The following are three examples of calculation strategies students might discover on their own if they were taught standard algorithms first and allowed to master them before moving on.
Common Core reverses the process by spending more time on these strategies than on standard math processes (see article on page 10).
Subtraction by ‘Counting Up’
The subtraction method known as “counting up” is suggested in the 2nd grade Common Core math standards: “Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.”
The counting up method used to be how cashiers made change before cash registers calculated the change due. It was also taught in schools, as evidenced by textbooks from previous eras (see Fig. 1).
Here’s how it works: If an item costs 34 cents and the customer hands the cashier a $5 bill, making change by counting up consists of first adding 1 cent and calling out “35,” then adding a nickel, then a dime (total at 50 cents), then a half-dollar (total now $1), then counting four $1 bills to bring the total to $5. The cashier has dispensed 1+5+10+50 cents, plus $4, for a total of $4.66 change handed to the customer.
I have found myself using the counting up technique in various situations. When reading about Albert Einstein recently, I noted he was born in 1879 and died in 1955. I calculated his age in my head by seeing that it takes 21 years to bring 1879 up to 1900 and 55 more to 1955, for a total of 21+55 = 76 years old.
Figure 1. Textbook teaching the “counting up” method (Source: Clark et al., 1952).
Although I did not actively practice the counting up method for making change, I eventually learn to use the method as my own personal shortcut for subtracting numbers. I’ve come to find I’m not the only one. Over the years, working with addition and subtraction has resulted in my being able to count up quickly.
Part of this is simply due to memory and part is an automated “carry the one” process that enables me to see 100–71 will yield a 9 in the ones place and a 2 in the tens place.
Addition by ‘Making Tens’
The addition method of “making ten” is another people will stumble onto if left to their own devices. It is embedded and explained by way of example in the 1st grade Common Core standards.
“Add and subtract within 20, demonstrating fluency for addition and subtraction within 10,” read the standards. “Use strategies such as … making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9).”
Similar to the “counting up” method, “making tens” has been used for years in subtraction, as shown in Figure 2, taken from a 3rd grade arithmetic textbook. Students had already mastered the standard method for addition when they were introduced to this method. The book did not insist students use this method; it was introduced as a possible help. Students were only required to use it in the set of exercises that followed. After that, it was up to the student whether to use it or not, which means it served more as a side dish than the main dish it has turned out to be under Common Core.
Figure 2. Textbook teaching the “making tens” method (Source: Eicholz et. al., 1968).
Place Value Strategies
Last but not least is Common Core’s 1st grade standard: “Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.”
This standard embodies many different ways to add two-digit numbers, including a method in which carrying is supposedly eliminated when adding two numbers. For example, to add 47 and 38, one first adds 40 and 30 to get 70 and then 7 and 8 to get 15. The subtotals of 70 and 15 are then summed to get to 85. This method has long-been a short-cut and mental-math strategy used by grocers who total customers’ purchases without aid from automated cash registers.
Figure 3 is taken from a 5th grade arithmetic textbook from 1948. It shows the method was certainly taught, but again, only after students mastered the standard method:
Figure 3 (Source: Knight et al., 1948).
Barry Garelick ([email protected]) has written extensively about math education in various publications, including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. Environmental Protection Agency and is teaching middle and high school math in California. Garelick is the author of Teaching Math in the 21st Century, which recounts his experiences as a long-term substitute in a high school and middle school in California.