Last Updated on October 10, 2023 by Editorial Team
In the beginning, teachers do not expect kids to be Einstein; but at the same time, they expect the latter to have a certain level of intelligence.
Very soon, things become frustrating when despite putting so many efforts, the results do not come out as desired.
At the onset, the instant reaction is to brand the child as a slow learner, or the one not being attentive in class. With the gradual study of patterns of mistakes, and using assessment tools like WIAT-III, learning disorders are found to be the real culprit.
Out of inquisitiveness, when I dug deeper into the cause of math scare or reading difficulties of children, I found some startling facts:
- About 8-10% of the US child population suffers from some or the other form of learning difficulty – Psychology Today[1]
- About 40% of the students enrolled in special education classes are diagnosed with learning disabilities of some kind – Child Development Institute[2]
- About 10% is children are suffering from specific learning disabilities such as Dyslexia, Dyscalculia – Science Daily[3]
Of these learning difficulties, a few are such that stay undiagnosed behind others who show similar symptoms. Dyscalculia is one such learning disorder. I have covered it extensively in various articles on this platform and now I intend to move deeper and give an insight into the specific number learning difficulties, such as number sequencing, faced by dyscalculic children.
Dyscalculia, as is established quite evidently, interferes with the ability to find the relation between numbers, sequence numbers correctly, identify patterns, or have an understanding of spatial parameters of any setting, as per the study[4] on Developmental Dyscalculia, by Ladislav Kosc.
In this guide, I would like to help you understand:
- What is number sequencing
- Different types of number sequences
- Importance of number sequencing in developing math intelligence
- Strategies to build number sequencing ability
- Best tools for teaching number sequencing
So, let’s take these pointers one by one!
What is number sequencing?
Number sequencing is nothing but the numbers arranged in a certain order or pattern. Children get an early glimpse of number sequencing in preschool days when they learn to dodge numbers by 1s, 2s, 3s, and so on.
Still not clear? Take a look at these number sequences: 1 4 7 10 13 16 19 This sequence is formed simply by dodging the odd numbers by 3
It is sort of playing with numbers and stimulating minds to think about operations.
So, basically, number sequencing follows a certain rule. Let’s move further and find what various types of number sequences are.
Different types of number sequences
We are surrounded by sequences[5] everywhere if we follow closely. However, to improve our perspective, we can differentiate sequences into certain categories, such as:
1. Arithmetic sequence
Demonstrated above was an example of an arithmetic sequence. In this sequence, the numbers are arranged with a common difference between them. It can be an increase or decrease by the same number.
Another sequence presentation looks something like this
2. Finite or infinite sequence
Number series 1,2,3,4 or 2,4,6,8 is an example of an infinite series. You can add on and on and on, in any manner.
The infinite sequence takes a finite garb when there is a condition attached to the process. For example, a sequence of numbers from 10 to 0.
10 9 8 7 6….0
It has a definite start (10) and a definite end (0)
You can also make a sequence saying to write even numbers between 4 and 12. So, your sequence becomes:
4,6,8,10,12
Here, ‘between 4 and 12’ becomes a condition and imparts finiteness to the series.
3. Geometric sequence
Simply put, this sequence is created when the first number and the resulting numbers are obtained by multiplying with a common number.
It looks something like this:
The numbers progress by a common factor x2.
Examples of geometric sequences
Depending upon the rule applied, you can coin as many sequences as desired. So, other kinds of geometric sequences are:
- Square number sequence: Each number in the series will be a square value. For example 1,4,9,16, and so on.
- Cube number sequence: Each number is a cube value, such as 1, 8, 27, 64, 125,….
4. Special Sequences
Many mathematicians in the past fiddled with numbers and identified a pattern, which was then named after them as well. The sequences discovered by them found real-time applications as well. Some of these sequences are:
a. Fibonacci series
0,1,1,2,3,5,8,13,21,34,55…
This sequence led to the formation of various patterns, arcs, etc. Economics, astronomy, and nature are some of the areas where this series proved its utility and helped in developing various theories.
b. Pascal’s Triangle
It is an unending series of numbers. It can be called a pattern rather than a sequence. The ends of the sequence are always 1. The numbers appear in the middle and adjacent to each other to create the next series of numbers.
Interestingly, the addition of a series of numbers comes out to be double of the upper series (see diagram given). This number sequence formed the base for expanding binomial series and became a trusted tool for advanced algebra learners.
All these categories of number sequences do leave us intrigued thinking about why exactly number sequencing is important and what role it plays in building numeric literacy. So, here is a little exploration of the purpose of number sequencing in a child’s math learning process.
Importance of number sequencing in developing math literacy
As the child progresses on the path of learning math, the concepts of estimation, probability, etc. take over simple math operations. The child can employ number sequencing ability in:
1. Sorting and arranging numbers in ascending or descending order
A practical use of this ability lies in keeping currency notes arranged according to their value. With number sequencing, the child can understand currency values better.
2. Developing number sense
The children engage further in number sense[6] and try to grasp the concepts of comparing numbers. They can identify relationships better
3. Forms basic premise for learning algebra and trigonometry
Various number sequences, like Pascal’s triangle mentioned above, help build a solid premise for preparing the mind to absorb algebraic and trigonometric identities.
4. Time-keeping
By gaining knowledge of number sequencing, children find it easy to do time estimations, and can also understand terms like half an hour later or an hour earlier, etc.
5. Enables geometrical thinking
Number sequencing also inculcates geometric thinking in children.
How?
The increasing number of edges and corners and the corresponding structures become easy to visualize when their geometric knowledge is empowered with number sequencing ability.
6. Place value manipulation
Multiplicative reasoning developed from number sequencing finds its best use in understanding place value.
Each place value is increased by a multiple of 10 while moving from units to tens, hundreds, and so on. Sequencing allows the child to comprehend the place value concept with ease.
Thus, sequencing is going one step up from simple counting. The conformance to certain rules converts the counting process into sequencing and helps children build a better understanding of decimal places and fractions as well.
Strategies to build number sequencing ability
It is certainly the tricky part – what could be the ideal strategies to build number sequencing ability in an early learner?
As it is evident that number sequencing develops abstract reasoning and allows kids to take a look at the world around them with a better math sense, why not let the world be the ground for learning?
This means the more kids are encouraged to identify patterns in the things around them, the better will they be able to relate to the sequencing process.
1. Use manipulatives
The first strategy will be to teach using manipulatives like blocks. Decimal manipulatives can give an amazing introduction to the sequence of numbers from 1 to 10.
Give exercises to kids to add blocks and arrange them in the order of their choice. You can label the blocks using numbers to give a tangible introduction to a number sequence.
2. Use number talk
Number talks offer a participative way of encouraging children to learn number sequencing.
Write the random numbers on the board. Ask the children about strategies to arrange them.
You can go the other way around as well. This means, writing the numbers in a sequence you want to introduce to the child. Now, encourage the child to identify rules and share strategies with the class.
For example,
You can use a similar kind of quiz to encourage children to identify patterns and also tell the next number in the series.
3. Make a ‘how to’ series
How to series allows children to try various ways of sequencing numbers. For example,
- Ask how to make a series of even numbers
- Ask how to grow numbers by doubling, or tripling them
- How to arrange the given objects according to the increasing size
These are very few basic exercises for stimulating children’s minds towards sequencing numbers.
4. Ask prediction questions
Counting individually is the start of the learning process, but sequencing helps in growing the knowledge further. A child’s mind can be stimulated to make a sequence by asking to predict the next number.
Do it a little easy way. You can help by telling the rule first.
For example, draw this picture or any one part of it on the board and encourage the child to guess the next number and share the strategy used to arrive at the answer.
5. Use mnemonics to memorize steps
We all have heard the name ‘BODMAS’. It is a classic example of the sequencing of operations. The same set of numbers can give different results when the BODMAS is not applied.
Similar mnemonics can be used to help the kids in memorizing the steps when the sequence involves multiple operations. For example, consider this series
5 12 26 54
Here, the common order is ‘nx2+2’, where n is the first term or the start of the series. To help children find the next three to four numbers, you can give them a cue to remember, m2a2, standing for multiply by 2, and add 2. Teachers can introduce the term Y2K2 (the year 2002) to the child to introduce this cue smoothly and help in association.
Best tools for teaching number sequencing
In any typical preschool class, one of the most common activities in math class is speaking and writing numbers. Teachers using books and boards, speak out the numbers and children repeat them. This simple activity gives the very first introduction of number sequencing to small kids.
However, children with learning difficulties may not keep up with the pace of their peers in the class. Thus, a better option will be to give a more tangible introduction of easy to grasp by all children. What could be the possible solutions? Listed here are a few.
1. Designing learning games/activities
Involving the child instead of forcing him to pay attention is always a better teaching strategy. Learning games allow children to engage focus and internalize the number sequencing concept in return. Here is an interesting activity that we all know as a lottery in which the numbers are kept in the bowl and then taken out to find the lucky winner. It can be modified as below:
- Take 10 sticky notes and draw dots instead of writing numbers. This means, one dot on one note, two on the other, and so on. Put all these notes in the fishbowl and shake it.
- Ask kids to take sticky notes. One kid will take out one sticky note. You can make it more dramatic by asking them to close their eyes while doing so.
- Now ask him/her to count dots.
- Also, make a table with 10 columns in it. Once the kid iterates the number of dots, you can ask him about its correct position on the table.
Similarly, you can design an outdoor activity where kids who have learned to identify numbers are asked to pick a placard with a number on it. To make it quick, you choose the numbers 1 to 5. ‘Run to your number’ – this is what I call this activity. Its important steps are:
- Paste the placard on the kids’ dress.
- Ask them to stand on one side of the playground randomly, and on the other, you can make slots that kids are supposed to occupy after running from the starting point.
- You speak out a number. The kid with that number starts running reaches the other end, identifies the slot, and stands there. Numbers can be iterated in any of these sequences – 1,2,3,4,5 – Kids learn counting as well as the first level of sequencing. Then you can move to 1, 3, 5 –odd number sequence or 2,4,6 – even number sequence
This activity offers a playful way of engaging kids in learning number sequencing. You can further increase the challenge by asking them who will go first, who will be following next, and so on.
2. Printable worksheets
While the activities mentioned above are suitable for preschoolers, printable worksheets offer an interesting tool for teaching number sequencing to students of grades 1 to 4.
Giving these easy challenges through worksheets can give a preliminary introduction. With an increase in the level of efficiency, the students can be made to identify the pattern.
For example,
a. Choose the common operation in these series and write the next three
b. Solve the problem
Give students some problems to solve. Like, I put 5 coins on day 1 in the piggy bank. And, add 5 coins daily. Write down the number of coins in the piggy bank each day. Also, tell me what will be the number of coins on 7th day
3. Multiplication chart
You can ask the children to make a chart of numbers by giving them an operation as a cue for arranging numbers in it. For example,
| x | 1 | 2 | 3 | 4 |
| 2 | ||||
Multiplication results or products also are a type of series with common operation ‘+n’.
4. Block activities
Give kids blocks to play with and ask them to keep on adding blocks in some number pattern. For example, start by giving 1 block. Now, ask to keep on adding one block (or 2,3,4) and write the resulting number on a piece of paper. Alternatively, you can simply ask for the number of blocks added to the resulting figure. With better practice and repetitions, the kids can become confident about how many blocks will be appearing in the next in the series.
5. Group activities/challenges
Give small balls to make geometric patterns or arithmetic patterns. Kids can ask each other to find the rule or identify the pattern.
6. Board games and dice activities
You can use a simple object like dice to teach number sequencing. Dice can give knowledge about sequences of numbers from 1 to 6. You can ask the child to roll the dice and count the dots appearing. The dice shows how numbers increase by +1.
According to Dave Moursund’s book, ‘Introduction to Using Games in Education: A Guide for Teachers and Parents’, board games prove to be an effective teaching tool. Bingo board games, sequence-making using cards, and other mind-stimulating games offer kids a perfect escape from struggling with numbers. You can use board games available online or even design a few of your own to make the learning process simpler.
7. Number sequence solving apps
Why not make mobile phones a tool to familiarize number sequencing? Drive the child’s interest in number sequencing by giving a readymade solution in the gadget of their choice. You can find apps that serve various purposes, such as:
- Give practice of number sequencing and teach concepts of before, after, place value, and others
- Introduce the child to number sequences and patterns like Fibonacci numbers
- Provide answers to number sequencing problems
- Pick a common order and arrange numbers in that order
- Logic reasoning problems and solutions based on number sequences and patterns
- Number sorting to understand ascending and descending order
8. Number sequencing books
Books are the ageless resource that allows kids to acclimatize with the number sequencing activities that stimulate minds to understand and experiment with numbers. – The article ‘Books Make a difference by SB Neuman‘ points out their importance beautifully.
I have come across a number of books that help homeschoolers to get their share of practice and introduction. Books make more practice possible through problem-solving puzzles, practice sets, quizzes, identification activities, counting/tracing activities, and several others.
While some of these are making a place in the curriculum, the other number sequencing books offer additional learning support and to practice more in free time. The results of number sequencing books show phenomenal change with the learner growing into an expert in abstract reasoning or math reasoning.
Introducing number sequencing books has to be made very interesting. You can cultivate a story around the activity first. A simple example will be – consider this number snake:
To attract the students to do a number sequencing series on this, you can start by telling a snake story or some interesting facts about the figure. Then, you can slowly bring attention further to the sequencing activity. Use sentences like:
“ Do you know this snake is growing every day? 2 inches, 3 inches…and so on” to encourage them to find the next number.
Teaching methods can include ideas as simple as using the fingers of the hand to achieve the purpose of number sequencing introduction. While moving to higher grades of learning, the strategies can be different. For example, the Abacus board offers a handy solution to enhance the sequencing abilities of the child while building number sense when a certain comfort level is achieved.
Final Thoughts
Number sequencing is a part of the process of developing number sense in a child. It involves computing, analyzing, and reasoning; therefore, the very first must-have is patience. Understandably, the kids can feel boggled over by the steps.
Thus, the main task is to befriend them to sequences by making computations as easy and enjoyable as games rather than forcing upon them some compulsive learning course. Once comfort with number sequencing is achieved, the students can move further to master math reasoning.
References
- Learning disability. (n.d.). Psychology Today. https://www.psychologytoday.com/us/conditions/learning-disability
- Staff, P. T. (2019). About learning disabilities. Child Development Institute. https://childdevelopmentinfo.com/learning/learning_disabilities/
- Learning disabilities affect up to 10 percent of children. (2013, April 13). ScienceDaily. https://www.sciencedaily.com/releases/2013/04/130418142309.htm
- Kosc, L. (1974). Developmental Dyscalculia. Journal of Learning Disabilities. https://doi.org/10.1177/002221947400700309
- Pasnak, Robert & Schmerold, Katrina & Robinson, Melissa & Gadzichowski, Marinka & Bock, Allison & O’Brien, Sarah & Kidd, Julie & Gallington, Deb. (2016). Understanding number sequences leads to understanding mathematics concepts. The Journal of Educational Research. 109. 1-7. 10.1080/00220671.2015.1020911.
- Jordan, N. C., Glutting, J., Dyson, N., Hassinger-Das, B., & Irwin, C. (2012). Building kindergartners’ number sense: A randomized controlled study. Journal of Educational Psychology, 104(3), 647–660. https://doi.org/10.1037/a0029018
An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’,