1) What, if any, Common Core standard does the problem address?
2) How will solving this problem (or similar ones) help and/or hurt students?
3) What is the more traditional way of teaching the concept?
4) Is there any research that supports either the old way or the new way?
When I did a search for horrible Common Core math problems, one of the first sites that came up was "The Ten Dumbest Common Core Problems." So I will start with their first problem and go from there. Here it is.
In case you can't see the problem very well, it asks students to use number bonds to help them skip-count by seven by making ten or adding to the ones. Here is my analysis:
1) What, if any, Common Core standard does the problem address?
I did searches for "number bonds" and "skip-counting" as well as a quick skim of the math standards for the elementary grades. There is no mention of "number bonds" anywhere in the math standards for any grade. "Skip-counting" is mentioned once in 2nd grade as such:
2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s.
In this context, it is clear that by "skip-counting," they simply mean to count by 5s, 10s, and 100s. But skip-counting by seven is certainly not mentioned, nor is anything called "number bonds." Now, I realize that the top of this worksheet clearly says, "NYS COMMON CORE MATHEMATICS CURRICULUM," but this goes to show that just because something is labeled "Common Core" does not make it so.
After searching some more, I found this standard in 1st Grade Mathematics:
1.OA.6. Add and subtract within 20, demonstrating fluency for addition and
subtraction within 10. Use strategies such as counting on; making ten(e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to
a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between
addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8
= 4); and creating equivalent but easier or known sums (e.g., adding 6 +
7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
I bolded the applicable section of the standard because I think the above problem is an example of making ten to add. However, I think it is also clear in the standard that students should be able to add and subtract and should know a variety of ways of doing so. This standard doesn't seem to dictate that this particular method has to be taught, but that various ways of adding and subtracting should be taught so that students become capable of adding and subtracting fluently.
As I researched a little more, I found that the term "number bonds" seems to come from Singapore Math, something from which the Common Core math draws heavily.
2) How will solving this problem (or similar ones) help and/or hurt students?
My mother learned to add by playing Dominoes - the kind where the ends of the strings of dominoes have to add up to multiples of five in order to get points. Then, when someone goes out, you add up the dots on your remaining dominoes and those points go to the person who played their last domino. We occasionally played growing up. I remember my mother teaching me that the quickest and easiest way to add up the points was to make groups of tens. So, if you had an eight, you would group it with a two, and so on. When we reached the last dominoes that no longer made tens, we regularly used the above method.
Of course, adding 7 + 7 is a simple problem that children should be able to answer from memory, but that doesn't mean that the concept of making tens is useless. What if we replace the problem with 38 + 5? When I do a problem like this in my head, I automatically break the 5 into a 2 and a 3, group the 2 with the 38, and know the answer is 43. I don't have to think of the individual steps in my head, but it is certainly the way I visualize the problem. And it helps me get to an answer much more quickly than thinking, "5 + 8 is 13, carry the one, add to the 3 in the 10s place, the answer is 43." Both methods work fine, but I like knowing both and being able to choose the one that works best and most quickly for me.
But these are both anecdotal evidence that tell us little about whether learning this method will help or hurt students. How does solving this problem help students? What can they learn from the problem? I would assert they learn the following:
- Numbers can be broken down into their parts and then added back together without changing the outcome.
- Grouping numbers in 10s can be a quick and effective way of adding.
- Adding in your head is simpler when utilizing groups of 10s.
- If you can't remember a simple math fact, you can figure it out quickly.
And how does it hurt students?
- Some students may not memorize their math facts because they know an easy way to come up with the answer without memorization. (Not sure this is entirely a bad thing. I would rather students know how to come up with the correct answer than just memorize it.)
- Parents who do not understand this type of problem become frustrated and are unable to help students. This can alienate parents from school involvement. This is a HUGE problem, but one that can be solved by better teacher-parent communication.
3) What is the more traditional way of teaching the concept?
Memorization. Rather than number bonds, fact families were used. Here is a site I found that explained some of the difference between the two.
4) Is there any research that supports either the old way or the new way?
In searching for research on the topic, I came across a study entitled "Teaching and Learning Mathematics." It is a fairly extensive list of things research can tell us about teaching and learning math. These are just a couple of the claims listed.
Practice toward mastery of basic skills and procedural algorithms should not occur until students develop the meaning underlying those skills or algorithms. Research results (and frustrated teachers) consistently suggest that if this practice occurs too soon for a student, it is very difficult for that student to step back and focus on the meaning that should have been developed at the very beginning (Brownell and Chazal., 1935; Resnick and Omanson, 1987; Wearne and Hiebert, 1988a; Hiebert and Carpenter, 1992).
Students trying to master the basic addition facts should be given experiences with the derived fact strategies. For example, 5+6 can be transformed into [5+5]+1, which can be solved by the sum of the easier double [5+5]=10 and 1. Because this strategy builds on a student’s number sense and meaningful relationships between basic combinations, it improves fact recall and provides a “fall-back” mechanism for students (Fuson, 1992a; Steinberg, 1985).
The above paper lists several studies that support the claims. I found additional studies to support these claims fairly easily, but none that refute them. All of the studies I found showed that student who understood concepts like number bonds or skip-counting were significantly better at mastering math facts. If anyone knows of studies to the contrary, I would love to know about them.
In summary, this type of problem is not dictated by Common Core, but is certainly in line with the standards. Solving problems of this type has several benefits for students. Research supports this assertion. The only negative is misunderstanding on the part of parents. This is a major issue that must be addressed. Parents must be given enough information to help students understand and complete homework assignments. They should also receive some sort of explanation of the reasons a new concept such as this one is being taught to their child. Getting parents and teachers on the same page is essential to student success.
