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Addition Strategy 2 Doubles & Near Doubles

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Addition Strategy 4 Make 10 & More +7,8,9

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Subtraction Strategy 1 Count Back  1,2,3

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Subtraction Strategy 2 Back Down Thru 10  2,3,4,5,6

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Subtraction Strategy 3 Build Up Thru 10  7,8,9

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Subtraction Strategy 4 Think Addition   Sums to 10, Part 1

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Subtraction Strategy 5 Think Addition   Sums to 10, Part 2

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Subtraction Strategy 6 Think Addition   Sums Greater than 10

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How to Listen to and Build On Student Thinking

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How to Listen to and Build On Student Thinking

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Angelina: First, I made 4 circles. Those were the baskets. Then I started drawing kittens in the baskets, but that got a little hard and messy so I thought, I will just put little squares for the kittens.

Dad: That was a great idea to use squares. How did you figure out that you had 25 kittens?

Angelina: I counted them.

Dad: Could I hear and see how you counted?

Angelina points to the kittens (boxes) one-by-one and counts them. This time she has 24 kittens.

Angelina: That is odd, I got 24 this time. Which is right, 24 or 25?

Dad: Why don't you count them again and see what you get?

Angelina counts and again gets 24.

Dad: So, which do you think it is 24 or 25?

Angelina: I want to count them again.

Dad: That's a great idea.

Angelina: I got 24 again, so 24 kittens.

Dad: Thanks for sharing your math thinking with me. You did a great job figuring out how to solve that problem!

Angelina: Now, can I get those kittens?

Dad: No, remember I said that this was pretend.

The most important thing to remember when engaging your children in mathematical problem solving is to support them to solve problems using their own strategies. Since we all want to help our children learn math, it is often tempting to say, "The way you solved that problem was great, but now let me show you a faster way." Unfortunately, this can give children the message that our strategies are better than theirs. They also might develop the belief that someone else has to show them how to solve problems. Believing that you are a person who understands mathematics and can generate ways to solve problems is essential to success in the mathematics one encounters throughout life. These beliefs are crucial outcomes of our children's elementary school mathematics education.

The problem about 4 groups of 6 worked well for Angelina because it wasn't a problem that she could immediately solve and it also wasn't a problem that was so hard that she couldn't solve it on her own with some work. Her dad supported her to solve the problem by changing the pennies to kittens and suggesting that she draw what was happening on the paper, but he didn't tell her how to solve the problem.

It's not easy to choose problems for your child. With some practice you will get better at it. Here are some problems that might work for your elementary school children. If you choose a problem is overwhelming your child, make the numbers smaller or move to an easier problem.

1. Leon had 8 cookies. He ate 3 of them, how many cookies does Leon have now? (This problem may be routine for some children - especially older elementary school children. For older children you could try 15 cookies, he eats 7; 35 cookies, he eats 12 or 50 cookies, he eats 21.)

2. Ms. Jones has 4 children. Each child has 2 stickers. How many stickers do they have altogether? (For older children try 6, 9, 26, 149, or 247 stickers for each child.)

3. Tyrone has 18 Yu-Gi-Oh! Cards. He wants to have 20 Yu-Gi-Oh! Cards. How many cards does he need to get to have 20 all together? (This problem will work best for children who are 7 or older. For an added challenge, try using these numbers, has 58 cards and wants 61, has 98 cards and wants 101, has 598 cards and wants 601, has 998 cards and wants 1001, has 3998 cards and wants 4002.)

4. Linda has 4 boxes of pencils with 3 pencils in each box. How many more pencils does she need to get to have 20 pencils all together? (This problem will work best for children who are 7 or older. For an additional challenge try 6 boxes with 12 pencils in each box and Linda wants a total of 100 pencils.)

More problems are available on our Mentor Teacher Website. (Look under the teachers' names for lists of problems.)
A personal note from the author:
If you are reading this blog during the COVID-19 crisis, I want to encourage you to be gentle on yourself and your children when it comes to academics. I feel it is disrespectful to the teaching profession and unsympathetic to the realities of family life to expect parents to fulfill the role of a teacher at any time, but especially under times of stress. Parents who are working outside the home to provide vital services that our society needs should be especially gentle on themselves and their families. My children are now grown, but if they were in elementary school and I was able to be home with them, my goal would be to have them solve 2 or 3 problems a week. The number of minutes I would set aside for each problem would be roughly double their age - for example about 10 minutes for a 5-year-old, 20 minutes for a 10-year-old. If you aren't able to set aside this time right now for any reason, please don't worry about it.
pdf file of this paper that teachers can distribute to parents
This blog post was supported by the U.S. Department of Education, through grant award number U423A180115 to Florida State University. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education.

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Doing Math with Your Child - Promoting Problem Solving; Suggestions for Parents by Linda Levi - CGI Math Teacher Learning Center

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Angelina: First, I made 4 circles. Those were the baskets. Then I started drawing kittens in the baskets, but that got a little hard and messy so I thought, I will just put little squares for the kittens. Dad: That was a great idea to use squares.

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The following six video segments introduce fundamental ideas young children learn about counting and numbers. Each video is about four minutes long, and each concept is specifically referenced in the Kindergarten mathematics curriculum standards currently adopted in most of the United States. Accompanied by videos of students showcasing their understanding of these principles, Rob Schoen and Zak Champagne briefly discuss these concepts in the following videos.

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The following six video segments introduce fundamental ideas young children learn about counting and numbers. Each video is about four minutes long, and each concept is specifically referenced in the Kindergarten mathematics curriculum standards currently adopted in most of the United States.

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By now everyone has seen the outrage-inducing image of a third grader's paper in which he is marked down for stating that 5 x 3 = 5 + 5 + 5 = 15. In case you missed it, here are videos and stories from several groups ripping Common Core for it: Business Insider, IFLScience, Huffington Post and mom.me - and I'm sure you can find many more as the photo of this kid's paper has gone viral.

You might also remember the photo of a check that went viral not that long ago - a man filled out a check to his son's school by attempting to write the check amount in ten frames. There have been countless other similar photos with accompanying derision that have gone viral via social media and email (more on these examples in a moment).

In the national discussion on America's perceived educational woes, the Common Core Standards have become a bit of a unifying punching bag, especially with respect to elementary school math. Everyone seems to love a photo of a test question, homework problem or corrected work that vilifies the Common Core. You know the type - the question asks the students to show a seemingly straightforward elementary math topic, but it requires the answer to be given in what seems to be an overly complicated way. We look at it and say, "Why can't they just do it the normal way?!?" We are alarmed at the representation of something that we see as so basic and elementary in a new and unfamiliar arrangement, and we are outraged when we see a student's work marked down when it appears to be correct.

The vast majority of the comments and coverage of these viral images and stories has been highly critical of the Common Core. Here's the thing though - all of these criticisms boil down to a fundamental misunderstanding of the Common Core State Standards (CCSS).

Virtually every example of one of these attacks on the Common Core fall into one of two categories:

1. The people who spread the example (and trash it) missed the point of the Common Core Standard in question

2. The educator responsible for the example missed the point of the Common Core Standard in question

Consider the ten frames check (which falls into the first category) - the father was frustrated by a representation of numbers with which he wasn't familiar and it fit nicely into his preconceived notion that the Common Core is terrible, serving only to confuse students and parents. Here is an article that does a more elaborate job of skewering his response, but in short, this father is upset because he doesn't immediately recognize and understand a concept being taught to his second grader. Rather than try to make sense of it and understand the purpose, he ridicules it, and other similarly frustrated parents jump on board.

In fact, ten frames are a way of visually modeling our counting system that help kids to better understand it. They were never meant to replace our current way of writing numbers - they are designed as a supplemental aid to assist in deeper understanding. It can be frustrating to parents, to be sure, to be initially stumped by your kids' homework, especially when they're in the earliest grades. Certainly, there are teachers out there who don't always hit the mark with an assignment, or who fail to provide resources for parents to understand something that may be new to them, but in the end, let's not forget that we're all looking for the best educational outcomes for our kids. And let's be honest, the way we've been teaching math for generations in America has not worked for everyone, which is why we have a very sizeable segment of our population who simply says, "I can't do math." So then why are we closed off to considering new ways of conceptualizing the foundational ideas of mathematics?

Now consider the 5 x 3 question. According to IFLScience (which I love, by the way), Reddit and Imgur commenters expressed outrage at "the overly pedantic 'by-the-book' thinking." The whole thing reads as an incrimination of Common Core as stifling mathematical thinking in favor of stringent and arbitrary definitions and algorithms. And yet, this is a completely wrong-headed interpretation of Common Core. The outrage is warranted, it's just misplaced - this example is of the second type I mentioned above, in which the educator has misunderstood and misapplied the standards with an overly narrow literal reading of them.

The standard in question says, "Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each." This teacher obviously read this standard as saying that the only way to view 5 x 7 (or in the case of the paper in question, 5 x 3) is as 5 groups of 7 objects each. So for 5 groups of 3 objects, that might look like 3 + 3 + 3 + 3 + 3. And yet, "e.g." means "for example," not "This is the only valid interpretation." A reasonable reading of the standard by a mathematically literate person should allow for interpreting 5 x 3 as 5+5+5, or 3 groups of 5 objects each, especially when you consider that four standards down on the list is the one about commutation (along with other properties) in multiplication, e.g., 5 x 3 is equal to 3 x 5 (note, the e.g. I just used means that this is just one example; the property applies to infinitely many other pairs of numbers as well - see how that works?). The ultimate goal of these standards is to help our children to develop their foundational understandings of our number system and of basic arithmetic, and so if a student intuitively knows that 5 x 3 is equal to 3 x 5 and that they can both be represented as 3 rows of 5 items or 5 rows of 3 items or 3 stacks of 5 pennies or 5 piles of 3 apples or ... well, you get the picture, then we've accomplished our goal!

Interpreting multiplication in the way described above is not remotely new; rather, it's pretty standard fare for understanding what multiplication is. Perhaps the idea of making students show the example on paper is more of a new phenomenon, and yes, the Common Core definitely advocates that educators encourage students to interact with ways of modeling the mathematical concepts they are learning so as to better master them. It does not, however, require stringent adherence to narrow, arbitrarily chosen interpretations of these models, and educators who focus their teaching in that way are doing it wrong.

For another example, consider this image, which I first received in a forwarded email (this particular version of it was apparently taken from the web site of David Van Sant, a recent Republican candidate for the Georgia State House, who lost to a fellow Republican), about a math problem that has gone viral, helping to fire up people against Common Core.

At first glance, the diagram of the Common Core approach might appear unnecessarily complicated, especially when compared to the setup of the standard subtraction algorithm that we all grew up with (not to mention that the image shows the setup of the standard algorithm but doesn't actually show the process, which actually isn't quite so simple with the borrowing that will be required). Most will look at the diagram of the number line, at a glance see a bunch of steps that don't seem to make a lot of sense, and accept it as further evidence to support an already burgeoning outrage toward Common Core, thanks in part to a healthy side dish of confirmation bias.

A closer look at the method, however, can reveal that the use of the number line (an important visual tool in arithmetic and algebra) allows this method to get at a different way of thinking about addition and subtraction and their relation to each other - a vital way of thinking for students who we would like to understand arithmetic on a deep enough level to facilitate the learning of higher levels of mathematics in a meaningful way (which should be essentially all students).

If you haven't yet made sense of the second diagram, think about the way that people used to give change at the store (perhaps a bit of a lost art these days). Suppose you purchased something that cost $8.27 and paid with a $20. The clerk would start at the value of the item purchased (in this case $8.27), then start with the change, bringing you first to $8.30, then to the 50 cent level, then to an even dollar amount, then a ten dollar amount, and so forth, until the value was brought up to the $20 you paid with:

"Okay, $8.27, 30 cents <putting three pennies in your hand>, and 20 more is 50 cents <putting two dimes in your hand>, and two quarters makes nine <dropping two quarters in your hand>, and ten <giving one dollar>, and ten more makes twenty <giving a ten>."

The approach is a perfectly sensible way to give change for humans - it focuses on round figures that we can add and subtract more easily, and it focuses on the true essence of subtraction - the difference between the two referenced amounts. In the case of the change, it's the difference between what you were supposed to pay and what you did pay (in other words, your change). The vertical subtraction algorithm we learned in school doesn't make this clear - it is a memorized algorithm that can be done efficiently with pencil and paper by someone who has practiced it and it certainly can be made to make sense through study of our base ten number system by emphasizing the places of the various digits and the concept of borrowing when needed.

When it comes to doing subtraction problems like this in your head, I suspect that most people who excel at this type of mental math use a method similar to the number line diagram that is shown (the supposedly laughable Common Core example). The ability to visualize and break up the problem allows someone to keep track of the values more easily and to more consistently and efficiently produce the correct result without putting pencil to paper.

Here's a hilarious example from a parent lampooning the Common Core (I found this in a Google image search, but I believe I remember seeing this one make the rounds through email or Facebook):

This problem is a bit cherry-picked in that it requires no "borrowing" in the standard algorithm, so this one is much more easily completed by that traditional method. And yet, the parent's humor aside, the point of the number line model is not to teach the most efficient algorithm to perform subtraction, it's to help students understand what subtraction is.

I'm certainly not suggesting that we shouldn't teach the standard subtraction algorithm that we all grew up learning, and neither is the Common Core. In the CCSS, it's actually referred to as the "standard algorithm" for addition and subtraction, and the CCSS require that students completely master it for multi-digit numbers by the end of fourth grade. I am also not suggesting that the primary goal of math classes should be to enable students to perform complicated mental math without using pencil and paper. The goal of math classes should be to foster a deep-level understanding of the mechanisms that we teach, and that's where compelling students to learn a variety of techniques for subtraction, for example, can allow students to approach a concept from a variety of different directions, using a variety of different tools, and tying it to other concepts they learn. This is, in a nutshell, what teaching mathematical understandings at a deep level should be doing. And this variety of approaches to build deep level understandings is exactly what the Common Core seeks to do.

For all the griping about the CCSS, and in particular the math standards that people love to hate with these images of unfamiliar methods and models or of work that is inexplicably marked down, they're actually pretty good. The idea behind them - in an attempt to improve on many of the learning standards that were out before - is to encourage depth of understanding, and they are definitely geared at that. They're not perfect - as a high school math teacher, I have some issues with how much is crammed into certain courses and certain topics that may be overemphasized. But these things can be adjusted as time goes on, without scrapping them entirely. Or even if they're not improved, a capable and competent teacher should find the CCSS to absolutely be a workable set of standards - an improvement over what we had before. What we need, in order to do the best job of teaching with these standards, is the best group of teachers possible, as well as a stronger national emphasis on the value of education. The last thing we need right now, in my opinion as an educator, is to start over again with a new set of standards just as we're getting used to the Common Core.

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You're wrong about Common Core math: Sorry, parents, but it makes more sense than you think

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By now everyone has seen the outrage-inducing image of a third grader's paper in which he is marked down for stating that 5 x 3 = 5 + 5 + 5 = 15.