"They're counting from the beginning again..." — How to teach children with ADHD who struggle with counting combinations
"I was drawing a tree diagram, but halfway through I lost track of where I was and started counting from the beginning again," or "My counting method is all over the place, and I get a different answer every time."—
I sometimes hear these comments from parents of children with ADHD regarding the unit on counting combinations. They can do it when working together, but when they are alone, they miss items or count them twice. Seeing them recount over and over, you might find yourself saying, "Count in order properly," and then regretting it later. Perhaps some of you have had this experience.
There is a reason related to developmental characteristics why counting combinations is difficult to master. It is not because they "lack focus" or "lack motivation."
In this article, I will explain the reasons for this and provide concrete teaching methods you can use at home.
What you will learn in this article
What is the unit on counting combinations (arrangements and combinations)?
Why do many children, and especially children with ADHD, struggle with this?
Skills needed to solve counting combination problems
Tips for teaching and encouraging your child starting today
What is the unit on counting combinations (arrangements and combinations)?
Counting combinations is a unit where you count "how many ways there are in total." It appears in the 4th or 5th grade of elementary school, and students learn about "arrangements" (arranging items while distinguishing order) and "combinations" (selecting items without distinguishing order). It is an important concept that serves as the foundation for probability in junior high school and beyond.
For example, there is a problem like this:
"Choose 3 people from 4 (A, B, C, D) and line them up in a row. How many possible arrangements are there in total?"
The tool used to solve this problem is a "tree diagram." A tree diagram is a chart where you branch out and write down all possible cases, such as "If the first is A, then the second is B, then the third is C..." If you decide on the first person, write down all the branches that come from that, then decide on the next first person, and so on, writing them down systematically, you can count without missing anything or counting the same thing twice.
However, this part about "writing systematically" is difficult for many children.
Why is counting combinations difficult? (Many children struggle regardless of developmental characteristics)
Counting combinations is a unit that many children find difficult, regardless of whether they have developmental characteristics or not. There are three main reasons for this.
The first is that there are many steps. Counting combinations requires counting "systematically, without omissions, and without duplicates." You must maintain a fixed method—"fix the first item, write down all cases that come from it, then move to the next first item"—without breaking it until the end.
The second is that it is difficult to distinguish between "arrangements" and "combinations." Do you count "A to B" and "B to A" as different things (arrangements), or as the same thing (combinations)? This distinction requires accurate reading of the problem statement and is a common point of confusion.
The third is that trying to process it all in your head causes information overload. Because you have to think about the next step while remembering "which branch I am currently counting" and "how far I have counted," it is easy for your brain to get congested.
It is by no means rare for children to restart counting halfway through or to come up with inconsistent answers.
Points that become even more difficult for children with ADHD
What I have explained so far are difficulties common to many children. In the case of children with ADHD, several hurdles are added to this.
The first is the load on working memory. Working memory is the ability to temporarily hold and process necessary information in the mind while performing a task. In counting cases, one must move their hands while holding onto information such as "which branch am I writing now?" or "what did I fix at the beginning?". Children with ADHD often have challenges with working memory, so this information can disappear while they are writing, making them prone to returning to the beginning, thinking, "Wait, how far did I count?"
The second is the challenge of continuing a procedure in a consistent order. A tree diagram is established by maintaining the order of "fixing the beginning and finishing writing it before moving on to the next" until the end. If attention shifts elsewhere midway, this order collapses, leading to writing the same branch twice or skipping a branch, which results in overlaps or omissions.
The third is the burden of remembering throughout the task whether one is counting by "arrangements" or "combinations." Maintaining the premise of "do I distinguish the order for this?" while counting also places a strain on working memory.
The increase in re-counting and the instability of answers are not due to a lack of effort on the child's part, but are due to the characteristics of their brain's information processing. There is no need to blame them.
Skills necessary to solve counting cases
For counting cases to be mastered, several skills need to be built up individually. Please use this as a guide to check where your child is struggling.
Working memory: The ability to move one's hands while holding onto where one is currently counting and how far one has progressed
Ability to continue procedures in a consistent order: The ability to continue without breaking the order of "fix the beginning and finish writing it -> move to the next"
Understanding to distinguish between arrangements and combinations: The ability to read from the problem statement whether or not to distinguish the order
Awareness of noticing omissions and overlaps: The awareness to check "have I counted everything?" or "have I counted the same thing twice?"
It is not necessary to have everything at once. When you can see where your child is struggling, you will know where you need to help.
How to teach children who are bad at counting cases [Step-by-step format]
From here on, I will introduce specific ways to teach at home. You don't have to do everything at once. Please try from the parts that seem doable while observing your child's condition.
Step 1: Understand the distinction between arrangements and combinations through familiar examples using diagrams
When the distinction cannot be made through verbal explanation alone, it becomes easier to convey by showing diagrams of everyday examples.
For example, if you are "choosing 2 colors from 3 colors—red, blue, and yellow—to decide the colors of a jacket and pants," then red jacket/blue pants and blue jacket/red pants are different combinations (arrangements). On the other hand, if you are "choosing 2 colors from 3 to put in a bag," then putting in red and blue is the same as putting in blue and red (combination). Repeatedly confirming whether "order matters" in familiar situations makes it easier for the concept to stick.
Step 2: Think about whether it is an "arrangement or combination" first, and write it on paper
Before you start drawing a tree diagram, first check, "Is this a problem where order matters?" While reading the problem together, try saying this to them:
"Is this a problem about arranging things, or just choosing them? If you count them as different when the order is different, it's an 'arrangement.' If they are the same even if the order is different, it's a 'combination.'"
If you think only about this first and write it down on paper, you won't have to wonder, "Do I need to distinguish the order?" while you are in the middle of counting.
Step 3: Draw the tree diagram while "saying it out loud"
Since counting only in your head can lead to information overload, have them say what they are writing out loud. By moving their hand and mouth together, "where they are currently counting" is brought out of their head, which reduces the burden on their working memory.
"If we make A the start, then next is B, then next is C—let's try writing it while saying it out loud."
If your child can say it out loud, acknowledging them by saying, "Good, that's the way," will make it easier for them to continue.
Step 4: Draw a line to separate each branch once it's finished
Once you have finished writing everything for "when A is at the start," draw a line to mark the end of that section. By making the separation visible, it becomes clear that "this part is finished" and "next, we start from B."
"You've written everything that starts with A. So, let's draw a line here, and then let's start from B next."
If there is a separator, even if their attention drifts, the progress remains on the paper, so they don't have to start counting from the beginning again.
Step 5: "Small-step practice" for one branch at a time
Instead of counting the whole thing at once, start with practice that only involves writing one branch, such as "only when A is at the start." Once they can reliably write one branch, gradually expand to two branches, and then the whole thing.
"Today, let's try writing everything that starts with A. Once you can do that, let's try starting from B tomorrow."
By breaking it down into small parts and accumulating successes, the feeling of "I did it" leads to motivation for the next step.
You don't need to feel like you have to do all of this every day. It is enough to just choose one step that seems doable and try it out.
Summary
Let's organize the struggles with counting problems and the countermeasures.
Many steps (counting regularly, without omissions, and without duplicates) → Draw a line to separate each branch once finished, and leave the progress on the paper
Hard to distinguish between arrangements and combinations → Think about whether order matters before starting to write and note it on paper / show familiar examples using diagrams
Load on working memory (difficulty keeping track of where they are counting) → Write while speaking aloud to get information out of the head
The sequence of steps is easily disrupted midway → Practice in small, branch-by-branch segments to build up successes
You do not need to do everything at once. Finding a way that makes your child feel, "I was able to do this myself," is the first step toward independent learning.
To parents who find it difficult to teach every day
Trying to put the measures introduced so far into practice—preparing problems, timing your prompts, and checking the stopping points together—requires a considerable amount of concentration from you as a parent. Honestly, continuing this every day is very difficult.
For those people, I would like you to consider the home learning digital teaching material, "Tenjin". It is a teaching material exclusively for Windows PCs, providing an environment where it is difficult to get distracted by games or videos during study, allowing your child to focus on their work.
In a study using Tenjin, it was reported at the Japanese Academy of Learning Disabilities that using it for just 10 minutes once a week for 3 months improved the visual working memory task scores of all participating children. I mentioned earlier that "the load on working memory is high for counting combinations," and Tenjin is designed to support learning that requires holding that information temporarily in a manageable way.
Dr. Yasuhiko Yamauchi, a leading expert in special needs education (Representative Director of the Association for Growth Support of Children with Disabilities), after actually reviewing Tenjin, evaluated it by saying, "You are missing out if you don't know about this" and "it is the most recommended teaching material".
With Tenjin, you learn counting combinations in the flow of Lecture → Key Points → Problem Practice.
The lecture uses animation to visually explain how to draw a tree diagram by "fixing the start and branching out from there." Because it is short and compact, it is designed to be easy for children who have difficulty maintaining concentration to work through to the end. Following this, the key points of the lecture are organized on a summary screen, and then you proceed directly to problem practice.
Problems are displayed one per screen, and you proceed to the next problem after checking the answer. There is no "getting distracted by seeing other problems." Also, because you can check the approximate time required on the table of contents screen, it is easier to plan "how much time it will take today."
When you get a problem wrong, a hint is automatically displayed, and you can work on that problem again. It is designed so that even if you are not sitting right next to them, your child can proceed while checking "what they should be careful about" on their own. When they get the correct answer, audio saying "Genius!" or "Amazing!" plays, and points are added.
"Even if I get it wrong, it doesn't hurt my feelings (because I can do it like a game). It's good that I know how to get the right answer after making a mistake." — From a post-Tenjin experience survey
We have also received comments like this regarding children who have difficulty maintaining concentration.
"There is nothing unnecessary, so it is easy for the child to concentrate. The lectures are thorough and easy to understand. The number of problems is just right." — From a post-Tenjin experience survey
Tenjin is structured by grade level in accordance with textbooks, and you can easily switch grades. Even when you want to go back and check units that are prerequisites for counting combinations, you can build up from where your child is currently at without strain.
In a study conducted by Associate Professor Harumi Sakamoto of R-University of Health and Welfare (Medical Doctor, Developmental Disability Specialist) targeting children with developmental disabilities, there is this report.
"A child who lacked confidence in everything showed a major change in motivation, such as expressing a desire to 'take lessons' on their own just one month after starting Tenjin. The confidence of 'I did it!' leads not only to learning motivation but also to the motivation to take on challenges in sports and daily life—in other words, 'independence'."
— Harumi Sakamoto, Associate Professor, R-University of Health and Welfare
*Results may vary by individual.
Tenjin is currently offering a free trial. We will deliver a PC dedicated to the trial for free, so you can check how your child actually operates it before making a decision. No credit card is required, and we will not strongly push for a purchase after the trial. Tenjin is a one-time purchase learning material, available from approximately 2,750 yen per month. Please check the details regarding costs and purchase plans when requesting materials.
→ Click here for Tenjin free trial and material request
https://www.tenjin.cc/lp/wp/hattatsu-s-02-price/
Frequently Asked Questions (FAQ)
Q. Even when I have them write a tree diagram, it falls apart halfway through. What should I do?
A. We recommend separating it branch by branch rather than asking them to "write everything correctly at once" from the start. Once they finish writing "when the first item is A," draw a line, confirm together that "we've finished this part," and then move on to the next. Leaving the separators on the paper prevents them from having to recount from the beginning if their attention drifts. Speaking out loud while writing is also effective for reducing the burden on their mind.
Q. How should I explain the difference between "arrangements" and "combinations"?
A. The key to distinguishing them is "whether or not you count them as different things if the order is different." Since it is difficult to convey with words alone, it is easier to understand if you show familiar examples in diagrams. For example, if you are "deciding the colors of a jacket and pants," the order matters (arrangement), but if you are "choosing two colors to put in a bag," the order does not matter (combination). Please try confirming this repeatedly in situations close to your child's daily life.
Q. My child has an ADHD diagnosis. Will Tenjin be a good fit?
A. Tenjin can be used by children with various characteristics. It is not about whether or not there is a diagnosis, but whether it fits your child's current way of learning. It is designed to be easy to focus on, with one question per screen and automatic hints when they make a mistake, among other features to help them engage with limited information. Please start with a free trial to see your child's reaction.
Q. In what grade of elementary school do they learn about counting possibilities?
A. Counting possibilities (arrangements and combinations) is a unit mainly taught in the 6th grade of elementary school. The timing and scope covered vary depending on the school and textbook. You can check the grade and unit your child should be working on using the actual screen during the free trial.
References
Ministry of Education, Culture, Sports, Science and Technology "Elementary School Curriculum Guidelines (Notified in 2017) Commentary: Mathematics"https://www.mext.go.jp/
National Institute of Special Needs Education https://www.nise.go.jp/
This article was created by the Tenjin Media Editorial Department. There are individual differences in the characteristics of developmental disabilities. If you have any concerns regarding learning, please also consider consulting a specialized institution.