"Bear in mind that the wonderful things you learn in your schools are the work of many generations, produced by enthusiastic effort and infinite labor in every country of the world. All this is put into your hands as your inheritance in order that you may receive it, honor it, add to it, and one day faithfully hand it to your children. Thus do we mortals achieve immortality in the permanent things which we create in common." - Albert Einstein

Sunday, March 3, 2013

Making Sense Out of Numbers: Math Common Core

Avoiding a mile wide but only an inch deep treatment of mathematics in schools sounds laudable. Of course, it sounds reasonable but a statement can only go beyond merely a platitude with the actual implementation. There are more than adequate quotations one can use to describe what education ought to be, but in the end, how teaching and learning actually occur inside the classroom is all that matters. It is therefore important to keep the lofty goals in mind, but one must likewise keep one's feet on the ground, and in the end, make the decision that best benefits the learner. For this reason, the teacher remains central in education. A curriculum is a guide, drawn hopefully with the correct intentions. One must remain faithful to the original intentions. This is similar to a society. More often than not, bills are passed and made into law, and it takes a Supreme Court to figure out if some of these laws are unconstitutional. Inside a classroom, when a curriculum gets down to the detailed level, it is only expected that some elements may actually be working against the original intentions, and it is up to the teacher to choose the right path appropriate to the needs of the learners.

The Math Common Core of the United States is designed to be both coherent and focused. To achieve this, the new curriculum is supposed to cover less topics but in greater depth. It is supposed to inform the learner not just the rules and processes in math, but how math actually works. The following is a video from the Teaching Channel introducing the Math Common Core:



The following is the kindergarten component of the Math Common Core:
  • Counting and Cardinality

    • Know number names and the count sequence.
    • Count to tell the number of objects.
    • Compare numbers.
  • Operations and Algebraic Thinking

    • Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
  • Number and Operations in Base Ten

    • Work with numbers 11-19 to gain foundations for place value.
  • Measurement and Data

    • Describe and compare measurable attributes.
    • Classify objects and count the number of objects in each category
  • Geometry

    • Identify and describe shapes.
    • Analyze, compare, create, and compose shapes.
  • Mathematical Practices

    1. 1. Make sense of problems and persevere in solving them.
    2. 2. Reason abstractly and quantitatively.
    3. 3. Construct viable arguments and critique the reasoning of others.
    4. 4. Model with mathematics.
    5. 5. Use appropriate tools strategically.
    6. 6. Attend to precision.
    7. 7. Look for and make use of structure.
    8. 8. Look for and express regularity in repeated reasoning.

For grade 1, the following is the guide:
  • Operations and Algebraic Thinking

    • Represent and solve problems involving addition and subtraction.
    • Understand and apply properties of operations and the relationship between addition and subtraction.
    • Add and subtract within 20.
    • Work with addition and subtraction equations.
  • Number and Operations in Base Ten

    • Extend the counting sequence.
    • Understand place value.
    • Use place value understanding and properties of operations to add and subtract.
  • Measurement and Data

    • Measure lengths indirectly and by iterating length units.
    • Tell and write time.
    • Represent and interpret data.
  • Geometry

    • Reason with shapes and their attributes.
  • Mathematical Practices

    1. 1. Make sense of problems and persevere in solving them.
    2. 2. Reason abstractly and quantitatively.
    3. 3. Construct viable arguments and critique the reasoning of others.
    4. 4. Model with mathematics.
    5. 5. Use appropriate tools strategically.
    6. 6. Attend to precision.
    7. 7. Look for and make use of structure.
    8. 8. Look for and express regularity in repeated reasoning.
Similar guidelines are provided for Grades 2-6 and the two years of middle school. In contrast, the high school guides are organized by concept rather than grade. These provide flexibility for the states or even at the district or school level. States and schools can decide how to shape instruction. These concepts are: Number and Quantity, Algebra, Functions, Geometry, Statistics and Probability, and Modeling. The last one, Modeling, is expected to be integrated in all years of high school. The authors of the Common Core note:
"These standards do not mandate the sequence of high school courses. However, the organization of high school courses is a critical component to implementation of the standards. To that end, sample high school pathways for mathematics – in both a traditional course sequence (Algebra I, Geometry, and Algebra II) as well as an integrated course sequence (Mathematics 1, Mathematics 2, Mathematics 3) – will be made available shortly after the release of the final Common Core State Standards. It is expected that additional model pathways based on these standards will become available as well."
The Mathematics Common Core provides specific content and practices, yet it allows for teachers to do the final finishing touch so that it will be effectively implemented inside the classroom. Is this for real? Barry Garelick wrote in the Atlantic back in November of last year the following article, "A New Kind of Problem: The Common Core Math Standards". First, here are his comments for the first 97 pages of the Math Common Core Standards:
Let's look first at the 97 pages of what are called "Content Standards." Many of these standards require that students to be able to explain why a particular procedure works. It's not enough for a student to be able to divide one fraction by another. He or she must also "use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9, because 3/4 of 8/9 is 2/3." 
It's an odd pedagogical agenda, based on a belief that conceptual understanding must come before practical skills can be mastered. As this thinking goes, students must be able to explain the "why" of a procedure. Otherwise, solving a math problem becomes a "mere calculation" and the student is viewed as not having true understanding. 
This approach not only complicates the simplest of math problems; it also leads to delays. Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them "understand" the conceptual underpinnings.
I guess the point of providing the student multiplication to help appreciate division by fractions is to overcome the limitations of manipulatives, of things we could actually hold, in explaining what happens when we divide something by a fraction. A child can easily relate to dividing by a number greater than 1. A pizza can be divided into four parts, each slice being a quarter of the whole pizza. But how does one divide a pizza by 3/4? Why is the answer 4/3? It is bigger than before. Making the student perform the reverse process brings the equation to something much easier to relate. 3/4 of 4/3 is one. This can be visualized.

Garelick does end the article with a hopeful view:
As the Common Core makes its way into real-life classrooms, I hope teachers are able to adjust its guidelines as they fit. I hope, for instance, that teachers will still be allowed to introduce the standard method for addition and subtraction in second grade rather than waiting until fourth. I also hope that teachers who favor direct instruction over an inquiry-based approach will be given this freedom.
At the end, for any curriculum, teachers provide the determining step. I do hope that teachers do make sense out of this curriculum.







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