Following on from last year’s successful series of blog articles on 5 ways to implement the science of reading, this year experts will be providing ready-to-use tips on the science of maths.

Inspired by Tom Sherrington’s Five Ways Collection, the posts have been edited and curated by Brendan Lee and Dr Nathaniel Swain.

The second blog post of the series comes from Karen Tzanetopoulos, co-author of the book, “How Children Learn Math: The Science of Math Learning in Research and Practice” (2023).

Five Ways to Start Teaching Fractions

Rational numbers (fractions, decimals, and percentages) are essential to algebra and higher-level maths, yet many students struggle to learn what they mean, how to use them, and how to compare them. Understanding rational numbers is counterintuitive because they only sometimes act like whole numbers, which of course children learn first.

Children and adults can have a whole number bias (Alibali & Sydney, 2015) leading to a belief that rational numbers will operate the same as whole numbers which leads to mistakes and misunderstandings. For example, when comparing fractions with different denominators, such as 1/10 and 1/100, children will pick 1/100 as being more than 1/10 since 100 is larger than 10.

To add further confusion, whole numbers have a set sequence that cannot be altered, such that 1 is always followed by 2, 2 is always followed by 3, and so forth, whereas rational numbers can always be further subdivided, such as 1/10, 1/100, 1/1000, or 0.1, 0.01, 0.001, etc. Developing a deeper understanding of the meaning and relationship of rational numbers from the beginning of instruction can have a long-lasting effect and make the journey less troublesome. So, the following are 5 ways to start learning fractions based on scientific research on math learning.

Putting it into Practice

Five ways to start learning fractions.

1. Use explicit language

2. Start with continuous unsegmented proportions

3. Start segmenting, writing, and combining fractions of continuous amounts

4. Use a linear model (number line) for fractions instead of segmented shapes

5. Simplify the conditions

1) Use explicit language

The English language of rational numbers is rather a nightmare and is phonologically, semantically, and morphologically complex. For instance, the term third is a confusing homonym, being both a very familiar ordinal number as part of a sequence of first, second, third, as well as a much less familiar number name that means it is one of three equal parts of a whole. Two-fifths is difficult to say, difficult to phonologically process, vague in its meaning, and a homonym, being both an ordinal number and representing two of five equal parts of a whole.

This challenging language does not depict what the fraction names really mean, which is a proportion of parts to the total number of parts. Confusion abounds. So, let’s make fraction language easier to say, say what it means, and make fractions more understandable.

The Science

East Asian languages such as Chinese and Korean use a clear way of naming fractions that focuses on the part-whole relationship. In Chinese, ¾ is named of four parts, three. In Korean, ¾ is named of four equal parts, three (Miller et al., 2005). Naming fractions in this way focuses on the relationship between the two numbers, whereas English-speaking children most typically think of fractions as two parts and focus on either the numerator or denominator to judge their magnitude (Braithwaite & Siegler, 2018). Research has shown that changing the language from, for example, three-fourths to three-out-of-four-parts, three-of-four parts, or, of-four-parts, three, helps children focus on the proportion and leads to greater and faster understanding of fractions (Paik & Mix, 2003).

The Action

When introducing fractions and decimals, begin with the descriptive language of explicitly naming the parts to the whole: 1 of 10 parts, or of 10 parts, 1; 3 of 4 parts, or of 4 parts, 3. Of course, students must learn the thorny English names as well because it is how we communicate about them. The descriptive terminology can come first, eventually followed by the name: 1 of 10 parts, the name is one-tenth.

• Always focus on the denominator first, in other words, the number of parts that make the whole. When naming a fraction, first ask how many parts make the whole.

• Practice writing fractions by always writing the denominator first. Ask, “How many equal parts make up the whole?” Then, write the dividing line and the denominator first, leaving the numerator blank. Next ask, “How many equal parts of the whole are there?” Then fill in the numerator.

• Explicitly naming the parts to the whole makes it easier when comparing fractions. Four of five parts is easier to visualize than four-fifths.

• Practice reading fractions using the explicit language. 1/10 as one of ten parts, or of ten parts, one. Read a wide variety of fractions using this explicit language.

• Add magnitude comparisons when reading fractions using explicit language, such as one of ten parts, close to none. Nine of ten parts, almost all.

2) Start with continuous unsegmented proportions

Teaching fractions most often begins with segmented, non-linear shapes, such as a circle (often representing a pizza) or a rectangle divided into parts, yet many scientific studies have shown that these shapes are hard for children to cognitively process and that there are better ways to start. One of these alternate ways is continuous unsegmented proportions.

The Science

Researchers have found that children benefit from first beginning their study of rational numbers by comparing proportions of unsegmented continuous amounts such as water or pictures of continuous amounts. Cognitively, continuous amounts are easier to process than segmented shapes. A study of third, fourth, and fifth graders revealed they had a greater understanding when comparing continuous rather than segmented amounts (Begolli et al., 2020., Park et al., 2021). When children are presented with segmented amounts, they focus on counting segments and not the proportion to the whole (Jeong et al., 2007).

The Action

An easy way to teach with unsegmented continuous proportions is by using water and clear plastic beakers, cups, or other containers, starting with at least four to five of them.

• Explicitly talk about how one full glass of water has been divided into two equal parts, then point to each and say, “Here is one of two equal parts and here is one of two equal parts. The name for each is one-half or half. Now, you are already doing fractions! Fractions are about dividing whole things into equal parts!” Pour the two half-full cups of water into the empty cup and talk about combining the two parts make one full cup again.

• Start the story again, and say that now two friends come along, so they need to divide the full glass into three equal parts. Do so and point to each one as you say that each is one of three parts, and the name for each is one-third. Talk about whether each cup has more water if they divide the whole glass into two parts or into three parts.

• Continue the story with more and more friends coming, dividing the full glass into more and more parts, and talk about how the more equal parts something is broken into, the smaller each part will be.

• Finally, talk about how when they pour the parts back into the full cup, they have all the parts, whether it is two of two parts, three of three parts, four of four parts, etc. When all the parts of the whole are there, it is the same amount as one whole, or one.

3) Start segmenting, writing, and combining fractions of continuous amounts

Having introduced fractions with continuous amounts without the distraction of segmentation, students will then be ready for segmentation and writing fractions. Repeat the exercise above, but now the teacher or students can mark the cups at the water level with a line and with the fraction.

If the full cup is divided into three cups, discuss that the full amount will be divided into three parts and then mark each cup as 1/3, saying both one of three parts and the name one-third. Once the students have marked a variety of fractions on the cups, they can begin combining some of the parts.

For example, combine 1/3 and 1/3 together and talk about then having two of three parts, or two-thirds and mark the cup with a line and the fraction. Children can combine a wide variety of fractions in this way, say and write the fractions, and have a lot of fun too!

4) Use a linear model (number line) for fractions instead of segmented shapes

This suggestion may be quite surprising since teaching fractions with segmented shapes is ubiquitous. The cognitive research in this area is abundant and clear, so at least consider the idea.

The Science

Researchers have shown that using a linear model of fractions results in a richer understanding of fractions than segmented shapes (Barbieri et al.) and impacts learning algebra and higher-level maths. The following are a few (but not all) reasons that number lines have an advantage over segmented shapes for understanding rational numbers and are a great way to begin fraction instruction.

• Rational numbers, like whole numbers, can be represented on a number line, and children as young as second grade can see and compare ratios of line lengths (Kalra et al., 2020).

• Success in fraction placement on a number line is a particularly strong predictor of subsequent math achievement (Vamakoussi et al., 2018).

• Comparing the magnitude of fractions on a number line is more like comparing continuous proportions which we have learned is easier for many children than segmented shapes.

• The number line also allows an easy transition to using rational numbers for measurement, much more so than segmented shapes. Measurement is a very common and important use of fractions and decimals.

• Segmented shapes require two dimensions (length and width), thus more to cognitively process, than the single linear dimension of a number line.

• The single-dimension number line also allows rational numbers to be ordered and their magnitudes compared. Fraction and decimal ordering on a number line strongly correlates to a student’s ability to compare their magnitude. Researchers asked students in grades 7, 9, and 11 questions about ordering fractions and decimals on a number line, and without having been trained in the number line, they treated fractions and decimals as whole numbers and were not successful in understanding their order, thus demonstrating a lack of understanding of these numbers (Varmakoussi and Vosniadou, 2010).

The Action

Provide students with a variety of fraction number lines that begin at 0 and end at 1. Provide students with paper strips that are the same length as the number line. Alternative materials could include playdoh, clay, or model magic which they can roll out to be the same length as the 0-1 number line. Students can then cut the paper or materials into different lengths that match the different number lines.

For example, using the number line with 1/3 segments, cut one segment that is 1/3 in length, another that is 2/3 in length, and then another that is 3/3 in length. If using the paper strips, label each strip with the corresponding fraction. Line them up and compare their magnitude. Continue in this way with different fraction number lines. Students can also combine like parts to make the whole.

5) Learn from mistakes.

The Science

Students often struggle with understanding the effect on magnitude when fractions are combined, and even struggle with understanding that when two fractions are added together, the result will always be more (Braithwaite et al., 2018). Children can become adept at the procedure of adding fractions, but they may not understand the change in magnitude or why the procedure works. They may learn to add or subtract numerators only and not add or subtract the denominators when completing the operation, but many do not understand why. This confusion will limit their success with rational numbers as they continue to higher-level maths. Understanding and comparing the magnitude of fractions is a mark of proficiency with fractions, and many students and teachers themselves struggle to learn these concepts.

The Action

Use a number line from 0 to 1, and first divide and label four parts (1/4, 2/4, 3/4, 4/4). Write a simple equation of 1/4+2/4=3/4 and plot it on the number line. Discuss that the result is more than either of the two individual fractions and is closer to 1, or 4/4. Next, let students know that you are going to make a mistake and add the denominators, then write the equation again but this time add the denominators so that the result is 3/8 . Further, subdivide the number line into 8 segments and label each segment, then plot 3/8 on the number line. Adding denominators does not work because the result is LESS than the starting point! Impossible! Adding denominators results in more parts to make the whole, and the more parts there are, the smaller each part gets! Do the same exercise with subtracting fractions, first following the rule of only subtracting the numerators, then mistakenly subtracting the denominators. The result will be MORE because the fewer parts there are, the larger they will be. Students can have a lot of fun making “mistakes” and will have a deeper understanding of fractions.

More info

• How Children Learn Math: The Science of Math Learning in Research and Practice: Nancy Krasa, Karen Tzanetopoulos, Colleen Maas

• Knowledge for Teachers Podcast: Karen Tzanetopoulos on the Science of Math Learning in Research and Practice