In my final in this Common Core Structure series, I want to just spend a little time discussing the three Key Shifts of the Common Core: Focus, Coherence, and Rigor. The CC are standards - states have always had standards. The difference here is a clearer set of aligned standards, throughout K-12, that ensured the standards built on each other within a grade, between the grades, and provided a cohesive set of understandings, skills and application. Hopefully the previous four posts have given a clearer understanding of how the structure of the CC was designed to support these shifts, so now lets actually look at these three shifts in depth. I am going to use some specific standards to exemplify each shift, as I think it helps make sense of them.
Focus
The CC is really focused on students conceptual understanding of mathematics and their ability to apply these understandings to real-world problems. So, within each grade, there are "less" standards,
and more focus to help them build strong foundations before they move on to the next grade level. Each grade goes deeper into fewer concepts, which then, as they move through the grades, they continue to build. So, for example, in grade 3, the focus is on multiplication (because in K-2 they built the foundation of whole numbers and addition & subtraction).
As an example, here is 3.OA.7 standards (Grade 3, Operations & Algebraic Thinking, Standard 7):
Coherence
If the idea is to deeply understand fewer concepts at each grade and continue to build and add on to
that understanding, then this means between the grade levels there must be connected standards. This is the coherence. Throughout each grade, the mathematical skills and concepts being focused on are building the foundation for the next grade, where students will continue to develop the understandings and add on to those understandings. What is learned in one grade impacts what comes next and is a result of what came before - coherence. Think of it as a ladder - each grade is a rung, and you must step on that rung (i.e. learn and understand the concepts) in order to get to the next rung, which will eventually get you to the top. Skipping a rung may make you fall because you have missed some important step and aren't secure.
Using the Grade 3 example from above, to demonstrate coherence, lets look at related standards from Grade 2 and Grade 4, so you can see how the standards fit together, or cohere, between grades.
Looking at the coherence - in Grade 2, students are working with rectangular arrays to develop an understanding of multiplication. In Grade 3, they take that further by being able to user properties to see the connection between multiplication and divisions. In Grade 4, they are able to expand on that commutative property and apply it to verbal and written equations. Each grade provides focused understanding that helps them build deeper understanding at the next level - the standards cohere between the grades to reinforce previous understandings and build the foundation for future concepts.
Rigor
Rigor is probably the most understood of the shifts. Unfortunately, many interpret rigor to mean harder problems or, worse, MORE problems. Rigor DOES NOT MEAN MORE PROBLEMS OR HARDER PROBLEMS. I repeat...RIGOR DOES NOT MEAN MORE PROBLEMS OR HARDER PROBLEMS!! It kills me when I go into classrooms, especially those labeled as "gifted" or 'advanced" and see students being assigned 50 homework problems every night because they are "smarter". No, no, no, a thousand times NO! All that does is make students HATE mathematics (I can attest to this with my own daughters, who always had ridiculous amounts of math problems, especially in the higher level classes). If nothing else, please stop the practice of assigning more homework and thinking this means you are providing rigor.
Rigor is understanding mathematical concepts, being able to explain your thinking, being able to apply your understanding to new and different situations. Three things - understand the concept, be able to use the concept to solve problems and explain thinking, and be able to apply the concepts to real-world situations and different situations. So, having a student multiply two digit numbers 50 times for homework is NOT rigorous, because you have left out explaining why and applying to new situations.
How do you create rigor in your math classroom? How do you get away from drill-and-kill and developing fear and hatred of math in your students? My simple suggestion is ask good questions and expect real answers. Here is where the Mathematical Practices come into play - read those again (or go back to my previous post related to the Practices). Ask questions. Expect students to explain and justify their answers using whatever method works - words, models, similar problems. Give students DIFFERENT types of problems that require them to apply their understanding in new ways. And ask questions!!! If you don't know what to ask, ask why? Or ask "can you show me what you did"? Or ask "is that true all the time? " To me, rigor means you expect them to explain what they did, why they did it, and can they do it differently or in another way?
Focus
The CC is really focused on students conceptual understanding of mathematics and their ability to apply these understandings to real-world problems. So, within each grade, there are "less" standards,
and more focus to help them build strong foundations before they move on to the next grade level. Each grade goes deeper into fewer concepts, which then, as they move through the grades, they continue to build. So, for example, in grade 3, the focus is on multiplication (because in K-2 they built the foundation of whole numbers and addition & subtraction).As an example, here is 3.OA.7 standards (Grade 3, Operations & Algebraic Thinking, Standard 7):
Under Cluster Heading of Multiply and Divide within 100: 7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g. knowing that 8x5 = 40, one knows 40/5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.As you can see, very focused. There is a footnote in Grade 3 that students "need not use formal terms for the properties of operations", because they are building understanding. Notice, grade 3 is only working with numbers less than 100. And that they are expected to know their times tables between all one-digit numbers. It's focused, specific, with the idea that if they learn these concepts and skills, and gain true understanding, then they will be able to continue to build on these as they go to the next grade level. Focus is few standards, learned deeply, as a building block for the next level.
Coherence
If the idea is to deeply understand fewer concepts at each grade and continue to build and add on to
Using the Grade 3 example from above, to demonstrate coherence, lets look at related standards from Grade 2 and Grade 4, so you can see how the standards fit together, or cohere, between grades.
Grade
|
Grade
2
|
Grade
3
|
Grade
4
|
Domain
|
Operations
and Algebraic Thinking
|
Operations
and Algebraic Thinking
|
Operations
and Algebraic Thinking
|
Cluster
|
Work
with equal groups of objects to gain foundations for multiplication
|
Multiply
and divide within 100
|
Use
the four operations with whole numbers to solve problems
|
Standard
|
4. Use addition to find the total
number of objects arranged in rectangular arrays with up to 5 rows and up to
5 columns; write an equation to express the total as a sum of equal addends
|
7. Fluently multiply and divide within
100, using strategies such as the relationship between multiplication and
division (e.g. knowing that 8x5=40, one knows 40/5=8) or properties of
operations. By the end of grade 3, know from memory all products of two
one-digit numbers.
|
1.Interpret a multiplication
equation as a comparison, e.g.,
interpret 35=5x7 as a statement that 35 is 5 times as many as 7 and 7 times
as many as 5. Represent verbal statements of multiplicative comparisons as
multiplication equations
|
Looking at the coherence - in Grade 2, students are working with rectangular arrays to develop an understanding of multiplication. In Grade 3, they take that further by being able to user properties to see the connection between multiplication and divisions. In Grade 4, they are able to expand on that commutative property and apply it to verbal and written equations. Each grade provides focused understanding that helps them build deeper understanding at the next level - the standards cohere between the grades to reinforce previous understandings and build the foundation for future concepts.
Rigor
Rigor is probably the most understood of the shifts. Unfortunately, many interpret rigor to mean harder problems or, worse, MORE problems. Rigor DOES NOT MEAN MORE PROBLEMS OR HARDER PROBLEMS. I repeat...RIGOR DOES NOT MEAN MORE PROBLEMS OR HARDER PROBLEMS!! It kills me when I go into classrooms, especially those labeled as "gifted" or 'advanced" and see students being assigned 50 homework problems every night because they are "smarter". No, no, no, a thousand times NO! All that does is make students HATE mathematics (I can attest to this with my own daughters, who always had ridiculous amounts of math problems, especially in the higher level classes). If nothing else, please stop the practice of assigning more homework and thinking this means you are providing rigor.
How do you create rigor in your math classroom? How do you get away from drill-and-kill and developing fear and hatred of math in your students? My simple suggestion is ask good questions and expect real answers. Here is where the Mathematical Practices come into play - read those again (or go back to my previous post related to the Practices). Ask questions. Expect students to explain and justify their answers using whatever method works - words, models, similar problems. Give students DIFFERENT types of problems that require them to apply their understanding in new ways. And ask questions!!! If you don't know what to ask, ask why? Or ask "can you show me what you did"? Or ask "is that true all the time? " To me, rigor means you expect them to explain what they did, why they did it, and can they do it differently or in another way?
Example: Student says 5x8=40, which means that 40/5 = 8. You ask, why? They start drawing examples of 8 groups of dots five times, showing it equals to forty. They then draw another example of 40 dots, and they circle groups of five dots, and show that their are now 8 groups. You then say, ok...does that work for other problems? (i.e. model it with different numbers) Or, can you give me a real example of this? (i.e. apply what you are learning to a real-world situation).The CC is not just about standards. Standards are standards. What is to me the most important aspect of the CC is the focus, coherence and rigor that are embedded throughout and that help students build their conceptual understanding, learn appropriate skills at appropriate times, and apply those understanding and skills to new, real-world situations. I know a lot of states are dropping out of CC or changing the name or creating their own standards, much of this due to ridiculous politics and ignorant people not truly understanding the goal and purpose of the CC. Fine. But - regardless of what state you are in or what mathematical standards you use, if you do nothing else, be rigorous in your instruction. Use the Mathematical Practices and you are helping your students.
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