The Common Core State Standards for Mathematics was the result of a state-led movement by the National Governors Association and the Council of Chief State School Officers. The common core standards for math consists of two sets of standards: practice standards, which are the same for all students, kindergarten through high school, and content standards, which are specific to each grade level. Both the content and practice standards reflect the standards in the NCTM (National Council for Teachers of Mathematics) book, Principles and Standards for School Mathematics (2000), and the book Adding it Up: Helping Children Learn Mathematics (2001)
It is my opinion that most everyone believes that students in the United States perform far below their peers in mathematics, an opinion which is supported by the Trends in International Mathematics and Science Study (TIMSS). One positive aspect of the Common Core Standards is that they are internationally bench-marked, which means that our standards will compare favorably to standards of other countries.
Strategies for subtraction
Another positive aspect of the common core math standards is that they will help deepen children’s understanding of mathematical concepts and help develop their number sense. Students will learn skills that are needed in all areas of life, such as critical thinking, making sense of problems, perseverance in solving problems, reasoning abstractly, attending to precision, looking for patterns and regularity, and then making generalizations to develop rules and formulas. (These behaviors and skills are expressed in the Standards for Mathematical Practice).
One parent recently posted a letter on Facebook expressing frustration at a math problem given to his son.
The problem illustrated in that letter is one that should only be assigned after students have studied several strategies for subtraction. Students are being asked, not for the correct answer to the subtraction problem, but to engage in reasoning, look at the pattern, and make sense of the problem. The strategy being illustrated in the problem shown in the letter is using a number line for subtraction (first image).
One of the parent’s frustrations was that the subtraction problem was so easy, that it was not necessary to go through that long process to find the answer. However, the question wasn’t asking students to find the right answer, but to explain what went wrong with the method being used. There are many methods that can be used to subtract and learning many methods helps build conceptual understanding, increase number sense, and provide a way to check your results by doing a problem more than one way. Teaching multiple methods for solving a problem also addresses the fact that we do not all think about things in the same way, so we don’t all learn the same way. The fact that so many adults are still struggling with basic math shows that just teaching the traditional way does not work for everyone.
The problem given in the assignment illustrates one method that can be used to help students understand the process of subtraction. Another method for subtraction involves decomposing one or both numbers. Decomposing a number generally means to break it apart, which can be done different ways, depending on the context. Image 2 shows examples.
Image 3 shows some other examples of how decomposition can be used as a strategy for subtraction.
Teachers will still teach the traditional methods
The traditional algorithms will still be taught. Here are a few standards from the Common Core Standards addressing this:
Adding and subtracting: In grade 3, students learn strategies that deepen conceptual understanding. In grade 4, they learn the standard algorithm
Grade 3 standard:
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. (3.NBT.2)
Grade 4 standard:
Fluently add and subtract multi-digit whole numbers using the standard algorithm. (4.NBT.4)
Multiplying: In grade 4, students learn strategies that deepen conceptual understanding. In grade 5, they learn the standard algorithm
Grade 4 standard:
…multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. (4.NBT.5)
Grade 5 Standard:
Fluently multiply multi-digit whole numbers using the standard algorithm (5.NBT.5)
Fluently means efficiently. The standard algorithm is the goal (see grades 4 and 5). Standard algorithms (the traditional methods) are very efficient, but teaching only using the standard algorithm creates holes in number sense and understanding the concepts.
Depth is important
Common core standards will mean that students will need to learn math in a deeper way than before. Look at a sample 6th grade problem from Illustrative Mathematics
Students will need to know the standard algorithm for dividing a fraction by a fraction, but also should be able to reason using a conceptual understanding of what it means to divide fractions. For example, when dividing 6 by 2 , we can ask, “How many groups of 2 are in 6?” The answer is 3. In the same way, when dividing 6 by ½, we can ask, “How many halves are in 6?” The answer is 12.
For the problem shown above, 7/8 divided by¸1/4, the question could be, “How many1/4’s are in7/8?
Conceptually, this can be understood by drawing a picture (Image 4)
Some of the frustration that many people feel regarding the common core standards is that the learning all of these different methods seems inefficient.. However, students need to develop conceptual understanding and to learn how concepts are interrelated. This will enable them to actually solve problems using math as a tool, rather than memorizing what seems to be an unrelated set of procedures and rules, and then trying to remember where to apply those rules. In general, people feel like math should be taught the way they learned it. In fact, many people never really learned it so that it can be remembered when needed. When you understand how concepts are connected, how to look for patterns and generalize to form rules, you can recreate the process for yourself if you’ve forgotten the standard algorithm.
For example, in middle school or high school, students usually memorize rules for multiplying powers and taking a power to another power. The rules are often memorized something like this: When multiplying powers, we add the exponents, and when we take a power to another power we multiply the exponents.
One problem with just memorizing this is that students frequently confuse when to add exponents and when to multiply. By having them look for patterns and generalize what they see to come up with the rule themselves, they can repeat that process when they forget whether they should add or multiply exponents. (See image 5)
When students are asked to write about an error in a problem, such as the one addressed in the letter, they are being asked to disclose a deeper understanding of a concept than just how to get a right answer. While a correct answer is still required, it is crucial that students develop the ability and confidence to explore, reason, and strategize as they are faced with problems in the field of mathematics. Equipping our students with these skills is a major part of the Common Core Standards for Mathematics.