# Fuzzy math and the related Common Core Standard

October 5, 2013

Remember the Prego commercials- It’s in there! While some would like to promote the false idea that CC doesn’t include fuzzy math, the standards clearly call for these methods. Here is a little example that a viewer sent to Rich Hoffman at News 12.

Here is the standard that illustrates these methods:

• CCSS.Math.Content.1.OA.C.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).

In grade two they call for the use of these strategies to show FLUENCY in addition and subtraction of numbers 0-20. I thought fluently meant quick, efficient and accurate, not the above.

• CCSS.Math.Content.2.OA.B.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.
• 2See standard 1.OA.6 for a list of mental strategies.

For the record, knowing all sums of two digit numbers if expected an entire year earlier in high performing countries. The CC delays this expectation for our students. They delay having kids committing these basic facts to memory so they will have to rely on other methods like the ones above.

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1. Betsy Harvey says:

What is the takeaway here? That mastery of a skill set is being delayed or that the example looks needlessly complicated? Decomposition is just another way of performing mathematical calculations. The same as partitioning and partial sum addition. That said, I wouldn’t attempt decomposition until my students were Higjly proficient with partitioning and partial sum. Decomposition is used in prime factorization which is a middle school skill.

• Erin Tuttle says:

Thanks for your post, I understand what you are saying, but it’s interrelated. Supporters of CC say that the method in the example is not required in the standards, and teachers are misinterpreting what they call a traditional set of math standards. They are not traditional, as they require these techniques to be used for several grades BEFORE the standard algorithm is introduced. Fuzzy (reform) math programs don’t encourage the use of the standard algorithm in favor of these strategies. To say the CC doesn’t do this is not true.

High performing countries require memorization of the facts earlier and do not have children using these strategies to solve addition and subtraction into fourth grade.

Parents are watching their children use these needlessly complicated methods which is leaving them frustrated and hating math.

• Smarter Than Y'all, Apparently says:

It just seems needlessly complicated because they’re using small enough numbers that their methods are unnecessary, but this is the exact process I use to perform mental math and I’ve never ever studied the common core curriculum.

• Josh Hulett says:

i completely agree. I have always done mental math this way but I don’t break it down to so many levels. For the 8+6 example I would just transfer 2 from the 6 to the 8 and add 10+4. I think the only downside for CC is that it is excessive beyond the way I have always done in my head.

2. Lynne Marquardt says:

How about 20+10=30. 7+6=13. 30+13=43. But I’m a math teacher. What do I know?!

• Erin Tuttle says:

Yes, that seems easier and to make more sense. I think it is a good method to explain place value, but by the beginning of fourth grade it shouldn’t be used to solve addition problems. By that time, students should be using the standard algorithm and not need to go through these steps anymore. Rigorous standards in high performing countries require this in second grade.

• John says:

Hey Erin,
I wonder where the students in the high performing countries attend college. Maybe the United States? I find that interesting, don’t you?

• Erin Tuttle says:

Yes, I do. It’s a known fact that are selective colleges are filling their spots in STEM degrees with foreign students. They are better prepared for higher level math.

• Kami Mercorella says:

That’s exactly how I would figure it out and teach it. I aced Math as a kid. This CC curriculum makes things too confusing.

3. Katie Thomas says:

Lynne, I immediately solved it the way you mentioned (well first I did it in my head, but then I tried to revert back to elementary school)! There is such a thing as over complicating something! I remember helping my now 4th grader learn by place value…which is basically what you did. Is there suddenly something wrong with that method?? I have to believe that putting one number on top of the other, adding the ones column, 13, carry the 1 to the tens place, so 1+1+2 =4, written next to the 3 in the ones column (43) is a pretty simple concept…..Should fuzzy and math be used in the same sentence? :/

• Erin Tuttle says:

Katie, You are correct. Lining up the numbers and adding in columns is NOT fuzzy math. That is the standard algorithm and is considered traditional mathematics. Under CC the kids, are not to use this method until two years after they have to perform two digit addition. In the meantime they use the overcomplicated methods as prescribed in my post, that is fuzzy math. Thanks for your comment.

• Kristin H says:

I am a math teacher too and I think Lynne’s method presented of partial sums using place value is a good one! However, I also see great merit in the method presented in the original post. The idea of these so called “fuzzy math” methods where students are not using the standard algorithm is to be able to do addition and subtraction of larger numbers in their head without putting pencil to paper as the algorithm would call for. I tried to think of another example where I would definitely use this method in my head. Try 209 + 38. Would you think it was too confusing to be able to say this is the same problem as 210 + 37? Being able to rearrange addition problem like that shows great number sense–something that is sorely lacking in most middle school students today. I am glad this type of number sense is now being explicitly taught in elementary school. Hopefully teachers will become more skilled at teaching it in a less confusing way as they get more training in teaching the Common Core.

4. Ben S says:

Gotta love when the shortcut is more complex than the actual problem!

5. Stephenie Loney says:

I am a second grade teacher and I have to say CC is definitely challenging. Teachers still need to remember we are accountable for teaching based on what our students need and what is the best practice for teaching. You must gauge your students like a thermostat and if they are not responding to your methods of teaching it is time to switch gears. I teach speed drills making sure my kiddos are fluently adding and subtracting 0-20. They also need to have mental strategies to quickly add and subtract numbers 0-20. I teach fact families, adding 10, finding a 10, flip-flop, adding 1 and subtracting 1, doubles, and place value. I know Common Core is critical to what they are mandating for us to teach but I also know we are here to make sure these kids can perform. We cannot forget that those high stakes tests are what teachers are evaluated on. I want to make sure that students enjoy what they are learning and not take away their joys of discovery. The one thing that is positive about CC is that it reveals the answers through discovery and not rote memorization.

• Erin Tuttle says:

Thanks for your comment. I’m encouraged to see that some teachers are adding to the prescribed curriculum of the math standards and encouraging fluency. I applaud you for tempering high-stakes testing accountability for what is right for the students. I don’t see how our old standards did not allow for conceptualization and mandated strict rote memorization. Discovery can be effective, but it can also slow the progression down quite a bit. While Common Core students are stuck in “deeper learning,” students in high performing countries are getting to the next level and are more likely to reach Algebra 1 in eighth grade. This is crucial for them to attain higher levels and courses of math in high school, the greatest predictor of college success. Under CC, most students will only get to Algebra 1 in 9th grade where it is placed. The progression is too slow because so much time has to be focused on conceptualization and application, not procedures. The only path for kids to do Algebra 1 in 9th grade under CC is to do all the seventh grade standards and half the eighth grade standards in 7th grade. In 8th grade, they do the other half of 8th grade standards and all of Algebra 1. This doesn’t allow for a comfortable progression and sets up a tracking system where only students with exceptional math abilities will be tracked into this option.

I guess a teacher could teach students the standards in the next grade level to quicken the pace to Algebra 1, teaching 3rd grade standards in 2 grade and so on. Has this option be explored at your school?

• Kristin H says:

I wonder, however, what is the problem with students not reaching “Algebra 1” until they are in the 9th grade? Is the point of speeding the progression of taking Algebra 1 so that our students can get out of it faster? Speeding through a course as important as algebra is not going to help the majority of kids in the long run. Common core has a focus on college and career readiness; in major surveys of college professors, they have overwhelmingly noted algebraic skills as the most important prerequisite math skills for success in college, but have said they find them severely lacking. Is it because so many students are being pushed through it at an earlier grade level than they are ready for? The middle school common core standards place a major focus on developing algebraic thinking in a much more meaningful way than in previous middle school standards. They are in effect getting what used to be Integrated Algebra in 6th, 7th, and 8th grade and the 9th grade curriculum is more challenging than it used to be as “Integrated Algebra” (New York State). Trying to cram multiple years of the math standards into 1 year will ultimately be detrimental to our students (and as an 8th grade teacher, I know it’s damn near impossible to do it well).

Is the point of speeding the progression of taking Algebra 1 so that students can get to calculus in high school? When you look at higher performing countries, their students do not see calculus until post-secondary education, period. Being high performing in math does not mean getting to higher-level courses earlier. It means having a deeper understanding of the building blocks of mathematics that then is a solid foundation on which all other math learning can be built. They are high performing in math because their students took the time when they were in 1st and 2nd grade to focus on addition and subtraction and ALL the different ways it can be done.

There is certainly much training for teachers that needs to be done on how to effectively implement the Common Core standards. Conceptual understanding is not the only tenant of the Common Core. I recently heard one of the authors of the Common Core standards speak and he continued to emphasize that instruction should be equally weighted between conceptual understanding, fluency, and application. So while conceptual understanding is being built in addition of double digit numbers, there is a focus on single digit addition fluency. The benefit I see in delaying the standard algorithm in favor of alternative methods is that learning how to break apart numbers in different ways will ultimately be a skill that leads students to greater number sense. A method like the one in the original post actually makes students think algebraically at a very early age (“what is the number that I need to add to 26 to get the next multiple of 10 which is 30?”). Algebra is so important that the Common Core standards are starting to include it as early as 1st grade! Once a standard algorithm is taught, it is very difficult to get students to think about alternative methods. As a middle school teacher, I deal with this every day. My students cling to standard algorithms they have been taught but do not understand and when they are asked to do anything that does not have a standard algorithm, they shut down. Their poor number sense makes learning the middle school standards near to impossible.

I am not saying that every teacher knows how to implement the standards properly, nor should they know without being given sufficient training and support. I am saying, however, that the problem seems not to lie within the standards themselves but instead within the haphazard implementation that has been expected of teachers without the proper training.

• Erin Tuttle says:

Where to begin? There is no need to rush children to get them to Algebra in 8th grade if the standards are scaffolded correctly. Early mastery of Algebra is the greatest indicator of success in college. Increasing the progressions a little each year is appropriate, not cramming it in. The CC does offer an accelerated path to Algebra 1 in 8th which does seem like a nightmare. They suggest students all receive the same track through sixth grade. They split in 7th, with a very select few receiving all 7th grade and half the 8th grade standards in 7th grade. In eighth grade, they get the other half of the 8th grade standards and all of Algebra 1. This is a bad idea and will indeed rush students through Algebra.

Yes, some kids need the opportunity to reach Calculus in high school. Many selective college and those seeking to enter STEM fields must have it in order to compete against our international competition. I don’t know what high performing countries you reference in your post, but all the top performing international students who come to US colleges are starting in Advanced Calculus. Not giving American students the same opportunity is unfair.

I believe students must understand number sense, but holding off on the standard algorithm until sixth grade is more than ridiculous. The CC takes it too far, and students progression to Algebra is made much too slow. Teaching algebraic thinking in 1st grade is a HUGE waste of time. I’m sure you must be unaware that 500 Early Learning Experts wrote a letter for the standards writers to redo the K-3 standards. Children don’t develop the mental processes to think abstractly until around 11. This is why there is so much frustration going on in the young grades with CC. Please watch this video to explain this in greater detail. http://www.youtube.com/watch?v=vrQbJlmVJZo

6. Sheree says:

My son always was a straight A student never had a B or lower on a report card until after they implemented the common core into the schools curriculum. I think the different methods are confusing the kids. I hate the way they are taking basic math and making so many more steps to it. The Lattice multiplication is crazy to me! Math was always my best subject love it and I love numbers.. Even I get confused with a lot of the methods

7. Debra Klinger says:

Many local school systems are using the Everyday Math curriculum… except parents refer to it as the every NIGHT program because every night they teach their children math. My middle schooler in honors math said the teacher asked at the beginning of the year how many kids did math with an enrichment center or tutor outside of school and all but TWO students raised their hand. High achieving math kids aren’t getting that way at schools with Everyday Math. This method requires students to utilize non-standard algorithms (such as partial sums for addition and lattice multiplication). It’s inefficient and confusing, at best. Reform math programs like Chicago Everyday are known as a “mile wide and an inch deep”. They purport the benefits of a spiraling curriculum but that plays out to mean that a second grader will touch on time, money, probability, place value, fractions, double digit addition and subtraction, measurement, geometry, the list goes on… and master none of it. The teacher must have children working in small groups, exploring the answers – sounds good, but limits direct instruction time to SHOW them how and demonstrate using many examples. The reformists would scoff at that idea calling it “no kill and drill”. Many local and national high performing schools are using Math in Focus (a Singapore Math curriculum) or Saxon math with highly quantifiable and immediately positive results. These programs, used by the most successful math countries in the world from Singapore to Finland (and the other dozens ahead of US) are the exact opposite in nature. They require standard algorithms and mastery of basic concepts in a successive common sense order. If you’re interested in more, have a look at Parents Against Everyday Math on Facebook – it’s full of great information on reform math and the effects of common core.

• Erin Tuttle says:

Debra, I enjoyed your post. I have often heard that Everyday Math programs have high numbers of students seeking help outside the class. Unfortunately, not all kids can afford to do this and many will fall behind. What really irks me, is that the school with over 50% of their kids in Kumon or another program, will claim Everyday Math is the reason their scores are high on the standardized math tests and become huge advocates for other schools to adopt EM. It’s such a racket.

Check out Barry Garelick’s Facebook or World Class Mathematics, he posts great info.

• Ray says:

I implored my child not to use fuzzy math to solve her addition problems. I asked her to solve three digit addition problems using common core fuzzy math. Explaining this new fuzzy math required extra steps. Had she followed traditional logical steps, as she knows how, these three digit problems would have been solved easily.

Although she understands the concept of solving two digit addition problems by tens and ones ( 20 + 32I 50 + 2 = 52 ), that method is more long winded for her. I implored her to continue to use the traditional method to solve addition problems. She is a second grader doing 4 digit division and multiplication problems in Kumon. Traditional arithmetic methods in Kumon helped her move at least 2 years ahead of most of her peers.

• Erin Tuttle says:

Keep up with the Kumon- it is one lifeline parents have to combat the fuzzy math. Your daughter is lucky to have an involved parent.

8. Michael says:

Honestly, it looks confusing at first, but after reviewing it, this is the way I have been doing math in my head since I was a child.

My math teachers never understood how I got to my answers, but I was correct in the final answer.

Im not a math teacher, or even knowledgeable of advanced math.
Its basically just balancing the equation.

Round one side up to make a ten, and subtract the rounded difference from the other side.

+4 to make 30, than balance that additional 4 from the other side, so 17 becomes 13.

30 +
is easier and quicker than 26 +

But there doesn’t have to be a specific order. You can choose to do this to the left side or the right side. adding up to the 10, or down to the 10.
its all the same concept, just a different preference.

• Terry Lewis says:

That’s the way I have done math since I was a child too (raised in the 60s) and I was the quickest child in class in coming up with the answers. I was helping my fourth grade grandson with some problems recently and how to do the problems using common core and I didn’t even realize that’s what I was doing. Didn’t find out until later that’s what it is called.

9. Blackraven says:

I have BS in Electrical Engineering and graduated with high honors. I also received a Masters in Engineering from Korea. They way they are teaching math here is complete wrong and adds more complication than necessary. It is confusing to the student and confusing to the parents of the student. It does not teach a short approach or quit mental methods, like when adding any thing ends in a 9 subtract -1 and move it over to the next spot.

Example

19 + 7 = 26… see 7 subtract one move it to the tens spot.

This is just a quick mental short cut.

It is better to teach math by going linear across starting with the very right in adding and subtracting and move left. They are making it so our students cant do basic math problems.