not using the crutch. The study only focused on the decomposition algorithm.  However, once the students learned how to complete the algorithm, they were no more proficient than students learning the traditional way. Basically students could learn how to solve a subtraction problem quicker, but they didn’t necessarily understand why they were borrowing  or regrouping the numbers.

It’s interesting that this new algorithm became popular around the same time as the Great Depression. Could the adults in charge of the Brownell study have been thinking differently about math, numbers, and money due to the financial stress and pressure of that decade? Why did the Brownell study only focus on the decomposition algorithm when using the “crutch”? And Why did it catch on so quickly? Many educators at the time disputed the study and many today feel the borrow and pay back method is much more efficient. Yet, textbooks today have continued to teach subtraction using the decomposition method with the “crutch.”

In my personal opinion, I think there was definitely a change in the way people viewed money and borrowing and this allowed the decomposition algorithm to catch on so quickly. The idea of a “crutch” or trick to solve the problems faster also helped speed up its popularity.

What are your thoughts? Did you learn subtraction through the borrow and pay back method or through the decomposition method? Do you think I might be on to something about the changing views of money, credit, debt, and borrowing? Or am I stretching this a bit?


  1. Stay at Home Mom CFO Reply

    Little House that is so cool (the methods and a female Math teacher)!!! I never heard of the borrow and payback method and it is so interesting how it plays psychologically. (Great tie-in with the great depression too.) I would love if my kids were taught the payback method instead of the current method. Do you know which method the Asian countries who kick our butt in Math use?
    .-= Stay at Home Mom CFO´s last blog ..Don’t Count Your Money Until it’s “There” =-.

    • @SAHMCFO – I’m not sure which method the Asian countries use, but I do know they are being taught math through problem solving. Instead of just figuring out calculations, they are being taught to use their basic number sense to invent strategies to solve word problems. In the end, these students learn how to do the computations though a variety of methods. They are taught the algorithms, or steps, last. If you think about it, it makes sense. Most standardized tests focus on word problems, not just computations.

  2. @Debra – When I first looked at the two math problems, I didn’t get it either. My math professor explained it step by step, then I got it. Basically you add a 10 to the top number if it is a number you can’t subtract from. By adding the ten to the top number, you add a ten to the bottom one as well. If you add a ten to 39, for example, that number becomes 49. I hope this makes more sense! 🙂

  3. I learned the borrow and regroup method. I didn’t know there was another method! Interesting!
    .-= Mrs. Accountability´s last blog ..APS Offers $30 Rebate On Your Old Fridge =-.

  4. Very interesting! I also didn’t know about borrow-and-payback method. Although I’d take straight-up algorithms over problem-solving learning any day.
    .-= RainyDaySaver´s last blog ..Fix-It Friday: Light Switches =-.

  5. Thanks! I was looking up the method that my 5th grade teacher taught me in the 70’s when I couldn’t for the life of me, get the regrouping method. I am now homeschooling my kids, and this is the method that I teach them. I was looking into this for some other parents whose kid’s weren’t getting subtraction either – you explain very well how to do it. Not sure if this plays into the money aspect of the culture as a whole, but maybe that is why it makes sense to me – I am a saver not a spender!

  6. Michael J. Read Reply

    Hi : I use the Vedic method for subtraction and can do any length subtraction problem from left to right.

    Email me at mread@csumb.edu and/or skype me at mjohnread to discuss it 🙂

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