Helping Students "Get" Symbolic Mathematics

Using ANS (Approximate Number System) to Build a Cognitive Foundation for Theoretical Math

What if I told you there was a way to help all children “get” the number system?  I recently discovered a fascinating study relating to how we go about teaching math.  In 2013 Joonkoo Park & Elizabeth Brannon studied how the approximate number system (ANS) correlates with symbolic mathematics. 
 
We have learned that humans have a unique ability to perform symbolic math, but humans share with animals an approximate number system (ANS) that allows for estimation and a rough calculation of numbers, without using symbols. 
 
There are many theoretical claims out there that ANS provides a foundation and cognitive basis for theoretical math.  The question is:, ‘Can ANS improvements transfer over to symbolic math improvements?’  Park & Brannon set off to find this.

The Study
 
Park & Brannon developed a hypothesis:  Complex math skills are fundamentally linked to rudimentary preverbal quantitative abilities.  This provides the first direct evidence that ANS and symbolic math may be causally related, and raises the possibility that interventions aimed at the ANS could benefit children and adults who struggle with math.  They developed 2 different experiments to test their hypothesis. 
 
            Experiment 1
           
In the first experiment, they used a training study with a pre- and post-test design to assess whether performance on a non-symbolic arithmetic task would improve over repeated testing and whether improvement would transfer to symbolic arithmetic.   Adult participants had to add or subtract large quantities of visually presented dot arrays without counting.   
 
They gave all participants a set of multi-digit addition and subtraction problems both before the first training session and after the tenth training session.  The approximate arithmetic training resulted in a substantial improvement in symbolic math performance. 
 
            Experiment 2
 
In the second experiment, to account for a possible placebo effect, they included a training group that received world knowledge training over the course of multiple lessons.  They also aimed to test the relative efficacy of the ANS-based training compared to other training based on symbolic numerical associations.  They randomly assigned participants to various training groups- Approximate Arithmetic (AA), Numerical Ordering (NO),and Knowledge Training (KT). 
 
As in Experiment 1, there was substantial improvement on the non-symbolic approximate arithmetic task.  To assess the transfer effects, they compared the standardized gain scores in symbolic math performance across the three training groups using one-way analysis of variance.  This analysis revealed significant differences on the math gain scores across the training groups; the AA training group showed higher increases than the NO and KT groups.

            Results
 
The study’s findings were important in a few ways.  First, the results were the first to show that ANS training improves symbolic arithmetic.  Second, there was a striking transfer between the approximate addition and subtraction task and the symbolic math test.  Thirdly, this link suggests important directions for math interventions.

How it Affects Students
 
So, in short, ANS training is directly linked to how we should approach math.  It shows that when students’ ANS improve, so does their symbolic mathematics.
 
            Visual Approaches
 
The findings from this study show the value of taking a visual approach to math.  Viewing large arrays of dots and quickly making estimates has a lot benefit to symbolic mathematical abilities.  That’s not to say symbolic math isn’t significant; ANS training is key to getting there.
 
How to Apply
 
There are a number of ways you can increase ANS training in your classroom, and, although it’s important to lay this foundation in younger students, the study shows us it can improve arithmetic of students of any age, even adults!
 
           Estimate Challenge
 
For students of any age you can hold a “Guesstimate Challenge.” Fill up a jar with marbles and have your students give their best estimate of the number of marbles.  It’s an easy, simple way to get kids to work on and develop their ANS!
 
Keep in mind that this will not take much time at all!  Consider making this a weekly challenge and allowing students to make their guesses whenever they have time during the week.
 
            Partner Cards
 
To quickly add a bit of ANS training to your classroom, incorporate partner cards with different amounts of dots. Randomly pass out sets of dot cards- Figur8 provided cards to download here, and have your students swiftly partner with the student who has the same number of dots.   They will have to estimate the number of dots by using their ANS. 
 
Partner cards are a great way to quickly partner up your students, and since you already are spending time partnering up the class you’re not wasting any time.  
 
I hope you found this study as cool as I did!  Do you have any great ways to train your students’ ANS?   Don’t forget to subscribe in the box below!