“Let’s go invent tomorrow rather than worrying about what happened yesterday.” -Steve Jobs
Sometimes we hear from parents and some teachers that they would love us to go back to teaching math the way they learned it. They say they can’t help their students because the work doesn’t look the same, it’s all in a new language. The kids complain their parents are telling them to do the math in a different way than their teacher did and it is confusing. Why is all this happening?
The Utah Core State Standards in Mathematics put a premium on understanding the why (conceptual understanding) as well as the how (procedural understanding, skill, and fluency) and the application of mathematics. All three aspects are equally as important. The standards don’t try to reinvent mathematics. But they may and do take a different path than they did in the past. Why? Because of a great deal of evidence that our past methods of teaching and learning math failed a great many of our students.
Marilyn Burns, in her book Math: Facing an American Phobia says, “Even in the face of widespread failure in learning mathematics, we seem to want to cling to educational methods with a nostalgia for them that has long outlasted their usefulness and has perpetuated failure. The way we’ve traditionally been taught mathematics has created a recurring cycle of math phobia, generation to generation that has been difficult to break. We start young children with counting and move them along through arithmetic, then on to algebra, geometry, trigonometry, and so on. The ‘and so on’ depends on whether or not the student sticks with math, which means not falling off the ladder of math progress in school. But an alarming percentage of the people in our country have fallen off the ladder and feel like mathematical failures. And once people fall off the ladder, there seems to be no way for them to get back on.
. . . Children must be helped to learn mathematics in a better way than we were, so that mathematical limits do not shut them out of certain life choices and career options.”
I agree with Marilyn, and with countless other mathematics educators and mathematicians, that we must “help children learn mathematics in a better way than we were…” That is what the Utah Core State Standards are constructed to do.
One giant step in helping children learn mathematics in the core is the evidence-based method of building understanding of and fluency in algorithms by using math strategies that are based on sound mathematical principles such as place value, properties of operations (e.g., associative, commutative, distributive, identity), the relationship between operations, and mental math. Make no mistake, great math teachers have been using these strategies for many years. They are not new.
The standards make heavy use of strategies instead of algorithms in the early grades when students are building number sense and making sense of operations like addition, subtraction, multiplication, and division. That the standards mention strategies eighteen times in grades K-3 but just seven times in grades 4 and 5 and only twice in grades 6 – 8 should tell us something about their use. They build understanding. Students then, at an appropriate time, with the correct understanding, begin using and become fluent with the standard algorithms. We want them to know those algorithms, but we are firm in our assertion that students must understand them before they use them. At each step of the game students are required to explain and justify their reasoning mathematically. Of course, as I said in my last post, many of us still use strategies we learned or devised for solving problems mentally. See that post for details.
In the document Progressions for the Common Core State Standards in Mathematics: K-5 Operations and Algebraic Thinking (2011) the authors note three levels of strategies for adding and subtracting one digit numbers used in the standards. They are:
Level One: Direct Modeling by Counting All or Taking Away
Level Two: Counting On
Level Three: Convert to an Easier Equivalent Problem
For more information on these levels, including definitions and examples, click here and scroll to page 36.
The standards also make good use of mental math strategies in the early grades. Those strategies, which are part of the Level Two and Level Three strategies, include:
Counting on: 8 + 4 = □ (8 …9, 10, 11, 12)
Counting back: 12 – 4 = □ (12…11, 10, 9, 8)
Making tens: 5 + 7 = □ (5 = 2 + 3 so 3 + 7 = 10 therefore 10 + 2 = 12)
Doubles: 6 + 6 = □
Doubles plus/minus one: 6 + 7 = □ (6 + 6 + 1 or 7 + 7 – 1)
Decomposing a number leading to a ten: 15 – 7 = □, so 15 – 5 = 10, therefore 10 – 2 = 8)
Working knowledge of fact families/related facts/number bonds: 3 + 9 = 12 so 12 – 9= □
When students begin to face addition and subtraction problems where regrouping is necessary they first explore different strategies of solving such problems rather than immediately diving into the confusing world of what has been called carrying, borrowing, regrouping, decomposing, and so on. Students can use their understanding of place value and the associative and commutative properties of addition to solve problems such as 47 + 33. Instead of carrying a one (which is actually a ten) students write the problem in expanded form as (40 + 7) + (30 + 3) (though they probably won’t use the parentheses). They can then use the properties of operations and come up with (40 + 30) + (7 + 3). Then they add 40 + 30 = 70 and 7 + 3 = 10 and then finally add 70 +10 = 80.
I am sure some of you are thinking that is not very efficient. Remember, at this point in a student’s learning we are not looking for fluency, we are looking for understanding. How does this help them understand? Because they recognize principles they have learned before. Place value tells then that 40 and 30 both have digits in the tens place so it makes sense to group them together. The same with 7 and 3; they have digits in the ones place. In addition, they are making sense of the problem and persisting in solving it, thinking analytically and reasoning mathematically, looking for patterns in their reasoning, and so on, all of which are contained in the Standards for Mathematical Practice in our standards.
Another strategy is to start adding at the left in a vertical problem rather than at the right, as follows:
Van de Walle (2013) lists the following advantages of using strategies, including those students come up with on their own:
- Students make fewer errors. When students use strategies they understand they make fewer errors than they do when using algorithms they have been taught without understanding.
- Less reteaching is required. Though developing student understanding through strategies is slower than simply teaching an algorithm students, through constructive struggling (read the article), gain “meaningful and well-integrated networks of ideas that are robust and long lasting.”
- Students develop number sense. Through using number rich strategies that are based in sound mathematical principles students gain an appreciation and understanding of the base ten number system rather than simply using an algorithm they don’t understand.
- Strategies are the basis for mental computation and estimation. When students record their thinking as they are working through their strategies for solving a problem they will find that they can more easily do the steps mentally.
- Flexible methods are often faster than standard algorithms. In one of my Elementary Principals Math and Science Leadership Academy meetings I asked the principals to solve a problem similar to this one: 4005 – 3997 = ? and to raise their hands when they were done. Some of the principals immediately wrote the problem down and began using the standard algorithm to subtract using regrouping. Others raised their hands almost immediately. When I asked one of the principals who was done quickly to share her method she said, “Well, I simply looked at the numbers and realized that 3997 is 3 away from 4000 and then just added 5. So the answer is 8.” When the only tool you have is a hammer…
- Algorithm invention is itself a significantly important process of “doing mathematics”. When students come up with their own means of solving a problem they are “intimately involved in the process of sense making.” They develop a confidence in their ability to do mathematics. And when they can explain what they did that confidence grows and leads to greater success.
At the end of this progression students become fluent in the standard algorithm. By then, they will understand it. We will make sure they understand it by having them explain their reasoning when they use it. That reasoning was developed through the strategies.
Van de Walle, J. A., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally, Pearson Education
Operations and Algebraic Thinking Progression, http://ime.math.arizona.edu/progressions/, accessed on 3/27/2014
Burns, Marilyn. (1998) Math: Facing an American Phobia. Math Solutions