Common Core State Standards for Mathematics: Does It Add Up or Down? Part 2

Concerns About the Math Content Standards

You are encouraged to actually look at the Common Core State Standards for Mathematics document.  Become familiar with the document and learn to use your pdf reader search function to find what you are looking for.  Check information out for yourself.

Many general concerns about the Common Core State Standards for Mathematics are well expressed on a number of different websites.  The Where’s the Math? webpage on the CCSS lays out many concerns people have about the standards.  Rather than presenting those concerns again, you are encouraged to explore that webpage and follow some of the links provided to learn about many more specific concerns people have expressed.

Standard Algorithms

Here are graphic depictions of the standard algorithm for each of the operations of addition, subtraction, multiplication, and division.

addition subtraction

 

 

 

 

 

 

multiplicationdivision

 

 

 

 

 

 

Is the standard algorithm for each operation required in the Common Core State Standards?  Yes!  The Common Core State Standards does require the standard algorithms for addition, subtraction, multiplication, and division.  Is the requirement delayed?  Yes!  But, this is not a good thing.
algores

The Common Core State Standards for Mathematics does delay the requirement for the standard algorithms.  During that delay, what is taking place?  Well, leading up to the standard algorithm the standards require the use of strategies based on place value.  This allows for teaching alternative algorithms that are not efficient.  Even though the standard algorithms are based on place value, the standards emphasize strategies based on place value rather than the standard algorithms.

The Where’s the Math? webpage called Standard Algorithms in the Common Core State Standards shows the standards that explicitly require the fluent use of the standard algorithms for addition, subtraction, multiplication, and division.  This page also shows the standards that lead to the development of the standard algorithms for addition, subtraction, multiplication, and division.  It is in many of these standards you will see the required strategies based on place value.

What do strategies based on place value look like?  Here are a couple of videos that do a better job showing some of these strategies than I could do explaining them.  I encourage you to take the time to watch both videos.  Even though both videos were made prior to the CCSS. They show the type of strategies the CCSS encourage and publishers are including in their CCSS aligned programs.

Fact Fluency

The CCSS for Math do call for fact fluency in addition, subtraction, multiplication, and division and memorization of sums and products.  Rather than this being presented as a concern, it is presented to alleviate the concerns of those who think the standards do not call for fact fluency or memorization.  In grade 2:

2.OA.2. Fluently add and subtract within 20 using mental strategies.2  By end of Grade 2, know from memory all sums of two one-digit numbers.

In grade 3:

3.OA.7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 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.

Area and Perimeter—Third Grade

There is a real discrepancy in the manner in which the CCSS addresses area and perimeter in the third grade.  Let’s examine this.  Here’s the standard for area:

Relate area to the operations of multiplication and addition.

–Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

–Multiply side lengths to find areas of rectangles with whole- number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

–Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Tiling?  Yes.  This does call for tiling.  Will students be assessed on their ability to find area by tiling?  If it is in the standards, it should be assessed.  Maybe the more important question is should tiling be in the standards?  How many contractors expect their snapshot-2011-06-13-07-07-19employees to use tiling to determine the square footage of flooring material needed?  Seems to me they determine the square footage to determine the number of tiles needed rather than tiling to determine the square footage.  I do like to pick on this standard even though there are some good things in it.  It is interesting to note that finding the area requires multiplication yet the standard algorithm for multiplication isn’t required until grade 5.  I guess students will only be expected to find the area when the side lengths are two one-digit numbers.  Or maybe this is just a great opportunity to use valuable class instructional time to practice and apply those inefficient strategies base on place value.

Considering the excruciating contortions and detail in the standard for area, one might expect similar treatment for perimeter.  This does not happen.  In fact, the word perimeter does not occur in the standards until this standard appears in third grade:

Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

The concept of perimeter and its formula are never developed.  Instead, students are thrown into real world problem solving involving perimeter without any background knowledge.  The treatment of area and perimeter might be considered a bit uneven. I guess they want to make sure to cover the topic of area while going around the perimeter topic.

In fourth grade students are expected to apply the perimeter formula for rectangles in real world and mathematical problems without the concept of perimeter or its formula ever having been introduced, addressed, or developed.  I guess with enough deep conceptual thinking students will automatically know what perimeter is as well as the formula.  In fourth grade:

Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

 

Will the Standards Pass the Test?

The CCSS math standards are one of a kind—unique among math standard sets. I have looked at a lot of sets of math standards—most of the best ones and enough others to recognize bad ones. I have not seen any like the CCSS. I have heard it said the CCSS math standards were patterned after the Indiana math standards. If that is true, they did a terrible job. Saying the CCSS were patterned after the IN standards sounds great but I bet you the authors of the math standards could not tell you the process they used to do this patterning.  If they even attempted to articulate the process it likely would not be step by step.  Instead it would be by leaps and bounds and resemble jumping to a conclusion of such patterning with no evidence   (they already have a long established no evidence pattern).

You will find similarities among most sets of math standards. The wording of the CCSS math standards is significantly different from the top rated standards.  The CCSS are worded in such a way as to dictate pedagogy for too many of the standards in spite of proponent’s claims to the contrary. Take the tiling standard, oh, sorry, I mean the area standard presented earlier and compare it with WA’s grade 4 standard:

Determine the perimeter and area of a rectangle using formulas, and explain why the formulas work.

The CCSS standard dictates tiling while the WA standard simply says get the job done and leaves the how to do it up to the professional judgment of the teacher. By virtue of the pedagogical ideas that are inherent in them, the standards may result in the adoption of severely deficient textbooks and programs that value process over content and emphasize a student-centered and inquiry-based approach.

The verbs and how they are used in a set of standards tells a lot about the standards and helps identify the instructional intent behind them (i.e., constructivism, explicit).  The March 10, 2010 draft of the CCSS used the verb “understand” as the leading verb 27% of the time in the K-8 standards.  That was cut down to a little more than 9% in the final standards (still too much).  “Understand” can be difficult to teach and assess so the fun really escalates when you strive for deep conceptual understanding.

The Indiana, California, and Washington standards were among the top rated standards in country. They were written in such a way as to pass what I call “The Man in the Street Test”.  That is where the average man in the street would be able to read a standard and know what was meant.  The CCSS will only pass that test if the street is Pi Avenue or Pythagoras Lane and populated by mathematicians and they likely will only recognize the content and not have a clue about the embedded pedagogy.

Excuse me now while I go tile my living room walls so I can figure out how many cans of paint I need to purchase.  How am I going to keep all those tiles up there?  I only have two hands.  Rather than trying to hold all the tiles up at once maybe I will outline them one at a time on the wall and see how many it takes.  Will that count?

This part of the series, Concerns About the Math Content Standards, only considered a few concerns.  There are many other concerns that were not addressed here.  Part 3 of this series will provide some information about the Standards for Mathematical Practice.

02may27-sale-if-you-suck-at-math

 

Even grocery store employees have to express their answers in more than one way.

 

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5 Responses to Common Core State Standards for Mathematics: Does It Add Up or Down? Part 2

  1. Cathy says:

    As for my child’s 6th grade CCS math class: the math teacher has quit and is “moving to be closer to family” and half the class is flunking or is near flunking the class.

  2. Cecilia says:

    Interesting to read part 1 abt perimeters and tiling. My first grader is doing this. Plus at the end of her worksheet it automatically assums she can do geometrical calculations to figure a perimeter when she doesnt get the concept of what they are teaching. Did I mention this is first grade? What 1st grader can do geometry? My 3rd grader is at the top of her class. Consistently for 3 yrs her diebel sept test shows her higher than when she left in june. Why? Because I teach basic reading, writing and math during the summer. At first I thought it was the teacher. This year I clued in and asked why from Sept to June does she have a downward spiral? After reading the timeline implementation it fits perfectly! It wasnt necessarily the teachers, its the curriculum! I am on a journey now to teach my kids the “stuff” in between. There just aeems to be no cohesive “glue” to this form of teaching.

  3. Rich says:

    I do not know if you are deliberately misrepresenting the facts or really believe what you write.

    For “tiling,” which you seem to think absurd, the standard states: “Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.”

    You cannot just read out the “and show that the area is the same as would be found by multiplying the side lengths” part of the standard. Learning that a 8X10 kitchen would require (and is equivalent to) 80 one-foot square tiles hardly seems absurd.

    (Tiling is so absurd that the WA standards you praise say this about it: “4th Grade 4.3.B: Determine the approximate area of a figure using square units.Draw a rectangle 3.5 cm by 6 cm on centimeter grid paper. About how many squares fit inside the rectangle? Cover a footprint with square tiles or outline it on grid paper. About how many squares fit inside the footprint?” Loony, right!)

    Nor is that the last word on area–you write as if once tiling is learned, it is the only strategy kids will ever be taught. But as you note the standards go on to say “Multiply side lengths to find areas of rectangles with whole- number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.” So, feel free to trace tiles before you buy paint, but my third grader will know how to measure the height and width of the wall, then multiply them together to get the answer.

    It is also disingenuous to write that “finding the area requires multiplication yet the standard algorithm for multiplication isn’t required until grade 5.” The 3rd grade standard requires kids to learn quite a bit of multiplication and division, including “Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?” which is practically Algebra and “By the end of Grade 3, know from memory all products of two one-digit numbers.”

    You seem to acknowledge that students will learn multiplication facts, but oddly you claim it is “presented to alleviate the concerns of those who think the standards do not call for fact fluency or memorization.” I cannot follow your logic–essentially you are claiming this was included so that people who thought it was not included cannot say it is not included. Looks like a blatant attempt to characterize the inclusion of something you apparently want as somehow insidious.

    You also state that students are asked to calculate perimeters “without the concept of perimeter or its formula ever having been introduced, addressed, or developed.” How you reach that conclusion is beyond me. You present the standard–“Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.” Your imagined class room must be one where teachers do not teach.

    In fact, in the WA standards you praise, to use your phrase, “the word perimeter does not occur in the standards until this standard appears in third grade: “Solve single- and multi-step word problems involving perimeters of quadrilaterals and verify the solutions. I guess under your standards, “the concept of perimeter and its formula are never developed. Instead, students are thrown into real world problem solving involving perimeter without any background knowledge.” (II also not that under the Common Core, students are finding area in 3rd grade, which does not happen until 4th grade under the WA standard you praise.)

    In addition, you denigrate “strategies base on place value,” but do not acknowledge that the WA standards you praise advise the exact same strategies. Read 3.2.D and 3.2.G in those standards http://www.k12.wa.us/mathematics/Standards/WAMathStandardsGradesK-8.pdf.

    Finally, I am not sure what the meaning of your last line is. “Even grocery store employees have to express their answers in more than one way.” If it is meant as a dig on the Common Core, it fails utterly. As I hope you realize, the sign is not presenting the same answer in two ways. It is supposed to reflect a price drop: Was $3. Now 4 for $12. Perhaps if the employee had learned, as the Common Core sets out for third graders, to “fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that $12 ÷ 4 = $3, one knows $3 x 4 = $12),” the employee would not have made such a silly and obvious mistake.

    Of course, there may be other reasons to criticize the Common Core.

  4. Momofmany says:

    Other reasons to criticize common core? Are you kidding? It’s delays in math are ridiculous! My dd and grand dd brought home 8 basic math problems. Third grade. Ok one problem is 99+63. Easy right. You add 9+3 carry the one. Ok so then you add 9+6+1. The answer is 162. Easy. Wrong. You need to find a “friendly number”. What friendly number? What is a friendly number anyway? The friendly number isn’t given you have to figure it out. In this problem the friendly number is 1. 99+1 and 63-1. Then you add and subtract. 99+1 is 100 and 63-1 is 62. So 100+62=162. It was much easier to carry the one! But the friendly number can change dependent on the math problem. I’m sorry but you seem ignorant Rich as to what is really happening here. My son didn’t understand math at all until I homeschooled him for the most of last school year. He went from that crap to knowing his multiplication tables. Not taught now! He learned how to divide. Now he speeds through problems. Common core is the worst curriculum I’ve ever laid eyes on and I had homeschooled for 26 years prior to putting my children back into public schools. No wonder homeschooled kids kick butt when it comes to college entry and why we have 15 year old homeschooled students with bachelor degrees!

  5. Pingback: Sophia Abelita—Common Core Addition | Stop Common Core in Washington State

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