This article is a continuation of my editorial Why do students find mathematics difficult? As brought out in the first editorial, there is no one reason why students find mathematics difficult, BUT there are many external reasons why mathematics is MADE difficult. Since I wrote that editorial, I have learned a few things that I would like to pass on.
When teaching a subject that is new to the student, the best approach to foster the student’s understanding of the subject is to first determine what the student already knows and build on that foundation by explaining unfamiliar material based on that foundation. Sounds easy but is not.
Dave Norris1, Adjunct faculty University of Missouri Kansas City Computer Science, related a story that demonstrates his struggle to fill that gap between what the student already knows, and the material being taught.
I had a college student who was doing poorly in my class. She was in my office all the time trying to “get it.” I spent hours going over it with her. I tried explaining things in as many ways as possible. I never had a student try so hard and yet fail. I asked how she was doing in her other classes. Everything was good there. Had she told me she was struggling in them I would have suggested re-examining her career choice. This was a class in an arcane programming language that is hardly used anymore. It was required for graduation and as a prerequisite for other required classes. Although little used, it is a terrific learning tool to help understand all languages. Her final grade came in just under C level. After a lot of soul searching, I gave her a C. I saw her a couple semesters later. She told me that she finally understood what I had been trying to teach her. I asked why she thought that. She explained that in the follow up course, she understood everything that was going on and that when her professor mentioned one thing now, she flashed back to “yes, that was what Mr. Norris was trying to explain.” So maybe I just explained things badly to her or maybe she had an epiphany after my class finished. I had that happen to me. A couple weeks after slogging through a course that drove me mad, the lightbulb came on. I suddenly saw everything my very patient prof had tried so hard to explain to me. Sometimes one needs a lightbulb moment when every fundamental thing is brightly illuminated. All the details become very easy then. There were a few times while lecturing I could literally see it on the face of a student when it happened. Very rewarding.
What Dave Norris and his student finally experienced is called an Aha! moment.2 Something I have experience many times over the years, and many times after finishing a class. Why is that? One reason may be the professor/teacher/instructor covers too much information for the student to process the information, or assigns too much homework.
Mark Magnacca, in the article It’s Time to Retire the Fire Hose Method and Embrace Modern Learning Techniques 3, stated the following.
When it comes to absorbing important information, all of us have almost certainly heard the expression “drinking from a fire hose.” The mind is not hardwired to absorb, learn and comprehend material in a meaningful way by the fire hose method.
Schools do not use the file hose method, but the pace at which the subject is taught may seem like it to some students. For the student to understand and retain the information from class, they must find the time to meditate on the information. You might liken this to digesting food. Once your food digests and provides the nutrients to your body, then you can take in additional nutritious food. Some may be overly full (e.g., a new mathematics concept with lots of homework) and then being urged to eat more food when they attend the next mathematics class when they have not digested the previous days lesson and homework.
When the student is unable to grasp the information, sometimes parents, or the student, may need to hire a tutor. Note what Chloe Cheney 4, an Educationist, says:
While teachers possibly don’t mean to forget about your child, they must hold with the tempo in their lesson plans. They also have lots of other students to control. Even in case, your child sees their math trainer for extra assistance, it nonetheless might not be enough to help them get better at math. Plus, it can be difficult for a teacher to identify the particular concept that your child doesn’t apprehend. This is particularly true if your child is on the shy aspect. A math tutor could be able to quickly discover problems, construct your child’s talent set, or even speak with you approximately the ways wherein you may improve those standards at home.
Here are some topics I have noticed in when tutoring students in mathematics.
- Failure to learn/understand the language of mathematics.
- A teacher instructs students in the mechanics of mathematics but does not include the mathematical concepts.
- A student’s failure to ask the teacher questions when a concept is not fully understood.
- A teacher fails to instruct their students to recognize patterns.
Mathematics is a Foreign Language
I contend that mathematics is a foreign language. Why? Mathematics has a unique alphabet and vocabulary, and can be translated from mathematics to any language and from any language into mathematics.
A language has an alphabet. The English alphabet consists of the twenty-six letters of the basic Latin alphabet: a, b, c, … x, y, z. The modern German alphabet consists of the twenty-six letters of the basic Latin alphabet plus four special letters; three are vowels accented with an umlaut (⟨ä, ö, ü⟩) and one is a ligature of ⟨s⟩ and ⟨z⟩ (⟨ß⟩; called Eszett “ess-zed/zee” or scharfes S “sharp s”), all of which are officially considered distinct letters of the alphabet, and have their own names separate from the letters they are based on.5
Mathematics also has an alphabet that of consists of the twenty-six letters of the basic Latin alphabet plus symbols that are used to describe mathematical numbers, expressions, and operations.
Like a foreign language, mathematics has a vocabulary6 where the meaning of some words may be difficult for a student to understand at first. Here are a few examples.
associative property: property (which applies both to multiplication and addition) by which numbers can be added or multiplied in any order and still yield the same value, e.g. (a + b) + c = a + (b + c) or (ab)c = a(bc)
Cartesian coordinates: a pair of numerical coordinates which specify the position of a point on a plane based on its distance from the two fixed perpendicular axes (which, with their positive and negative values, split the plane up into four quadrants)
linear equation: an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable, and whose graph is therefore a straight line, e.g., y = 4, y = 5x + 3
modulus: a number by which two given numbers can be divided by integer division, and produce the same remainder, e.g., 38 ÷ 12 = 3 remainder 2, and 26 ÷ 12 = 2 remainder 2, therefore 38 and 26 are congruent modulo 12, or (38 ≡ 26) mod 12
quadratic equation: a polynomial equation with a degree of 2 (i.e., the highest power is 2), of the form ax2 + bx + c = 0, which can be solved by various methods including factoring, completing the square, graphing, Newton’s method, and the quadratic formula
Math to English
Like a foreign language, the mathematical language can be translated into English, or any other spoken language.
The notation articulates, “The set of all x’s, such that x is greater than 0“.
English to Math
English, or any other spoken language, can also be translated into the mathematical language.
Sarah earns $10 an hour selling calculators, and every time she sells a calculator, she earns an additional $3 commission. Jamie also sells calculators, and earns $30 an hour, but only earns an additional $1 commission for every calculator she sells.
How many calculators per hour on average would Sarah have to sell to be making as much as Jamie would per hour, if Jamie sold the same number of calculators?
First, set up the equations. Their base pay is constant per hour, so the variable is the number of calculators sold multiplied by their commission rate, so the earning equations will be:
10 + 3x = y
30 + 1x = y
These are Sara’s earnings and Jamie’s earnings respectively, with x representing calculators sold and y representing earnings.7
I know that one of the hardest aspects of learning another language, at least for me, was translating. (I took French in high school and German in college.) I could say the words and I knew their meaning, but speaking in sentences or reading a passage and understanding that passage in the “foreign” language was difficult for me. The same with word problems for many math students. Note what StephenwithaPhD says about the usefulness of mathematical word problems.8
One of my favorite uses for word problems, and one of their most useful applications, is they allow educators to see how a student would attempt a question, and even if they do not completely solve it, it can show us two important things:
- They act as a sort of diagnostic question and inform us of the student’s ability. Can they interpret the question correctly? If so, can they then carry out a correct set of calculations to solve it? If they couldn’t correctly interpret the question, can they carry out a correct set of calculations to solve the problem if they are given the correct interpretation? In being able to answer these questions about the student we are able to narrow down exactly where the student is having problems and then address these more directly, which is much more beneficial to the student.
- Looking at the path the student takes to solve a question can give an insight into how they think and, in turn, allows us to make a better judgement of their abilities. We will discuss this in more detail below when comparing the three different solutions to question one.
I also enjoy what Philip Lloyd says concerning word problems.
Obviously, word problems can be very confusing especially when you are given lots of facts and you have to change them into mathematical statements.
One simple answer to your problem is that many people are better at performing mathematical processes than they are at reading and understanding the context of a paragraph of technical information!
Let’s just consider an algebraic word problem suitable for someone just starting to learn about quadratic equations.
When I squared a certain number I found to my surprise that it was equal to 10 less than 7 times than number. What number did I start off with?
A person with a slight comprehension problem would probably not know where to start. This is quite common.
Some people would just start guessing by just trying out numbers.
I did this in classes that I taught and I loved when I got the following responses…
Bill says, “Sir, the number must be 2!”
Ann says, “No it’s not! The number is 5!”
The rest of the class are just stunned or still trying out numbers…
Then I encourage the class to just make an equation!
So both Bill and Ann were partially correct!
Actually it is great when young students find out that equations don’t just have to have ONE answer! (They will only have seen linear equations at this stage)
If students can be gently introduced the word problems it can take away this overwhelming fear of them. 21
Some may say that mathematics is not language, but a language can be mechanical when spoken or written, and when read, the concepts or meaning may not be understood by the reader. The same is true of mathematics.
Mechanics Versus Concepts
I have had several high school teachers, and college professors, who taught the mechanics of mathematics but not the concepts. What do I mean by this? Let’s use the quadradic equation and formula, and the slope of a line to illustrate this point about mechanics and concepts.
Quadratic Equation & Formula
The quadratic equation is written as ax2 + bx +c = 0. To solve the quadratic equation, students are told to plug the values of a, b and c into the quadratic formula to solve for x.
When I was taught the quadratic formula it was simple plug in the coefficients and the constant of the quadratic equation into the quadratic formula and find the value of x. However, there is much more to the quadratic equation and formula that may help the student.
- The graph of the quadratic function is a U-shaped curve called a parabola.
- The x -intercepts are the points at which the parabola crosses the x -axis. If they exist, the x -intercepts represent the zeros, or roots, of the quadratic function, the values of x at which y = 0.
- What are parabolas used for? “Parabolas can be seen in nature or in manmade items. From the paths of thrown baseballs, to satellite dishes, to fountains, this geometric shape is prevalent, and even functions to help focus light and radio waves.”9
- “The … x-intercepts are especially useful. These intercepts tell you where numbers change from positive to negative or negative to positive, so you know, for instance, where the ground is located in a physics problem or when you’d start making a profit or losing money in a business venture.”10
- Do any teachers explain why it works? As Finn Frankis11 points out, the quadratic formula always works because it is simply another form of the equation ax2 + bx + c = 0, solved for x.
Slope of a Line
In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. The slope formula refers to the formula used to calculate the steepness of a line and determines how much it’s inclined. To calculate the slope of the lines, the x and y coordinates of the points lying on the line can be used. In other words, it is the ratio of the change in the y-axis to the change in the x-axis.12
For me this was another calculate the slope using the formula m = (y2 – y1)/(x2 – x1) . I do not remember being informed by any teacher or professor that the slope of the line formed the tangent of a triangle as depicted above. Knowing this fact can help a student if they will take precalculus/calculus. Especially when they learn that if f is a real-valued function and a is any point in its domain for which f is defined, then f(x) is said to be differentiable at the point x = a if the derivative f'(a) exists at every point in its domain.
Just because the word tangent is used in connection when differentiation, connecting the slope to the tangent of a triangle when performing differentiation may not be evident to the student. (It was not evident for me when I took calculus.)
When a student understands the concept, then the mechanics can be easier. Concepts are best applied when testing because they will let the student know which math formula or rule to apply. When only the mechanics is presented in the classroom, then it is up to the student to raise questions about the math, e.g., ‘What is this used for?’
Ask Questions (Teachers and Students)
Math can be intimidating, and I agree. I have seen it on the faces of many students I went to school with and in the students I tutor. Note what Alex Eustes13 writes about mathematics being intimidating.
I think the essential problem is that most students view mathematics as some sort of “contest” to get the answer right, because it’s a subject that actually has right answers. The other thing about math is, it’s readily apparent who’s good at it because they always get the answers right — immediately — when given problems they’ve already mastered.
So, students who haven’t yet mastered those problems get intimidated by the students who have. And also by the teacher/TA, who has. When average students see those people in action, they think to themselves, “I’m not that good,” which then slips into “I’ll never be that good.”
Why Don’t Students Ask Questions in Class
“Your students have questions, but they rarely ask them—especially at the beginning of the semester. They feel awkward or embarrassed, or maybe it’s just inertia. Whatever the cause, the vast majority of student questions go unasked. For teachers, this is wildly frustrating because we can’t answer the questions they don’t ask (though some questions can be anticipated). In many cases, the unasked questions represent anxieties and uncertainties that negatively affect students’ performance in class and inhibits their learning. This is a particular problem in the sophomore composition class I teach. It has a reputation as a difficult class, so many students arrive intimidated and nervous.” 14
Another reason why students do not ask questions deals with pluralistic ignorance. What is pluralistic ignorance?
Simply put, pluralistic ignorance occurs when individual members of a group (such as a school, a team, a workplace, or a group of friends) believe that others in their group hold comparably more or less extreme attitudes, beliefs, or behaviors. When many members of any one group hold the same misperception about the group norm, this norm ceases to represent the actual composite beliefs and attitudes of the group. In other words, there is an actual group norm, comprised of the actual average attitudes, beliefs, and behaviors of all individuals in the group, and there is a perceived norm, which is the group-wide assumption of extremity in the attitudes, beliefs, and behaviors of other group members. 15
Said in another way, we guess at the group members’ beliefs and norms based upon our observations, and our guess is wrong, i.e., everyone else understands what the teacher just said but I do not. What happens is that many others in the class do not understand what the teacher just said and do not ask any questions either.
The man who asks a question is a fool for a minute,Confucius
the man who does not ask is a fool for life.
Do Teachers Ask the Right Questions?
Why is asking questions important?
“Both asking and answering questions are important parts of effective learning and teaching. The types of questions you ask should capture the students’ attention, arouse their curiosity, reinforce key points, and encourage active learning.”16
However, many classes are still taught by lecture. This leads to such articles as It puts kids to sleep — but teachers keep lecturing anyway. Here’s what to do about it. and Lectures aren’t just boring, they’re Ineffective, too, study finds. This is not to say that lecturing is not an effective teaching tool, but it is NOT the ONLY way to teach. For instance, the article Lecture Method Advantages And Disadvantages | Teaching, Definition, Types, Merits, Demerits, Pros and Cons, relates that a teacher should know the benefits and downfalls of the lecture method before deciding to use the lecture method in your classroom. Other articles, e.g., 5 Benefits of Classroom Discussions, provides teachers how discussion can benefit the students. For example, discussion helps students process information rather than simply receive. Discussion gets the students to practice thinking about the course material rather than cramming, taking the test and possibly forgetting what they learned.
Am I against lecturing and for discussion? No. I think each teacher should weigh the pros and cons of each to determine the best method to be used so the students will understand the material. And students should have the ability to ask questions at anytime including, “I do not understand what you just said?” That should lead to a discussion to clarify what the student does not know and provide the teacher with feedback on how to better present the information.
[Lecturing is the] best way to get information from teacher’s notebook to student’s notebook without touching the student’s mind.George Leonard
Questions not only lead to more questions but they also lead to answers and understanding. A good way to approach mathematics is by studying patterns, which may lead to many questions but a better understanding of mathematics.
Why talk about patterns? Are they important? Absolutely! Ashley Seehorn writes: “By studying patterns in math, humans become aware of patterns in our world. Observing patterns allows individuals to develop their ability to predict future behavior of natural organisms and phenomena. Civil engineers can use their observations of traffic patterns to construct safer cities. Meteorologists use patterns to predict thunderstorms, tornadoes, and hurricanes. Seismologists use patterns to forecast earthquakes and landslides. Mathematical patterns are useful in all areas of science.” 17
Almost every subject taught, I do not know them all, is about finding patterns. Patterns in speech, patterns in the writings of authors, patterns in writing (sentence diagramming), patterns in physics, and mathematical patterns, e.g.,
- Arithmetic Sequence
- Fibonacci Numbers
- Figurate Numbers
- Geometric Sequence
- Harmonic Sequence
Why do we need to look for patterns?
“Once we have decomposed a complex problem, it helps to examine the small problems for similarities or ‘patterns’. These patterns can help us to solve complex problems more efficiently. Finding patterns is extremely important. Patterns make our task simpler. Problems are easier to solve when they share patterns, because we can use the same problem-solving solution wherever the pattern exists. The more patterns we can find, the easier and quicker our overall task of problem solving will be.”18
Let me use an example when tutoring a student on scale factors and ratios.
In the textbook, the student is asked how many red trapezoid blocks does it take to build a scaled copy of Figure C using various scale factors. The last question for Figure C is to make a prediction of how many blocks would it take to build scaled copies of these shapes using a scale factor of 5 and 6. Then the best question of all: “Be prepared to explain your reasoning.” Basically, does the student see the pattern?
The student I tutor did not see the pattern. Even if the teacher did, or did not, point out the pattern, the student failed to grasp that patterns are important. The lesson summary then makes the following statement.
“Lengths are one-dimensional, so in a scaled copy, they change by the scale factor. Area is two-dimensional, so it changes by the square of the scale factor. We can see this is true for a rectangle with length l and width w. If we scale the rectangle by a scale factor of s, we get a rectangle with length s ▪ l and width s ▪ w. The area of the scaled rectangle is A = (s ▪ l) x (s ▪ w), so A = (s2) ▪ l ▪ w. The fact that the area is multiplied by the square of the scale factor is true for scaled copies of other two-dimensional figures too, not just for rectangles.” 19
Great statement! But a 7th grade student may not read or understand this summary statement. This information is essential to a students comprehension of patterns, and reading mathematical statements with understanding.
Patterns are important!
Some mathematical ideas are so fundamental that even if you didn’t discover them, someone else would have. Mathematics is the language of science and its structures are innate to nature. Even if the universe were to disappear tomorrow, the eternal mathematical truths would still exist. It is upon us to discover it, understand its functioning and build on our knowledge to find solutions to the physical event we seek to control. 20
We covered the following four topics that can make mathematics difficult to understand.
- The language of mathematics.
- Instruction in the mechanics but not the mathematical concepts.
- Failure of the teacher and student to ask questions.
- Failure to recognize patterns.
To conclude, I would like to quote Bugs Bunny when he says, Ehh, What’s Up, Doc?
1 “A question for teachers and professors – Have you ever had a student who worked really hard in your class and yet still did badly (maybe even failed the class)?” 2022. Quora. https://qr.ae/pvWgGh.
2 “Eureka Effect – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Eureka_effect.
3 Magnacca, Mark. 2019. ” It’s Time to Retire the Fire Hose Method and Embrace Modern Learning Techniques “. Training Industry. https://trainingindustry.com/articles/sales/its-time-to-retire-the-firehose-method-and-embrace-modern-learning-techniques/.
4 Cheney, Chloe. ” Why you should always hire a math tutor for your child? ” 2022. Medium. https://chloecheney44.medium.com/why-you-should-always-hire-a-math-tutor-for-your-child-dc0cb0e6ccae.
5 “German Orthography – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/German_orthography#Special_letters.
6 “Glossary Of Mathematical Terms & Definition”. 2022. Story of Mathematics. https://www.storyofmathematics.com/glossary.html/.
7 “Word Problems with Two Unknowns – Pre-Algebra”. 2022. varsitytutors.com. https://www.varsitytutors.com/prealgebra-help/word-problems-with-two-unknowns.
8 StephenwithaPhD. “Word Problems: A Student Nightmare But A Teacher’s Best Friend”. 2022. Medium. https://medium.com/y-math/word-problems-a-student-nightmare-but-a-teachers-best-friend-d60803d155d1.
9 Dotson, J. Dianne. “Real Life Parabola Examples”. 2022. Sciencing. https://sciencing.com/real-life-parabola-examples-7797263.html.
10 Sterling, Mary Jane. “Applying Quadratics To Real-Life Situations – Dummies”. 2022. dummies.com. https://www.dummies.com/article/academics-the-arts/math/algebra/applying-quadratics-to-real-life-situations-149412/.
11 Frankis, Finn. “Why does the quadratic equation work all the time?” 2022. Quora. https://qr.ae/pv9nAI.
12 “Slope Formula – What Is Slope Formula? Equation, Examples”. 2022. CUEMATH. https://www.cuemath.com/slope-formula/.
13 Eustis, Alex. “Why does math intimidate people a lot?” 2022. Quora. https://qr.ae/pve7rZ.
14 Crawford, Meriah. 2017. “A Simple Trick For Getting Students To Ask Questions In Class”. Faculty Focus | Higher Ed Teaching & Learning. https://www.facultyfocus.com/articles/effective-teaching-strategies/a-simple-trick-for-getting-students-to-ask-questions-in-class/.
15 ” General Overview – Pluralistic Ignorance – Reed College”. 2022. reed.edu. https://www.reed.edu/psychology/pluralisticignorance/.
16 “Asking Questions: Six Types”. 2022. Mathematical Mysteries. https://mathematicalmysteries.org/asking-questions-six-types/.
17 Seehorn, Ashley. “Types of Number Patterns in Math”. 2022. Sciencing. https://sciencing.com/height-rectangular-pyramid-8507285.html.
18 “Why Do We Need To Look For Patterns? – Pattern Recognition – KS3 Computer Science Revision – BBC Bitesize”. 2022. BBC Bitesize. https://www.bbc.co.uk/bitesize/guides/zxxbgk7/revision/2.
19 “Grade 7 Mathematics, Unit 1.6 – Scaling and Area”. 2022. Open Up Resources. https://access.openupresources.org/curricula/our6-8math/en/grade-7/unit-1/lesson-6/index.html.
20 “Our Physical Universe Is Based On Patterns In Mathematics”. 2022. Mind Matters. https://mindmatters.ai/2022/12/our-physical-universe-is-based-on-patterns-in-mathematics/.
21 Lloyd, Philip. “Why can I understand math like algebra, geometry, Trigonometry, etc… but not word problems?” 2023. Quora. https://qr.ae/prmsRi.
⭐ Devansh. “Math Is A Language. This Is How You Should Learn It.”. 2022. Medium. https://medium.datadriveninvestor.com/math-is-a-language-this-is-how-you-should-learn-it-cbc7aac45a8b.
Helmenstine, Anne Marie. “Why Mathematics Is A Language”. 2022. ThoughtCo. https://www.thoughtco.com/why-mathematics-is-a-language-4158142.
The featured image on this page is from the article “These Are The 10 Hardest Math Problems That Remain Unsolved”. 2022. Popular Mechanics. https://www.popularmechanics.com/science/math/g29251596/impossible-math-problems/.
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.