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Did you know that number lines should not be used before second grade? Number lines are very prevalent in kindergarten and first grade classrooms. However, it's actually recommended that in kindergarten and first grade, we instead use number paths. I hope you'll stick around as we investigate how to use a number path in our quest to help us build our math minds, so we can build the math minds of our students.

Watch the video or read the transcript below.

What is a Number Path?

First off, let's talk about what a number path is. There are so many times when I say that, and people are like, "I've never even heard of a number path. So what is it?" A number path is isolated objects in a path that allow kids to count things. A number line is a wonderful tool, and you've probably seen some of my last posts that are all about the power of number lines, but for our young kiddos, a number line is actually a length-based model. You're counting the length of something. Our young kids need something to count, not just the length, not a measurement. So a number path helps our kids to see the objects in a line and be able to count individual objects instead of having to count a length, which is weird to have to count for our young kiddos. If you've ever seen a kid struggle with number lines, it's because there's not a thing to count. When they do count, they end up counting the little tick marks on the number line instead of the length between those tick marks. So they'll often, if they're trying to add or subtract on a number line, be incorrect because there's not a thing to count. So a number path gives our students something to count because in kindergarten and first grade, they're still in that direct modeling or counting on phase, that I've talked a lot about, and they need something to count.

Now, the first place that I ever realized that I shouldn't be using number lines was quite a few years ago when our state was in the process of reviewing our math standards, and trying to come up with better standards. I was on a team of teachers investigating our standards and trying to decide what good standards were; what else was needed; did the standards go to a different grade level, etc. Well, after our first or second meeting, that is when the original talk of the Common Core Standards came about. So, our job as this group became not one of creating the standards (it became very prevalent that our state was pretty much going to adopt the Common Core), our state wanted us to review those and give feedback to the writers of the Common Core State Standards of what we thought.

One of the things that I remember seeing was a line in one of those very first original documents of Common Core Standards that said, "...number lines should not be used before second grade," and I started questioning it. What are you talking about? In every kindergarten or first grade classroom, people are using number lines because kids need those. So I started digging and, it took me a while but, I found a couple of books that talked about it and they're called, "Focus" from the NCTM. There's " Focus in Kindergarten "; " Focus in First Grade "; "Focus in Second Grade"; there's one for each grade level. It's all about teaching with the curriculum focal points. These were before Common Core ever came about. Our National Council of Teachers of Mathematics put together their Focal Points. These were the major standards that we really needed to focus in on at each grade level. So those books talk about what are those focal points and how do we do it, how do we teach those concepts? Both of those, the kindergarten and the first grade book, talk about number paths. And they have an image similar to this:

This image shows how a number path that gives kids a count model instead of a length-based model. It's not to say that the length-based model is bad, kids need it. It's just a little too abstract for our young kids in kindergarten and first grade. So that brings me to the second piece that I want to mention which is where the number path fits in this CRA model.

How The Number Path Fits Into The CRA Model

If you've heard me talk about that before, that is the Concrete to Representational to Abstract. Underneath this video, if you don't have number paths, there's a link to be able to download my version of a number path that goes up to 20. You can buy them on my website as well. I created the document at first for the teachers that I worked with because there was nowhere for them to get number paths. Then teachers would say, "Can I buy these somewhere?" So, I have them specially made with a coating on them, so kids can write on them with whiteboard markers and, then just wipe it right off. They're wonderful things to be using in your classrooms. There's a link to download it and use however you want. Or, if you want the pre-made ones that come with the special coating, you can go to my website, buildmathminds.com/shop . Or there's a link above this video that says Shop . You can go to my shop and the number paths are there. Alright

So let's dig into where the number path fits on the CRA model. So, again, CRA stands for Concrete-Representational-Abstract.

Concrete is when children are actually physically touching, moving objects around. If you have manipulatives, that's when they're working in the concrete phase. They're counting out 6 things. That's concrete.

Representational is when they're drawing a representation of those 6 things. Now the hard part with our young kids is that they will draw very intricate drawings. If we are telling a story about 6 teddy bears, they want to draw the teddy bears, and color the teddy bears, and make sure that the teddy bears have ears and eyes and include all of the details. But really we want kids to not have to get bogged down in drawings and instead use models. I talk to kids all the time about using a model in mathematics, not a drawing. There's a time when we want intricate drawings, but there's a time when we just want a model. And a number path is a great model for students to use. And, so, a number path falls in the representational phase.

Abstract is when they just have the digit. So if they just have the digit '6', that's very abstract. You don't see 6 things. So, the reason again that number paths are better is that the number path has the 6 individual things that kids can see. They can see each object there. In a number line, it's still a representation, but it becomes more abstract because where are the 6 things on a number line? It's not obvious to our young kids. The 6 things are actually the 6 lengths in between the tick marks. And that's hard for our young kiddos to see visually. So the number line is still a representation, but it's more abstract. The number path is a more concrete representation than a number line. It's still a representation, but it's falling more towards a concrete representation for our students.

Building Number Sense With Number Paths

Alright, so let's talk about my two favorite ways to use a number path. How do we actually use a number path? One of my favorite ways is to help our kiddos build number sense. For a lot of our students, when they think of 6, the only thing they think of is the digit 6. We want them to picture 6 things. So let's say I give the number path to a student and I say circle 7. Most often, the number one way that we see kids circle 7 is they just circle where the 7 is.

In reality, that's not what 7 is. 7 is 7 things. So I want students to understand that when they're showing 7, it's not just the digit. It's 7 things. On a number path, we encourage kids to circle 7 things when they're showing 7.

This also leads into doing comparison-type problems where kids will show an amount and then show the other amount and talk about which one is more, which one is less. That can lead into questions like, how much more, how much less; this is the precursor to building the idea of adding and subtracting.

Adding & Subtracting With Number Paths

So, when kids are adding and subtracting using a number path, the one thing I want you to know, you don't need to teach them how to show it on the number path. It's up to them. The number path is a model, it's a tool for them to use. There's no specific way a child needs to show their thinking on a number path. The one thing I do want them to do is to show their thinking. I should be able to look at the number path and tell what they did. So if we were solving a problem like 7 + 8 and I wanted them to show their thinking on a number path, I would not want them to just circle 7, and then circle 8 things, and tell me it's 15. Unless they just knew that 7 plus 8 is 15. If they had a thinking strategy similar to something like, I know 7 and then 3 more would make a 10, and then another 5 more gets me all the 8, that is what I want to see on the number path.

However they thought about the problem, I want them to show that thinking. Then, eventually, the cool part is we can add in the symbolic piece as well, that abstract numbers, and attach it to the picture that they made with their number path.

Now I'm going to show you an example with subtraction because subtraction always seems a little funky for people. People are a little unsure how to show subtraction, but the big idea for students to understand that we want them to get around subtraction, I want them to understand that we have 14. So, somehow I want them to notate that they had 14. They might circle 14. They might put a star above the 14. They might put a little line after the 14. Somehow to notate that that's what we started with. And then if they thought about 14 minus 6 as 14 and I'm going to take away 6, somehow I want them to show me how they took away 6. If they took away 6, one by one by one, then that's what I want them to mark off on their number path. But if they took away 6 all in one fell swoop, then I want them to show that on their number path.

Again, remember it's just all about how are they thinking about the problem. That's what I want to see. So once they've taken away the 6, then I need to know where your answer in this is. How do I know what's left? Again, they could show it in lots of different ways symbolically or with abstract symbols. Some of the kids might see that as 6 plus something gave me 14. Or they might think of it as 14 minus 6 gave them the 8t that was left there. It's all about helping them to use the number path to model their thinking. It's not about how we would use the number path. It's not about our strategy for using the number path. It's a tool. It's a model to help them show what was happening in their brain.

To wrap up, the best way on how to use a number path is to just let the kids use it. Let them show you their thinking around things about numbers and quantities and comparing and adding and subtracting. Then help them attach that to the concrete if they're actually counting out blocks, and attach it to the abstract symbols. Use the number path as that representational piece when you're working with kids in kindergarten and first grade. Even preschool kiddos can be using number paths. At about part way through first grade, we want to start moving towards a number line and helping kids see the connection between a number path and a number line, but before that point, it's all about helping them see the individual objects and build their number sense and their understanding of addition and subtraction.

Don't forget to get your download underneath this or you can go straight to buildmathminds.com/shop to buy some pre-made ones and let the kids start writing on them and showing their thinking. I hope that this has helped you build your math minds as you go out and build the math minds of your students.

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Did you know that number lines should not be used before second grade? Number lines are very prevalent in kindergarten and first grade classrooms. However, it's actually recommended that in kindergarten and first grade, we instead use number paths.

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Math Manipulative K-6+

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What could we do with them

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How to Use Cuisenaire Rods: Cuisenaire Rod Activities | ETA Hand2mind

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Learn how Cuisenaire Rods can help your students develop a solid understanding various math concepts by using visual tools to help them assess their own work. From Pre K to 6th grade, order your set of Cuisenaire Rods today from ETA hand2mind!

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Used for K-6

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Needed in every elementary math classroom

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What do they look like?

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LINK
Rich resource for the mathematics classroom

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Set of 7 geometric shapes - originating from China

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How do 'tangrams' support learning in mathematics?

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How do 'tangrams' support learning in mathematics?

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* develop understanding of spatial relationships
* learn geometric terms
* strengthen problem solving skills
*develop logical thinking skills
*perceptual reasoning (non-verbal reasoning)
* creativity

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ETA Hand2Mind - resources

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Resource Link

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Tangrams for kids: A learning tool for building STEM skills
Like building blocks, tangrams can teach kids about spatial relationships. They may help kids learn geometric terms, and develop stronger problem solving abilities. They might even help children perform better on tests of basic arithmetic. But what is a tangram?

Invented in China approximately 200 years ago, a tangram is a two-dimensional re-arrangement puzzle created by cutting a square into seven pieces - seven geometric shapes called "tans" (Slocum et al 2003).

What are the seven shapes in a tangram? Each tangram puzzle contains the following:

2 large right triangles

1 medium-sized right triangle

2 small right triangles

1 small square

1 parallelogram

Arranged correctly, these tangram shapes can be fitted together as a large square, rectangle, or triangle. They can also be arranged in a variety of complex shapes, including fanciful ones.

There are many ways to play with tangrams. The simplest way is to let kids create their own complex shapes. But traditionally, tangrams are treated as puzzles.

The player is shown a target shape in outline, so that the "seams" between the composite tans are concealed. Then the player attempts to recreate the shape using the seven pieces.

In essence, it's an exercise similar to structured block play, where the challenge is to create an exact copy of a structure depicted in a diagram. But there's a key difference.

In structured block play, the diagram provides you with explicit, visual information about where each piece goes. In a tangram puzzle, you're left to figure that our for yourself.

As noted by Zhen Yuan and his colleagues, solving tangram puzzles appears to activate parts of the brain associated with creative thinking and trial-and-error problem-solving (Hu et al 2019).

The educational benefits of tangrams

When we play with tans, we consider the shapes from a variety of angles and perspectives. How would the shapes look if we stuck them together? Rotated them? Slide them around into different positions?

Experiments suggest that thinking about such things - visualizing the spatial relationships between shapes in your "mind's eye" - can boost our visual-spatial skills.

Research hints that it can boost arithmetic performance, too. When Yi Ling Cheng and Kelly Mix asked kids, aged 6-8, to perform a series of tangram-like spatial tasks, the practice session seemed to prime the brain for mathematics.

Kids who spent 40 minutes solving shape rotation puzzles performed better on a pencil-and-paper arithmetic test immediately thereafter. Compared to tangram-like activities, crossword puzzle warm-ups had no such effect (Cheng and Mix 2012).

So there is reason to suspect that playing with tangrams has educational benefits, and many educators recommended their use in the classroom (Bohning and Althouse 1997; Krieger 1991; National Council of Teacher's Mathematics 2003; Clements and Sarama 2014).

In particular, these teachers argue that playing with tangrams may help kids

classify shapes

develop positive feelings about geometry

gain a stronger grasp of spatial relationships

develop an understanding of how geometric shapes can be decomposed

hone spatial rotation skills

acquire a precise vocabulary for manipulating shapes (e.g., "flip," "rotate")

learn the meaning of congruence

In addition, researchers have argued that tangram play can be an opportunity for kids move away from simplistic (and incorrect) ideas about shapes. Young children, for example, might struggle to explain what features are necessary for a shape to be a square. With guided questioning, a teacher can help kids determine why some attempts to make a square with tangrams are unsuccessful (Nic Mhuirí and Kelly 2021).

Children can also learn new vocabulary to describe the shapes they assemble (e.g., rhombus, trapezoid, hexagon). And Tom Scovo demonstrates how tangrams can help kids calculate areas without formulas. For the details, see his excellent activities using tangrams for kids in grades 4-6.

What about free play with tangrams? That's good, too. But take note: Kids probably stand to learn much more if their play includes discussion and teamwork.
Indeed, studies indicate that spatial play - like playing with tangrams - is particularly educational when kids interact with others who use spatial vocabulary.
For example, toddlers tend to develop bigger spatial vocabularies if, during play, their parents expose them to a greater range of spatial words, like "triangle" and "line." As I explain elsewhere, such kids are also more likely to develop stronger spatial skills.

So it's likely we can boost the value of tangrams by actively engaging kids in conversation. We can introduce them to words they might not know, like "angle" and "parallelogram."
It's also likely that kids will benefit from being asked to make predictions and explain tactics.
Research suggests that kids are more likely to master concepts when they explain them to others. What will happen if you rotate the triangle? What will happen if you flip the parallelogram? How must we move this shape in order to make it fit?

If we can get children to explain their ideas, we may help them improve their spatial skills and comprehension of geometry (Lee et al 2009; Clements and Sarama 2014).
So while kids may benefit from solitary play with tangrams, some of the best educational experiences may arise from playing with a (talkative) partner.
Any other teaching tips?

Tangrams offer kids an excellent opportunity to test out different geometric manipulations, and become familiar with the properties of the shapes they use.

But notice the triangles- big, medium, and small - are all the same shape. They represent a special kind of right triangle - an isosceles triangle with two 45-degree angles, and one 90-degree angle. And if you put together two triangles of the same size, you can make a square.

These properties aren't found in all triangles. On the contrary! But it's easy for children to come to that conclusion if they don't get exposed to a variety of triangles - equilateral, isosceles, and scalene. So it's important to expose kids to that variety, and call their attention to the ways in which triangles can differ (Clement and Sarama 2014).

Getting started: How to make a tangram

You can make your own tangrams by following the instructions on Tom Scovo's site, or by printing and cutting out the tangram shapes below.

If you right-click on the image here, you can save it on your computer for printing. Alternatively, after right-clicking, you can open the image in a new tab, and use your browser's print function.

Looking for more durable tangrams? You can buy some. Classic Tangoes includes two plastic tangram sets and a deck of puzzle cards. But the corners are a bit sharp. You can also find tangrams made from other materials, such as wood, foam, and thick cardboard.

Tangrams for young children

My favorite introduction to tangrams for younger kids is the book Three pigs, one wolf, seven magic shapes by Grace Maccarone. Unfortunately, it's out of print, but you might be able to pick up a copy used. It includes story (based on the folk tale of the three little pigs), a teaching guide, a set of tangrams to cut out, and some activities created by a math teacher. Although the publisher recommended this book for kids in grades 1-2, the book can be enjoyed by preschoolers.

For another, fanciful story featuring tangrams, see Grandfather Tang's Story (Dragonfly Books).

Virtual tangrams for kids

You might wonder if computer games are as educational as playing with real, physical tangrams. The National Council of Teachers of Mathematics (NCTM) recommends both.

According to the NCTM, computer games may offer special benefits because "the computer environment is likely to encourage [kids] to think about how they need to manipulate the tangram pieces rather than approach the task mainly by trial and error."

Read more and try out this online game by the National Council of Teachers of Mathematics.

Ideas for tangram activities

The classic approaches are either (1) to present kids with images of target shapes (and have them attempt to reproduce those shapes with their tangrams), or (2) to encourage kids to be creative, and come up with shapes of their own.

But there are additional games and activities to try as well. You can choose a theme (like "fish" "spaceships" or "houses") and see how many different examples of each type kids can come up with. For instance, how could challenge kids to create a series of cats.

More to read about the development of STEM skills

Interested in learning more about activities that boost STEM skills? In this article I recommend a number of STEM books and learning resources for kids, including some free ones.

In addition, to read about the many benefits of construction play, see my article, "The benefits of toy blocks." If you're specifically interested in structured block play, check out my article, "Can Lego bricks and other construction toys boost your child's STEM skills?"

And for more information about the development of spatial reasoning in children, see these Parenting Science articles:

References: Tangrams for kids

Bohning G and Althouse JK. 1997. Using tangrams to teach geometry to young children. Early childhood education journal. 24(4): 239-242.

Cheng Y-L and Mix KS. 2012. Spatial training improves children's mathematics ability. Journal of Cognition and Development. Published online DOI:10.1080/15248372.2012.725186.

Clements D and Sarama J. 2014. Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.

Hu Z, Lam KF, Yuan Z. 2019. Effective Connectivity of the Fronto-Parietal Network during the Tangram Task in a Natural Environment. Neuroscience. 422:202-211.

Kriegler S. 1991. The Tangram: It's More than an Ancient Puzzle. Arithmetic Teacher 38(9) 38-43.

Lee J, Lee JO, and Collins C. 2009. Enhancing children's spatial sense using tangrams. Childhood Education 86(2):92-94.

National Council of Teacher's Mathematics. 2003. Developing geometry understandings and spatial skills through puzzlelike problems with tangrams: Tangram challenges. www.nctm.org.

Nic Mhuirí S and Kelly D. 2021. Making squares: children's responses to a tangram task. In: Eighth Conference on Research in Mathematics Education in Ireland (MEI 8), 15-16 Oct 2021, Dublin, Ireland (Online).

Slocum J, Botermans J, Gebhardt D, Ma M, Ma X, Raizer H, Sonnevald D, and Van Splunteren C. 2003. Tangrams: The world's first puzzle craze. Sterling: New York.

Title image of child's hand's manipulating tangrams by Gwen Dewar

Line drawings of triangles and tangrams by Parenting Science

Image of many different tangram cats by Sveta_Aho / istock

Content of "Tangrams for kids" last modified 9/2022. Portions of the text derive from an earlier version of the article, written by the same author.

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Tangrams for kids: Educational tips and a printable template

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Tangrams for kids: A learning tool for building STEM skills Like building blocks, tangrams can teach kids about spatial relationships. They may help kids learn geometric terms, and develop stronger problem solving abilities. They might even help children perform better on tests of basic arithmetic. But what is a tangram?