A rekenrek is one of my favorite early elementary math manipulatives. First designed by Adrien Treffers from the Freudenthal Institute in Holland, a rekenrek is a set of 20 beads organized into two rows of 10, with 5 red and 5 white beads on each row.
You can easily create and recognize each number from 1 – 20 using the groups of 5 beads. For example, if I push the whole first row to the left and 2 beads on the bottom row, I can quickly recognize the number 12.
The ability to “see” numbers rather than counting them is called subitizing and is an important early indicator of number sense. Normally we are only able to subitize small numbers (1-4), but the rekenrek helps children quickly see larger numbers as combinations of sets. For example, I can quickly see that there are 8 beads if I see 5 red beads and 3 white. I could also show this with linker cubes, but it takes a lot longer to count out and construct stacks of linker cubes.
The rekenrek is also a quick way to demonstrate and practice addition strategies. You can demonstrate counting on with the problem 6 + 3 by first sliding over 6 beads as a group, saying “6,” and then sliding over three more beads individually, “7, 8, 9.” Remember to say “7, 8, 9” rather than “1, 2, 3” as you are sliding over the beads.
You can practice near doubles (or doubles plus one) by first creating a doubles problem such as 4 + 4, by sliding over 4 beads in each row. Then slide over one more bead on the top row to create 5 + 4 or one more bead on the bottom to create 4 + 5.
Demonstrating near tens is easy on the rekenrek. Let’s say you’re trying to solve the problem 8 + 4, first slide over 8 on the top row. Then slide over 2 on the top to complete the 10 and 2 more on the bottom to make 12.
Ever feel your fingers move unconsciously when you’re figuring out simple math problems? It may be you’re sensing the strong neural connection between our fingers and numbers. Scientists became interested in this topic when noticing that the same areas of the brain that control the fingers light up when solving math problems even though people were not actually moving their fingers.
Researchers have found a strong connection between math skills and finger gnosia, or knowledge of your fingers (e.g. Penner-Wilger & Anderson, 2008). They test young kids by asking them to put their hands in a box with an open side. The researcher then touches one or more of the kid’s fingers, removes the box, and asks which finger was just touched. Children’s ability to recognize which finger has been touched predicts how well that child will perform on a variety of math tests over years.
Yet anytime we talk about how one skill predicts another, as researchers we need to figure out if there is a causal relationship between the skills. Is it that kids who are already predisposed to be good at math have great finger gnosia when they are young? Or can we teach finger gnosia to young children and see effects later in math performance?
There is some early evidence that direct finger awareness training can affect students’ later math performance (Gracia-Bafalluy & Noel, 2008; Jay & Betenson, 2017). For instance, in a recent study of 137 six and seven year olds, researchers worked with students for 4 weeks either playing finger gnosis activities or having them play number games or both, to see the effects on students’ quantitative skills. They found that while finger gnosis skills and number games alone did not improve quantitative skills, when combined together, they did significantly improve students’ quantitative skills. Here are a few of the games, they played with students:
Finger Gnosis tasks:
-Counting by 1s, 2s, 5s, and 10s on fingers
-Teacher asks students to “show me” a specific number on fingers
-Holding up fingers to represent operations (Show 3+ 4 on your fingers)
One Saturday morning, I decided to try out an estimation task with my five year old daughter while we were eating pancakes. I gave her a piece of paper and drew a blank number line on it and labeled the ends with a 0 and 10.
I started by asking her to place where the 5 should go. I wanted to give her a landmark to see how that would shape her placement of the other numbers. Here’s what she did:
I was curious whether the number line would prompt her to connect that 5 is half of 10 and should therefore go in the center. It did not.
Then I asked her to fill in the 9. I was trying to be strategic about first asking her to place 5 and then 9, so she would be more inclined to space the numbers throughout the number line. From there, I asked her to fill in the rest of the numbers. Here’s what she did:
She quickly realized that her number line didn’t look right, and she wanted to fix it. So, she did another number line below and ran into trouble between 9 and 10.
I asked her to do a third number line, and this is what she did next:
You can see from this number line that she placed 5 at the midway point. She also corrected herself more than once to make the number line more accurate and linear.
Having kids experiment with number lines is an easy and fun way for them to think more deeply about number relations and estimation. There are some fascinating studies about how kids start with a logarithmic pattern of estimating (e.g. my daughter’s first number line, where the majority of the numbers lie towards the left of the number line) and with practice and age get more linear (evenly spaced).