Fortunately the authors of the Common Core State Standards for Mathematics heard this message loud and clear. They made a concerted effort to bring more focus and coherence to the standards. In the introduction to the Massachusetts Curriculum Framework for Mathematics (2011) it is stated that, "For over a decade, research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards must address the problem of a curriculum that is 'a mile wide and an inch deep.'"
Focus and Coherence
What is it that we are talking about when we use these two terms? The developers of the standards recognized that previous standards here in Massachusetts, and in states across the nation, have addressed a wide variety of disjointed standards. Coherence speaks to the interconnected nature of the new standards. The developers made a deliberate effort to build a logical progression of skills and knowledge into the standards as student progress from one grade to the next. We can see an example of this when we examine the Operations and Algebraic Thinking domain. In kindergarten students are asked to " represent addition and subtraction with objects, fingers, mental images, drawings, sounds...(K.OA.1)" and then to "solve addition and subtraction word problems, and add and subtract withing 10, e.g. , by using objects or drawings to represent the problem (K.OA.2)." In the first grade this progresses to "add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use mental strategies such as counting on; making ten; using the relationship between addition and subtraction; and creating equivalent but easier or known sums (1.OA.6)." The idea of addition and subtraction is further developed in second grade. The demand for fluency is increased, "Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers (2.OA.2)." Coherence is also developed within each grade level. In second grade fluency of facts is then applied in standard 2.OA.4, "Use addition to find the total number of objects arrange in rectangular arrays..." This standard is also, no-so-coincidentally, coherent to the third grade standards for multiplication (rectangular array models for teaching multiplication). This arraignment of concepts continues throughout the entire document. The standards are developed in a way to model the thinking the developers are hoping to instill in students. A student that sees the connections between mathematical ideas and understands the hierarchical nature of math. The strong foundation for this development begins in Pre-K.
Focus within the standards is addressed in the limiting of concepts students are meant to learn. Focus is built upon an underlying goal of a greater amount of mastery of fewer, albeit powerful, mathematics concepts. On the first page for each grade level within the Common Core State Standards for Mathematics (adopted into MA Frameworks as is) there are no more than 4 areas identified for instructional focus. They pinpoint critical areas upon which teachers should spend the vast majority of their instructional time. We should be using this page to help us in curriculum writing, unit planning and lesson planning. Unfortunately, these pages are often overlooked as teacher dive into the content specific nature. I strongly urge teachers to spend time with this introductory page. Actually, I would ask that not only do we examine this page for our grade level, but the one above us and below us as well This will give us a deeper understanding of where our students are coming from and where they are going.
The concepts of Focus and Coherence are further accentuated in the Standards for Mathematical Practice which will be a topic for later posts.
From the Source
The videos embedded in this blog are from the Hunt Institute. They interviewed the authors of the Common Core State Standards on a variety of topics. These summarize what I have tried to spell out on Coherence and Focus. Enjoy!
