After writing my wildly-popular What Common Core is NOT post last week, I realized something: most people are developing opinions about Common Core State Standards based on worksheets and work samples being passed around on Facebook. Instead of allowing a few examples of how Common Core math standards are being implemented to shape your views, here are additional examples of worksheets from a variety of publishing companies that aim to target these standards. I collected these images by typing the search terms “awful common core math” into Google and viewed the images. As these have been spread over and over again, I do not know how to give proper credit, but feel free to comment below if you know where credit should be given.

**Reminder:** Common Core is a set of standards, not a curriculum. These are merely interpretations of how those standards can be taught.

Without each worksheet displaying the grade level, I will attempt to sequence these from most simplistic to most complex and provide a description.

**Worksheet #1:** I would imagine this worksheet is part of a Kindergarten or 1st Grade math curriculum. This particular image seems to be causing a great stir on social media. Here’s what most people fail to recognize – You did not sit in class while the corresponding lesson was taught; your child did.

This appears to be an extension activity, an engaging way to continue a hands-on learning experience the children engaged in during class. I’m picturing tables set up in groups with a variety of manipulatives for children to explore. The idea appears to be to teach children that 1.) you can count anything (fingers, small blocks, etc.) and 2.) you can use many different methods to add up to the same sum of 5. This particular worksheet seems to be giving the children the freedom to create their own methods for arriving at 5. Using the basic concept of one-to-one correspondence (something that has been taught as far back as I was in Kindergarten for sure), children are matching the number of colored blocks to the number of fingers being held up. I would imagine if you colored 3 blue and 2 red, you’d likely hold up 3 fingers on your left hand and 2 on your right hand. When you ‘bond’ those squares, fingers, etc. together, you have a sum of 5. Giving children the freedom to select their own numbers with a sum of 5 allows them to think about the problem multi-dimensionally, teaching different number families.

**Worksheet #2:** This one is pretty straight-forward. It’s exactly what we were taught as children when learning to make sets of 10. Only difference is that they are laying on their side (we generally saw these sets of 10 positioned vertically), and they are drawn in two rows of 5 each. I personally would not choose to represent 10 as two sets of 5, but I’m guessing it ties in with previously-learned material (like skip-counting by 5). This worksheet demonstrates that when we subtract, we can begin by showing our groups of tens, draw circles inside each small box indicating the larger number (14), shading in the number we are subtracting (8) and counting how many circles are unshaded (6). Rather than asking the child to solve the problem, the worksheet is asking which image represents the problem visually. This will prepare children for creating their own visual representations for more complex problems. **Worksheet #3: **This is the type of math worksheet I remember completing as a child to learn place value. To understand that each digit in a 3-digit number represents ones, tens, and hundreds, children will expand the numbers by basically creating an addition problem. This is very similar to the method I illustrated to do addition in my original Common Core post. Again, children were probably given manipulatives in class to practice this concept.

**Worksheet #4:** My nerdy math self must confess something – I *really* like this worksheet! I’m guessing this is first grade math, and I actually had to think about it. I genuinely appreciate being shown all of the different ways to approach the same math problem. Since this is titled “Solving Another Addition Story Problem”, these concepts have been taught previously, so this is a review. The text clearly describes each method being used and the numbers represent the method visually. I kinda think it’s brilliant! Using number lines, hundreds chart, skip-counting, Unifix cubes, rounding off – these are all skills that have been taught in isolation, integrated into one assignment. It got me excited about helping my children with their math homework one day. Many critics of Common Core suggest that the curricula being created to target the standards is “dumbed down”, but to be honest, this is the type of work I remember doing in my gifted class in elementary school as it’s actually thought-provoking, relying little on rote memory to solve a problem.

My husband made a good point when we discussed this the other day – “Since when is a child supposed to earn an A on every assignment?”and that applies to this worksheet. Perhaps your child will earn a 50% trying to solve problems one way, an 80% solving problems a different way, and 100% solving problems the third way. Clearly, it demonstrates what method works best for your child and how he/she will probably solve problems independent of schoolwork.

**Worksheet #5:** This is an incomplete worksheet, probably 3rd grade math. When we use the standard algorithm for multiplying a 3-digit number by a 1-digit number, we have to both understand what it is that we are doing and keep tabs on what place we are in as we are multiplying. By expanding the problem as it is done below, it’s much simpler to solve and easier to keep your place as you go. Now students can also apply the multiplication facts they have memorized to quickly solve the problem. Great way for teaching more complex multiplication problems.

**Worksheet #6:** Final example from a third grade workbook. This simply teaches the relationship between multiplication and division. The terminology being used is the same terminology I remember using as a child: number sentences, whole numbers, and product. Again, instead of teaching multiplication and division separately, they are being taught concurrently. Makes perfect sense.

After seeing all 6 examples of math worksheets reflecting the Common Core standards, do you still take issue with the methods currently being used to teach math? If so, why? Keep in mind, you were not present when these lessons were being taught; your children were. Homework is an extension of what is taught at school; it is not the instruction itself. However, I am certain if you asked the teacher to demonstrate the methods being used, you could easily schedule a conference to learn more so that you can help your children if they struggle to grasp the concepts.

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Thanks for your rational and informative post. If the purpose of Common Core is to provide a variety of methods by which a child can learn, why are teachers such sticklers for the particular way a child is taught? Wouldn’t the method a child’s parent presents during homework help be as effective? Yet I cannot tell you how many tears and tantrums I encountered because I was “NOT doing it the way Mrs. SoandSuch does!” My children are aged 23-27, so I am definitely not in the midst of the Common Core fray, just curious, as it seems the insistence on methodology is as strong today as it was 15-20 years ago.

I am not sure about your location or time frame, but most of us accept whatever method works for the child to successfully solve problems. We teach different strategies because not all methods work for all children.

I don’t know any teacher who doesn’t accept a child-selected method, other than just an answer listed with no supporting work.

The problem I have with the common core ideology is that it ISN’T what we taught or HOW we were taught in school. I think we, as parents, struggle with it, because as you state, we weren’t in the class when it was taught. It makes it very difficult to help with homework assignments when we as adults, who have been doing these things the same way for many years have to try and grasp what the teacher is trying to show. Young children don’t exactly have the knack for explaining things clearly if they’re confused to begin with.

I also have issue with the mutliple variations to a point. Most manufacturing jobs now don’t want “free thinking”. They have a standardized set of instructions and require everyone to do everything exactly the same way every time. Not sure if this new methodology is going to help or hinder once this group of kids hits that point.

Just my thoughts

In the long run, I think that “free thinking” will help us to re-establish ourselves as a nation of innovators on this planet. No one is advocating that children get wild and crazy. Part of the teaching is determining what works/keep it, and what doesn’t work/discard it, and what is just way too over the top/let it go.

Kathleen, perhaps you will accept it, but the child does not necessarily believe that to be true, otherwise there would not be the drama at the homework table. As I said, my days in childrearing are long over. I did my homework helping duties through multiple school districts in Texas, as well as overseas with a US based curriculum and certified American teachers.

I have worked intimately with both the adults and the children of the countries where our manufacturing and technology industries have been moved. I have seen the focus which the children of those countries apply to their studies. I have spoken to parents and students about the pressures placed on them to succeed scholastically. The reality is, that if they don’t do well, they don’t advance to the next grade…period. These children have tutoring after school for every subject, not because they are not capable of working at grade level, but because they must perform at grade level or above in order to get the grades necessary for advancement. Parents will work multiple jobs and do without many things in order to make this possible. There is an expectation for excellence for these children…in schoolwork and in behavior. It was refreshing to work with them. It was also eye-opening.

I have asked a few business owners what we could do (as a STEM center – Science, Technology, Engineering, Math) to prepare students for success in their businesses. They said they could train them to do the work. What they needed were problem solvers and collaborators. This is true for executives and those who work in the field in construction.

Brian, you should read the other article he wrote :”http://huppiemama.com/common-core/”. My son is in 5th grade and went through all that other / newer way of doing math. It is NOT Common Core, it was called Everyday Math. Common core is just the set of standards that students should be able to reach at each level of school.

For this country to succeed and progress the kids have to reach higher goals. Teaching them to be able to work at a manufacturing plant is not a higher goal. Getting them to invent what the manufacturing plant is building is the goal.

Sean, I beg to differ, but who do you think is going to PRODUCE the product at the manufacturing plants? You can’t have a manufacturing facility with nothing but a board room full of engineers and inventors going to meetings. This is one of the big problems right now in this country and one of the reasons so many jobs are being shipped over seas. Parents for the last 30 years have been pushing their kids into getting desk jobs and now there aren’t enough skilled trade folks left to actually make the products.

I’ve done some additional reading on the common core subject and now understand the difference between that and the teaching methods. Unfortunately, that still doesn’t solve the problem of helping our children with homework assignments when we as adults weren’t in the classroom when these new methods were taught. How does that help them succeed reach higher goals?

But even in the manufacturing plant, things may go as planned one day, then run into a problem the next day. We want our students to be able to work that problem out, rather than having to run to someone else to fix the problem. That makes them a more valuable employee.

“I also have issue with the multiple variations to a point. Most manufacturing jobs now don’t want “free thinking”. They have a standardized set of instructions and require everyone to do everything exactly the same way every time. ”

Sorry, i have to disagree. 1) I’m in a manufacturing job, and we do want free thinkers… and 2) In a meeting where someone says, “oh uh, so we’ll need 418 – 168 lbs of aluminum, and that’s, uh…” the person who uses the above skills being taught to subtract math in their head is totally going to be regarded as more skilled than the person who uses the traditional methods and whips out paper or a smartphone. Just saying.

Cassie, once again, the topic of “in a meeting” comes up. We need reinforcement in the actual manufacturing of the products that are being designed. And as for free-thinking, I don’t think you’re following the same track as I am. Standard Work is a way to get everybody doing everything the same way every time. It directly affects, quality, safety, velocity, and cost in a manufacturing environment. It also helps with traceability to a point. Free thinking IS a great asset, to a point, but there are many many areas where it does more harm than good.

In this day and age, people aren’t expected to do the kind of math you refer to in their heads. Yes, many of us still can, but it’s not a requirement any more. Personally I don’t see how these new methods are any faster than the way it’s been done for hundreds of years. (And I’ve been getting a healthy dose of training in it from my youngest the last few months, believe me! lol)

I’m also a department trainer in my CNC shop, so I completely understand that not everyone learns the same way. Unfortunately, the schools aren’t teaching individual students the best way that they learn. They’re picking a style/ method and cramming it down everyone’s throats the same way we learned when we were kids. The only difference, is now they’re doing it with an unconventional method (not that that’s all a bad thing) that only the kids in that particular curriculum at that particular school are learning.

As I said in my response to Sean, one of the big problems we have with retaining work in this country in the manufacturing industries is a lack of skilled labor because It everyone is pushing their kids to go after the desk jobs and nobody wants to get dirty anymore. No one to run the machines, ok…. shut the plant down and move it to Mexico, or China, or Thailand, or any other number of places.

I’ve been a machinist since 1991 running CNC mills and lathes. I work with the kinds of numbers on a daily basis that are being discussed in these articles. And I can tell you first hand, you won’t find any counting sticks or screens with squares, dots, and penguins on them to help with the math involved. You use computers to do the math. It’s much more efficient and less prone to goofs than someone trying to quick change numbers in their head on the fly.

Sorry this turned into such a long rant, but I’ve been watching our manufacturing base and skilled labor pool in this country all but disappear for the last 20 years because nobody wants to get dirty anymore. It’ sad, as this is what made our country great and it’s all but disappearing.

In response to “I’m also a department trainer in my CNC shop, so I completely understand that not everyone learns the same way. Unfortunately, the schools aren’t teaching individual students the best way that they learn. They’re picking a style/ method and cramming it down everyone’s throats the same way we learned when we were kids. The only difference, is now they’re doing it with an unconventional method (not that that’s all a bad thing) that only the kids in that particular curriculum at that particular school are learning.”

I refer back to the articles… Common Core is a set of standards. How a teacher interprets and chooses to prepare students for success in those standards is what I think you are referring to. There is nothing in Common Core that says to do any math in any particular way. The difference that you may be referring to is that the responsibility of learning has been placed back on the student, rather than the teacher, through the 8 mathematical practices (pp. 6-8 of the Common Core State Standards).

I do agree that we need to prepare students to be “white collar” and “blue collar” workers. In order to do that, we have to prepare for a wide variety of futures (even some careers that haven’t been invented yet!) in the same classroom – thus the need for what we call differentiated instruction. This allows students to learn at their own pace through the mathematical connections they make before exposing them to the more “traditional” methods, which are generally the most efficient methods. We don’t know how they will need to use the skills we teach them, so we have to provide students with opportunities to make more connections and gain deeper understandings. The goal is not just to make students college ready. The goal is to make students college AND career ready.

I’m sorry but if you cant subtract 168 from 418 in your head common core isn’t going to help you to do so. The most inane problem I have seen is where the kids have to do 5 addition problems to determine what 32 – 12 is. Doesn’t that defeat the purpose of learning subtraction?

You are correct that a set of standards is not going to help a student subtract 168 from 418. That has to come from teaching and learning. As far as the rest of your comment, I refer back to the article. We don’t realize that we are already doing “5 addition problems”. In order to subtract 32-12, we have to realize that 32 is not a “3” and a “2”, it is 30+2. And 12 is not “1” and “2”, it is 10+2. Then we have to realize that we are subtracting 2 from 2 and 10 from 30. There is a lot of math that we are doing in our heads. As adults, this is a simple task. But we already learned it. What many teachers are trying to do, in order to build understanding, is allow students to think like a student and work their way up to thinking like an adult. Imposing abstract thinking at too young of an age surpasses the “understanding” part and is why students struggle in later grades, like algebra. Building number sense in K-2 is critical to success in algebra later on.

So, rather than defeating the purpose of learning subtraction, it is helping the student make connections and build foundations for future learning.

She states in her other article on Common Core that there are many more different types of students in our classrooms now than ever before. We have English Language Learners from all over the world, Special Needs children, and behavior needs children who all co-exist in the classroom. Basically, this is what/how ELL and SPED teachers have been doing for years. They teach their students to solve a problem, which may be in a different way because they lack skills in other areas.

I had to teach a K girl her alphabet in sign language, English, and Spanish because she needed the physical movement to remember her letters and had processing problems that prevented her from remembering things long term. Using several methods allowed her to remember ‘A’ in Spanish, ‘B’ in English, or ‘C’ in Sign.

However, just because you have many ways to solve a problem, does not mean you use them all. Students simply need to be able to explain how they got the correct answer by using a method that makes sense to them. In most classrooms, on the assessments, they do not HAVE to use a particular method. They simply have to be prepared to explain the method they used.

All I can say is educate urselves. Read “controlling education from the top” google Dr. James Milgram and see what he has to say about common core math and also Google Sandra Stotsky.

I will do that. But I work with 22 districts in our state and have watched students grades K-12 come alive in mathematics. It is a very exciting time in education. Good instruction is good instruction, regardless of what the standards are. But if we can raise the level of expectations for our children – AND they CAN meet those expectations, why would we want to lower them? American students are just as capable as students in other countries – unless we, as adults, hold them back.

Research is great. Listening to or reading other perspectives is great. BUT, getting out there and watching these kids in classrooms is what speaks the loudest to me. The issue is not the standards. The Common Core standards are appropriate. But we are starting to identify some weaknesses in instruction. That is a GOOD thing! We can’t fix it if we don’t know it’s there. Part of the transition to Common Core is recognizing strengths and weaknesses (in curriculum, in instructions, in assessment, etc) and attending to those. I wish my daughter could have experienced this. She is a junior in college (strong student, ambitious) and I did some investigative problems with her that we would give to a junior in high school (math). I am a math specialist (K-12). We both had to think pretty hard to do the work and enjoyed the struggle. She said that she wish she could have learned math that way.

I won’t go point-by-point, but I have to comment on #1 under Executive Summary (II)… that’s as far as I got before running upon an issue. The standards ARE state-led. The main writers are employed by colleges of education, not a government agency. Then, the standards were made public for K-12 teachers to review and make recommendations for adjustments. Each state may choose to adopt the standards or create their own, which is evidenced by states which have opted out. I will read the rest of this, because I like to consider multiple sides of the issue, even if I disagree. Thank you for the resource!

Sadly, my children’s grades are based on mastering all the various math models. They learned 4 ways to multiply. They are not simply graded on getting the correct answer. They are graded on getting the correct answer with the targeted model. My girls had A’s in math before CC was implemented. Now they have C’s and D’s in math because not all models make sense to all people. I told them to pick the method that works best for them, so their answer would at least be correct. This earned them their first “Needs Improvement” in Work Habits, because they were not using the right model. As a committed overachiever, grade oriented nerd – it hurts to tell my crying girls, that the grades don’t matter. They just need to master one way that works for them.

Before my children’s school implemented CC, I was able to show my girls multiple ways to do any given math problem. We stuck with what worked for each child. Now that is not acceptable. Maybe my children’s school is just not implementing CC correctly, but as it stands CC is hurting my girls.

Wow, no wonder the youth of today are so incredibly stupid.

When I see some of the comments like those above it hurts my head. I have seen with my own eyes where suddenly getting the right answer is not good enough…and as others (here and) before me have said A students become C/D students. What does this do to a child’s self esteem?

Anyway you may want to search Crewton Ramone and see a way of teaching math that is quite effective using manipulatives…if the child understand concepts, they usually can do the math required of them…ie jump thru the artificial hoops common core has created…

I am telling you all, in as polite a manner as I am able to, NC teachers from kindergarten upwards WILL NOT ACCEPT math work shown in any form but how THEY tell their students. My 8th grade daughter and I have to go through this EVERY year. A problem not solved in the manner deemed fit by the teacher or curriculum will lose points, and some with correct answers were COUNTED INCORRECT due to my daughter doing them as I showed her, which clicked, vs. how the teacher had shown. What is that doing mentally to our kids? You’re way isn’t good enough, or not MY way, therefore you lose? What kind of educational curriculum are these people being forced to stick to? Because every one of her math teachers swears its not them, it has to follow the curriculum or it does not count. Period. That, happy mommy it does not make!

i am looking at this. I am a CPA. I went through calculus in college. Algebra was always easy. Geometry was hard. After I figured out what the pictures were, which formula’s to use with what I could crunch the numbers all day long. I think in terms of numbers, not shapes, not distances, and not objects. This stuff gives me a headache now looking at it. I am not a visual learner and never have been. This stuff seems geared only to visual learners.

I think that learning different ways is good, even as i went through the different ways here i got it and understood the reasons behind it and i support it whole heartedly.

The biggest complaints seem to stem from parents with children receiving lower grades after their children switched to Common Core. They’ve valid and frustrating concerns, but it’s a mistake to leap to the conclusion that the change in curriculum is at fault for the drop in grades and is therefore simply bad. Dr. Wells has provided examples of work that show how subtle many of the changes really are (as well as intuitive — I wish I’d learned expanded form multiplication in elementary school instead of college).

Other reasons for declining grades include:

– Teacher may have trouble with new curriculum and difficulty teaching it

— as well as difficulty grading it

– Teacher may be resistant to new curriculum, trying to prove a point with grades

– Student may be in adjustment period, trouble adapting to new method

– Student may not be used to having to actually think about math

– Student may just be really good at testing and not as good at the subject.

– Students, Teachers, Parents, Administration, Everyone — may be uncomfortable with change.

James, I absolutely agree with what you have said! Thank you for these thoughts. I’m glad to know I’m not the only one who feels this way.

I wrote this. http://crewtonramoneshouseofmath.blogspot.com/2014/10/my-two-cents-on-common-core.html

I love this post. I really appreciate you taking the time to explain what is behind the different worksheets.

As a former school-based mathematics coach and district math specialist (now school-based administrator), its great to hear the thoughts of a parent who is open to the new standards. I have to say that the ideas presented in the common core are not new. I have taught math this way since I was a classroom teacher and that was over ten years ago. In my district we implemented the Investigations in Number, Data, and Space curriculum (the sheet you highlighted “Solving Another Addition Story Problem” comes from the student handbook that comes with the curriculum) to a lot of push back from teachers and parents. I find that students who struggle the most are usually not accustomed to having to think their way through a math situation (it’s okay to struggle a little) or they have not had math instruction in the primary grades that focus on math concepts (i.e. skill and drill). As for parents and the homework struggle, I get it! I have two children of my own. I encourage my teachers to carefully choose homework assignments that students can complete independently, or with little assistance. After all, homework should be for review and maintenance and not for learning new material!

My belief is that students need to learn the material before being taught how to think. I am a freshman in high school and i went through a program with similar worksheets as were above, and I never got the quality practice on one method than my parents did. My school participated in a trial program when i was in grades 3-5, and 7, and it was very confusing trying to learn 5 different methods for everything and then scrambling at the last minute to memorize one for the test. For example, in prealgebra in seventh grade, We tried to “discover” the Pythagorean Theorem by working with squares and triangles. We finally learned the theorem and how it was derived the day before the test and learned 4 different ways of completing it. I received a “C” on the test, not because of my lack of study, but because I had to learn a ridiculous method. Looking back at my old worksheets, I can easily solve the problems given in all of the different ways presented to us. This is not because i learned them, but because I had gone through enough math to develop the critical/free thinking skills that can only be gained through life experience to solve the problems in the methods given to us. Thinking cannot be taught, it is learned through years of study and interest. We are forcing small children to learn in a way their minds do not conform to yet. In high school, I am taking advanced geometry, and i am no longer participating in the trial run, and it is a much more straight forward course- we learn one way to solve, we practice it a lot, we build on it, we get tested. Having a standard routine is sometimes the best thing for students, and because material can be taught faster and more efficiently (not wasting time), the struggling students can receive more help in the classroom, a necessary experience for growing minds. Based on my evidence, I would conclude that the “new math” and strategies used to teach students are not suitable to students who have not yet been taught the facts (algorithm). -Emily, a PA high school freshman!!

Emily I actually think that your post might prove that learning in this manner has helped you and not hindered you. You seem to believe that the worksheets are easy now due to your extensive experience in math when in fact you are only in high school. There are many people that have graduated from high school that cannot understand these methods or solve them in the multiple manners that you were taught. Perhaps the fact that you were taught those concepts have benefited you in the way that you actually understand the concepts. The fact that they were hard for you then may just be that math can be difficult to understand. The fact that you are capable of now understanding with ease can likely be credited to a true requirement to not only be able to do the work but to have to fully understand these concepts. A lot of people can still not understand these worksheets and these are for very young students. You are now in an advanced math class which may or may not be the result of the education that you have received but I am sure that your grade school education does probably play a role in that. It’s easy to see fault, such as having received a “C” on a test at some point, but maybe it works both ways. You might get a C if you do not understand the concepts fully, you might get into advanced math classes once you do.

Apropos of Worksheet 5:

7 x 14 = 7 x (10+4) = (7 x 10) + (7 x 4) and so forth illustrates the (right) distributive property of multiplication over addition. If a student in, say third grade, can learn to visualize the break down the steps of a number multiplication in this way, in eighth grade, when faced with 7 x (10 + x), they have a much easier time knowing to move on to … = (7 x 10) + (7 times x) = 70 + 7x.

As a newly retired university professor who saw plenty of students who were mystified by the distributive property of multiplication over addition I’m all for giving a week or two to writing out the notation to illustrate these principles.