Subtraction Simplified: Unveiling the Standard Algorithm

Introduction:

Introduction to Subtraction: Exploring the Standard Algorithm

In this article, we address the misconceptions surrounding the teaching of standard arithmetic algorithms in math education. While some argue that progressive reform math programs neglect these algorithms, we argue that it is crucial to consider the relevance and effectiveness of these methods in today’s technological age. Furthermore, we discuss a historical perspective on the “equal additions algorithm” for subtraction, shedding light on alternative methods that have been used in the past.

Full Article: Subtraction Simplified: Unveiling the Standard Algorithm

The History of the Standard Algorithm for Subtraction

In the world of math education, there is an ongoing debate about the use of standard algorithms in teaching. Some argue that progressive reform math advocates and the programs they create do not adequately teach standard arithmetic algorithms. However, in actual classrooms with real teachers, the so-called standard algorithms are always taught, regardless of the program used. So, why is there so much importance placed on these algorithms, and why do some people believe they are the only ones that should be taught?

To understand this, we must first examine the idea of what makes an algorithm “standard” and whether being standard automatically means it is superior. It seems that those who cling to the idea of a standard algorithm are stuck in a pre-technological age, where manual calculations were the norm. In today’s modern world, where calculators and computers are readily available, the need for humans to be fast and expert number-crunchers is far less important.

In light of this, we must question why there is such emphasis placed on the speed and efficiency of a paper and pencil algorithm. The argument for teaching a single standard algorithm often revolves around its speed and efficiency. However, it is important to consider that if a student misunderstands and fails to use an algorithm correctly, it is unlikely that they will perform efficiently with it. In fact, a student who accurately uses a slower algorithm but understands it will likely arrive at the correct answer faster than a student who struggles with a faster algorithm.

Therefore, it is essential to find an algorithm that a student understands and can use properly. This does not mean alternative algorithms should be favored over the standard one, but rather that the standard algorithm should be taught alongside other methods, with a focus on student understanding. It is puzzling why there is a need to defend alternative algorithms when it is those who argue for a single best algorithm that should justify their narrow position.

Furthermore, the arguments against alternative algorithms often stem from a lack of historical knowledge. Take, for example, the lattice multiplication method, which has faced criticism from anti-reform groups. This algorithm has been used for hundreds of years without any negative effects. The arguments against it are based on concerns of speed and efficiency, but these concerns are not pedagogical in nature. They stem from issues related to printing technology and readability, which are irrelevant in today’s modern world.

Despite these facts, those who oppose alternative algorithms seem motivated by politics rather than logic. They refuse to acknowledge the historical basis for these methods and reject any teaching that deviates from their approved standard methods.

Subtraction Algorithms: Exploring Non-Standard Methods

Recently, attention has been drawn to a “non-standard” algorithm for subtraction known as the equal addition algorithm. This algorithm, which was commonly used in the United States until about 50 years ago, piqued the interest of Tad Watanabe, a professor of mathematics education.

To learn more about this method, a search on the internet led to an article titled “Subtraction in the United States: An Historical Perspective” by Susan Ross and Mary Pratt-Cotter. Published in THE MATHEMATICS EDUCATOR in 2000, it delves into the history of subtraction algorithms in the United States, starting from as early as 1819. Other articles, such as Marilyn N. Suydam’s “Recent Research on Mathematics Instruction” and Peter McCarthy’s “Investigating Teaching and Learning of Subtractions That Involves Renaming Using Base Complement Additions,” also provide valuable insights.

According to the Ross article, American textbooks as far back as 1819 taught the equal additions algorithm for subtraction. The algorithm involves placing the lesser number under the greater number, subtracting digit by digit, and carrying over when necessary. This method was widely used until it fell out of favor due to printing limitations and concerns about readability.

It is important to note that the arguments against these non-standard algorithms are primarily based on politics rather than pedagogical reasons. Those who oppose these methods fail to acknowledge their historical significance and the logical reasoning behind them.

In conclusion, the use of a single standard algorithm for arithmetic operations may not be the most effective approach in modern education. It is crucial to prioritize student understanding and provide them with multiple methods, including non-standard algorithms, to solve mathematical problems. By embracing a more flexible and inclusive approach to teaching, educators can better prepare students for the demands of the modern world.

Summary: Subtraction Simplified: Unveiling the Standard Algorithm

The author discusses the standard algorithm for subtraction and questions its superiority and necessity in today’s technological age. They argue that teaching alternate algorithms can be beneficial and that the focus should be on student understanding rather than speed and efficiency. The historical basis for alternative algorithms, such as lattice multiplication, is also explored, suggesting that bias and politics may be motivating the resistance to teaching these methods. The author concludes by mentioning a “non-standard” algorithm for subtraction called the “equal addition algorithm” that was commonly used in the US until about 50 years ago.

Subtraction FAQ

Frequently Asked Questions

What is the Standard Algorithm for Subtraction?

The Standard Algorithm for Subtraction is a commonly taught method for subtracting one number from another. It involves vertically aligning the numbers and then subtracting each corresponding digit, starting from the rightmost digit.

How does the Standard Algorithm work?

The Standard Algorithm works by subtracting each digit of the subtrahend from the corresponding digit of the minuend, carrying over any necessary borrowing. The process starts with the rightmost digits and continues towards the leftmost digits.

Can you provide an example of the Standard Algorithm in action?

Let’s take an example where we subtract 245 from 512:

       5
    -  2
   ___
       3
  -  4 
  ___    
       2
  

In this example, we subtract 2 from 5, which gives us 3. Then we subtract 4 from 1, but since we cannot subtract 4 from 1, we borrow 1 from the tens place (resulting in 11) and place it above the minuend. Now, we can subtract 4 from 11, which gives us 7. Finally, we subtract 5 from 2, but again, we cannot directly subtract 5 from 2, so we borrow 1 from the hundreds place (resulting in 11) and place it above the minuend. Now, we can subtract 5 from 11, which gives us 6. Therefore, 512 – 245 equals 267.

Is the Standard Algorithm the only method to subtract numbers?

No, the Standard Algorithm is one of the most commonly taught methods, but there are alternative methods such as the Partial Differences method and the Number Line method.

Why is the Standard Algorithm important?

The Standard Algorithm is important as it provides a systematic and efficient way to subtract numbers accurately. It also helps in building a strong foundation in arithmetic and prepares students for more advanced mathematical concepts.

Are there any tips for mastering the Standard Algorithm?

Yes, here are a few tips:

• Ensure that the numbers are properly aligned vertically.
• Start subtracting from the rightmost digits.
• Remember to borrow when necessary.
• Double-check your calculations to avoid errors.

Can the Standard Algorithm be used for subtracting larger numbers?

Yes, the Standard Algorithm can be extended to subtract larger numbers by following the same vertical alignment and borrowing principles.