"I've been told to draw a graph, but what should I do?"—Why children with developmental disabilities struggle with linear functions and how to teach them

The night before a second-year junior high school exam. If your child is looking like they are about to cry, saying, "What is y=ax+b? I don't know how to draw the graph," your heart as a parent might ache.

However, there is a reason based on their characteristics for why they struggle with linear functions. It is not your child's fault, nor is it because the teaching method is bad.

In this article, I will cover the following content.

• What kind of unit is a linear function? (A simple overview)

• Reasons for struggling by ADHD, ASD, and LD characteristics

• 4 steps for teaching at home

• Learning tools you can rely on when teaching every day is difficult

What kind of unit is a linear function?

A linear function is a unit that appears in second-year junior high school mathematics. The basic form is expressed as y=ax+b, where 'a' is called the 'slope (rate of change)' and 'b' is called the 'y-intercept'.

A major feature of this unit is moving back and forth between tables, equations, and graphs. You are required to derive an equation from a table, draw a graph from an equation, and read an equation from a graph—this mutual conversion is required.

Proportionality (y=ax), which is learned in the first year of junior high school, was a straight line where the graph always passed through the origin. In a linear function, by adding '+b' to that, the graph shifts up or down. It is only a small difference, but because the amount of information to process increases all at once, many children struggle here.

From here, let's look at the reasons why children with developmental disabilities are particularly prone to finding linear functions difficult, according to their characteristics.

Why do children with developmental disabilities and ADHD struggle with linear functions?

Struggles with linear functions often cannot be explained by "lack of study" alone. Because the way the brain processes information differs depending on the characteristics, the points where they struggle differ even for the same unit.

In the case of children with ADHD: Overload on working memory

In linear functions, it is necessary to process three stages in order—reading a table, setting up an equation, and drawing a graph—while carrying over information from the previous step.

Children with ADHD often have limited working memory capacity. Even if they can calculate halfway through, the moment they move to the next step, information from the previous step may drop out.

Also, it is easy for their attention to shift to other things during calculation, and it is not uncommon to see situations where they stop moving their hands before finding the y-intercept, even though they found the slope.

In the case of children with ASD: Obsession with the rationale and difficulties with spatial cognition

Children with ASD (Autism Spectrum Disorder) may not be able to move forward unless they are convinced of the fundamental reason why it takes the form y=ax+b. Even if they are told, "That's just how it is, so memorize it," they are not satisfied.

Furthermore, there are cases where they have difficulty with spatial cognition when drawing graphs. They may find it difficult to visually grasp movements like “1 to the right, 2 up” on a coordinate plane, which can lead to points being placed incorrectly.

For children with LD: Unable to grasp the abstract concept of “rate of change”

For children with LD (learning disabilities), the term “rate of change” is a highly abstract concept. Even if they understand the meaning of “how much y increases when x increases by 1,” they often hit a wall when trying to translate that into a mathematical formula.

Also, if they have difficulty with reading and writing, they may use a lot of energy just reading word problems and organizing the conditions, becoming exhausted before even starting the actual calculations.

Some of you may feel, “So my child wasn't just being lazy.” After understanding their characteristics, let's check “what skills are needed to tackle linear functions.”

What skills are necessary to learn linear functions?

Before tackling linear functions, please check if the following skills have been acquired.

▢ Do they understand proportionality (y=ax)?: Can they create a formula from a table and know that the graph passes through the origin?

▢ Can they read coordinates?: Can they plot a point on a coordinate plane when told (3, 5)?

▢ Can they hold 3 or more steps in their memory?: Can they follow the sequence of “look at the table → create a formula → draw a graph”?

▢ Do they understand the meaning of rate of change?: Can they answer when asked “how much does y change when x increases by 1”?

▢ Can they perform calculations with negative numbers?: Can they smoothly perform calculations like (-3) + (+5) = 2?

The items marked with an X are the starting points to go back and review. It is often the case that they don't misunderstand linear functions, but rather that the foundational skills are not yet developed.

Once you have checked these, let's move on to specific teaching methods. You don't need to do everything at once, so please try incorporating them while observing your child's progress.

How to teach children who struggle with linear functions

Step 1: Go back to proportionality and check the foundation

Before linear functions, first check if proportionality (y=ax) is firmly established. Give them a table and ask, “What multiple of x is y?”

If you move on to linear functions while this is still vague, they won't understand the meaning of “+b” and their confusion will only deepen. Once they can do proportionality smoothly, bridging to linear functions by saying, “Now, let's add +3 to this,” makes it easier to connect their understanding.

Step 2: “Learn with the body” for graphs

If their understanding of the coordinate plane is weak, it can be effective to create an x-axis and y-axis on the floor or a large sheet of paper using masking tape, and have them confirm the position of points by moving their fingers while saying “2 to the right, 3 up.” By appealing to their senses, it becomes easier to solidify spatial images that didn't click on paper alone.

Please try anything that allows them to confirm things using their hands, such as preparing larger graph paper, using coordinate stickers, or moving pieces around. If they are open to it, doing it together with your child is also a good idea.

Step 3: Fix the order of Table → Equation → Graph and repeat

Linear function problems have many variations, and they often look like different patterns every time. Therefore, there is a method of writing the order of "(1) Make a table → (2) Write an equation → (3) Draw a graph" on a procedure card and sticking it to the desk.

Since they can proceed while looking at the card in front of them without relying on their working memory, this is often particularly effective for children with ADHD. By externalizing the procedure, the burden on their mind is significantly reduced.

Step 4: Rephrase the rate of change as "If x increases by 1, how much does y increase?"

The mathematical term "rate of change" is abstract and difficult to grasp. Simply rephrasing it into everyday language as "If x increases by 1, how much does y increase?" can sometimes lower the barrier to understanding.

Furthermore, it is also effective to replace it with a familiar example, such as, "Every time you buy one juice, the price increases by 150 yen. This is the rate of change."

I have introduced four steps so far, but you do not need to feel that you must do all of these every day. Please focus on the point where your child is struggling the most and try them one by one. Even just 10 minutes once a week is enough.

Even so, some of you may feel that "teaching every time is honestly exhausting" or "my relationship with my child worsens when I teach them." I will introduce some options you can rely on in such cases next.

Summary of causes and countermeasures for struggles by characteristic

Here is a summary of the reasons for struggles and countermeasures for each characteristic.

• ADHD: Steps are missed due to working memory overload → Externalize with procedure cards and proceed one step at a time

• ASD: Cannot be convinced of the basis of formulas / Difficulty with spatial cognition → Carefully explain "why it happens" and use sensory-based coordinate understanding

• LD: Abstract concepts (rate of change) are hard to grasp → Rephrase into everyday language and go back to proportionality

Struggling with linear functions is not due to a lack of effort on your child's part. By finding a way of learning that suits their characteristics, there are many cases where they can move forward little by little.

To parents who feel it is difficult to teach every day

"I understand how to teach, but it's impossible to be with them every day while working and doing housework"—it is natural to feel that way. If you, as a parent, become exhausted, it will affect your relationship with your child.

Therefore, I would like you to know about the research results regarding working memory for the digital teaching material "Tenjin". In a study presented at the Japanese Academy of Learning Disabilities, it was reported that when Tenjin was used for 10 minutes once a week for 3 months, the visual working memory task scores of all target children improved. Considering that working memory issues are often directly linked to struggles with linear functions, this is data worth noting.

Mr. Yasuhiko Yamauchi, Representative Director of the Association for Support of Growth of Children with Disabilities (specializing in school psychology and education for children with disabilities), also recommends Tenjin, saying, "It is the teaching material I recommend the most" and "You are missing out if you don't know about it."

Tenjin's lectures (explanations) use animation to provide visual explanations. Since you can check the approximate time required (e.g., about 9 minutes) on the table of contents screen, it is designed to make it easy to plan ahead before studying. It is designed to be easy for children who have difficulty maintaining concentration to engage with.

The question screen displays only one question per screen. Because it is a format where you move to the next question after checking the answer, you will not be overwhelmed by the amount of information. When you make a mistake, hints are automatically displayed, a mechanism that ensures you are not left behind without understanding.

If you are struggling before reaching linear functions, it is often a shortcut to go back to previous units and study. You can proceed with learning according to your child's level of understanding, regardless of grade level.

Here are some comments from parents who are actually using Tenjin.

For children who have difficulty reading and writing, the optional voice reading feature is useful. Since it reads out the question text, hints, and explanations, you don't have to use energy on reading and can concentrate on the math content. You can also adjust the reading speed.

Tenjin's free trial does not require a credit card. A PC dedicated to the trial will be delivered to your home, so you can try it as soon as it arrives. Of course, there is no pushy solicitation.

In a study conducted by Associate Professor Harumi Sakamoto of R-University of Health and Welfare (Ph.D. in Medicine, specializing in developmental disability research) on children with developmental disabilities, there is this report.

*Results may vary by individual.

First, please request materials to see if it is right for your child.

→ Click here for Tenjin free trial and material request
https://www.tenjin.cc/lp/wp/hattatsu-s-02-price/

Frequently Asked Questions

Q. How much does Tenjin cost?
A. Tenjin is a one-time purchase teaching material, starting from approximately 2,750 yen per month. You can check the detailed pricing by requesting materials. No credit card is required, and siblings can also use it at no additional cost.

Q. My child is in the gray zone, will Tenjin suit them?
A. Tenjin can be widely used by children with various characteristics. It is not about whether there is a diagnosis, but whether it suits your child's current way of learning. Please check your child's reaction with a free trial first.

Q. My child is not motivated, can they continue?
A. It is not that they are "not motivated," but in many cases, they are in a state of "not being able to start because there are too many things they don't understand." Because Tenjin allows you to start from "what you can do" through backtracking learning, it is designed so that the accumulation of small "I did it!" moments leads to motivation.

Q. I have an image that children with ASD are good at math, are they also good at linear functions?
A. Children with ASD are not necessarily good at all aspects of mathematics. Even if they are good at calculation, there are cases where they struggle with spatial cognition of graphs or abstract concepts such as "rate of change." Strengths and weaknesses differ for each person, so it is important to observe your child's situation carefully.

References

• MEXT "Definition of Learning Disabilities (LD)" https://www.mext.go.jp/a_menu/shotou/tokubetu/mext_00808.html

• MEXT "Definition and Characteristics of Attention Deficit Hyperactivity Disorder (ADHD)" https://www.mext.go.jp/a_menu/shotou/tokubetu/mext_00897.html

• National Institute of Special Needs Education "Understanding Learning Disabilities (LD) and Difficulties in Arithmetic" https://cpedd.nise.go.jp/rikai/about/ld

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.