######

*Note: This communication was originally sent to the National Governors Association Center for Best Practices and the Council of Chief State School Officers.*

I am in strong support of all efforts to improve the mathematical education of our students, PreK–Grade 12, and endorse many of the ideas in the Common Core. I hope that my comments/suggestions will prompt educators to rethink some of the key ideas, their introduction, and their development across grades.

**1. ****Algebra and Algebraic Reasoning**

When NCTM released their 2000 *Principles and Standards for School Mathematics*, I was pleased to see the strengthening and expansion of the algebra strand to Pre-kindergarten. I particularly applauded acknowledgement of the importance of algebra and algebraic reasoning in the PreK–8 sequence as supporting the learning of other domains of mathematics (number, measurement, geometry, probability, and statistics) as well as preparing young learners for the formal study of algebra. The fact that most instructional PreK-8 programs have never offered a well delivered sequence of instruction in algebra is amazing to me. I was hopeful that the Common Core would reinvigorate that important part of mathematics. As I read the Standards, I see no algebra strand, or at the very minimum, a proportional reasoning strand through the early grades.

In the Common Core, the first identification of a major section dealing with algebra is in Grade 6. Before that, there are isolated items that have the special algebra symbol. But the use of that symbol is not consistent. For example, in grades 1, 2, 3 and 4, *solving word problems* is considered algebra. In other grades, it is considered as part of the problem solving section. Explaining why some of the computational algorithms "work" is considered algebra. But at other times, those same standards are included in sections entitled “work with the base ten system,” “solving problems with addition and subtraction,” or “operations on fractions.” From reading those standards with the algebra symbol, I don’t have a clue about the meaning and goal for instruction in algebra. And I am equally confused about the meaning of and development of problem solving.

We all know that successful completion of Algebra I is the gateway to the study of higher level mathematics and the sciences. We all have heard horror stories about the great numbers of students failing Algebra I. As educators, we recognize that good education builds on earlier learned and mastered concepts and skills. In a couple of years, California will be requiring that all students enroll in Algebra I in Grade 8. I have no doubt that other states will follow suit. How are we educators planning to get students ready for that eighth grade challenge?

**2. Statistics**

There is limited attention to this important domain of mathematics in the Common Core Standards in the elementary and middle school grades. Statistics is the primary support of the inquiry cycle in the design and conduct of experiments. It is useful in understanding how to deal with the collection, display and analyses of data in order to identify trends, to make predictions, and so on. It is central to success in the study of the sciences, business, economics, medicine, and of course higher level courses in mathematics. For me, statistics should be up there with algebra as a major strand throughout PreK–12 curriculum.

**3. The Presentation of the Standards**

I have great difficulty knowing what is expected from students, in terms of knowledge or skill to be demonstrated, when a standard begins with "Understand." I get it when standards require students to explain, draw, construct, organize, measure, compute, or solve. What do we expect students to be able to do with that information? The “Understand” standards should be clarified.

**4. The Developmental Sequence**

With regard to the order of skills and "rigor" of demands for performance at the various grade levels, I see several items that I would delete or move to a different grade level because they are developmentally "out of the ballpark.” For example, counting back by ones in K is not a useful skill. Some say that it prepares students for subtraction, when in practice, subtraction defined as removing a subset and identifying the quantity that remains “is left” is much more age appropriate for 5 year-olds. “Understanding the distributive property” in Grade 3 should be moved up a grade to where students may be able to use that skill and gain insight into how and why it works. “Understanding that fractions give meaning to the quotient of any whole number by any non-zero whole number" in Grade 4 is far too complex for 9 year olds. I have teachers who don’t “understand” this.

**5. Alternative Instructional Approach**

Many students continue to have difficulty with mathematics despite attempts of various groups (NCTM, NCSM, and the various State Frameworks) to suggest new standards for instruction. Some researchers have identified Grade 4 as the beginning of the downturn in math achievement. Others have suggested that the start of declining achievement begins in Grade 2. I suspect that much of the difficulty students have with the learning of mathematics may stem from their lack of enthusiasm for the subject because of the great amount of memorization and lack of application of acquired skills. High school, and even college students tell us that they are often bored with math and don't see the relevance of the subject to their futures. In 2009, the American Association for the Advancement of Science produced their Benchmarks to reiterate and expand on several of the points made in their 1993 document, *Science for All Americans: Project 2061.* Several of the benchmarks suggest a different approach to teaching mathematics, one that shows the utility of mathematics for solving problems as a way of enhancing understanding.

One of the drawbacks of teaching mathematics entirely as a separate subject is that mathematics is taught before real-world problems are identified, so the related experiences may have mostly to do with learning the procedures rather than solving interesting problems. (AAAS, Chapter 11, p. 5)

Carole E. Greenes

Associate Vice Provost for STEM Education

Professor, Mathematics Education

Arizona State University

*Carole Greenes is co-author* of two ORIGO series: ZUPELZ and The Think Tank (Problem Solving). * You can read reaction to the Common Core standards from co-author Carol Findell here. **These articles are the opinions of the author and do not necessarily reflect the views of ORIGO Education. *