Zimba, a theoretical physicist, delved into the history of the common core during his address and clarified a lot of the questions regarding how the core was constructed. One of the driving forces in his research was the staggering amount of topics omitted by countries that are leaders in the TIMMS report. Hong Kong for instance omits 48% of topics by 4th grade, and what they do focus on is deeper, richer engagement with concepts. You would find the same about Singapore, Norway, Japan, etc.
As a result of the evidence based and research driven process, three major shifts have occurred in mathematics education under the Common Core.
Shift 1: Focus
In the early grades the focus is on arithmetic. K-2 students will become masters of addition and subtraction. No more pattern problems. 3rd-5th graders will be masters of multiplication and division (whole numbers and fractions). Throw away your bar charts. Middle school concepts will include advanced geometry topics and algebra preparation and high school will look at college and career readiness concepts.
A good question to ask is, what exactly does college and career readiness concepts mean? In Zimba’s research, the most desired skills for high school students is proficiency in algebra and functions. Students also need to be able to persevere in following a number of steps to solve a problem as opposed to plugging a formula.
How does this change teaching in high school mathematics classrooms?
Classroom time should be spent solving problems that go deep, that push students to keep asking questions, searching and collaborating to unlock strategies, and have solutions that may not be so clear-cut. Problems should not just target one concept or standard. The best strategy is to link concepts across grade levels. High school teachers should be weaving in middle school concepts such as percents, ratios and proportions in their algebra assignments. This leads to the second shift.
Shift 2: Coherence
Upon arrival in middle school, students find themselves surrounded by the dark and mysterious world of fractions. Up until this time, their world has been safe, even fun, as they have progressed through the major operations with whole numbers. Fractions have been kept at a safe distance from them. This is no longer the case.
We will be introducing material throughout the grades bit by bit. In 4th grade students will multiply a whole number by a fraction. In 5th grade they will divide a whole number by a fraction. By the time 6th grade rolls around, multiplying and dividing fractions with fractions should not be as foreign, and students have solid foundations on what fractions are.
Coherence in high school is similar. Instead of fractions, it could be linking the equation of a circle, trigonometric identities and the distance formula together. All three of these concepts follow similar structures. If students can link them together, then they can be more likely to remember and utilize them.
Teachers need to find connections between concepts and not be afraid to teach them in tandem.
Shift 3: Rigor
In major topics of mathematics, students will be pursuing with equal intensity the following three goals.
1. Increasing conceptual understanding
2. Increasing fluency with concepts
3. Applying understanding and fluency in concepts.
Conceptual Understanding
One of the biggest differences between assignments in the USA vs. Hong Kong was examining the way we asked questions. Take these two questions for example.
1. Write 4 fractions equal to 5
2. Write a number that is greater than 1/5 and less than ¼.
These two questions really hit the nail on the head. How many different answers could there be for each question? What are all the different forms a solution can take? These types of questions can be a major difference between having students perform rote skills, and thinking about a major concept and develop their own skills and strategies. I think too often in the past we have been asking for the one perfect answer for each problem.
Fluency
Again, the idea is to restructure problems and add an extra dimension to the problems. We need to stop providing a series of problems in which the answers are readily apparent or formulaic, and ask students to continue to think deeper.
9x8 = 80-8
54 / 9 = 24 / 6
This last problem is a simple multiplication and division problem, but what do we do? Throw in a kicker that has students substitute values into an expression, this accomplishes our second goal I referred to earlier, coherence. Students familiar with this type of progression will be a bit better prepared for algebra.
Application
We all know what this looks like, right? The opening statement in the question below is simple enough. Just a quick glance tells us the rate, but look at the questions that come with it. Stop asking the easy questions and start asking the hard questions.
A bird flew 20 miles in 100 minutes at constant speed. How long would it take to fly 6 minutes? How far would the bird fly in 15 minutes? How fast is the bird flying in mph? What is the bird’s pace in minutes per mile?
Conclusion:
The Common Core standards are not a radical departure from the material we have been teaching in the classroom. It’s asking us to teach with more focus, and to develop richer problem sets that engender more thinking and analysis.
I recommend the following for further information and for example problem sets.
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