The purpose of this explication (I just love to use sesquipedalian words.) is to alert you to the classroom problems I experience as a substitute teacher. I would hope that these issues may motivate some teachers to make changes in the way they teach, or parents to take action to help your student or school make the needed changes.
In the 1967 Paul Newman movie classic Cool Hand Luke, the Captain of Road Prison 36 has just told Luke that he is wearing chains for his own good—to which Luke cleverly responds, “Wish you’d stop bein’ so good to me, Cap’n.” The Captain then speaks the memorable line: “What we’ve got here is failure to communicate.”
You may be asking yourself what does Cool Hand Luke have to do with teaching and learning mathematics. Let me explain.
When substitute teaching, I attempt to present mathematical concepts in an easy to understand manner. The problem I am presented with is when I ask students questions about a problem they are trying to solve, the most common response from some students is “I don’t know.” Either the students do not understand the mathematical terms (e.g., PEMDAS, coefficient or quotient) or the mathematical concepts (e.g., division of a fraction by another fraction is the same as multiplying the fraction by the reciprocal of the second fraction) I am using, or most likely they were not paying attention when the terminology or concept was taught. The students should know the terminology and concepts since their teacher covered the information in class. Some students fail to learn the language of mathematics.
Language of Mathematics
Communication is simply the act of transferring information from one place, person or group to another. Every communication involves (at least) one sender, a message and a recipient. This may sound simple, but communication is actually a very complex subject. Note what Sunil Singh [1] says about the complexity of the language of mathematics.
Mathematics is a complex language. The interest/curiosity/need to learn this language is first and foremost. If that’s not there, why the hell are you learning this language in the first place? It’s also a slippery slope towards this:
The love for language — the love for the language of mathematics — has to be put in place first. Or else, you’re building a house a cards with fluency. Why do we do “review” every September in every grade in mathematics? Once you learn a language, you don’t forget it?
He goes on to suggest the following, and I have heard similar words come out of many student’s mouths.
Kids forgot the math they learned because they simply don’t care — and, they don’t care about whether we care about that.
Math being a complex language means not easy! Not easy means boring! And students do not care to be bored!
“So many of the things that we do in math education — and maybe more generally in education — are giving students answers to questions that they would never think of asking. By definition, that’s what it is to be boring. If you’re sitting at a bar and someone’s telling you stuff that you’re not interested in and you would never think of asking about — what is more boring than that? That seems to be the model of our educational system: ‘Here’s the formula for the cosine of the double angle.’ ‘Well, I don’t care about that.’”
Steven Strogatz
Comparable to learning a foreign language, many students taking mathematics ask “When will I ever use this?” (I asked the same thing when I was in college taking German. Answer. IF you decide to get a PhD, then you MAY have to translate German mathematics papers for your dissertation. As Bill and Ted would say: BOGUS!)
Like other subjects taken in school, I do not know how useful math will be to the student once they graduate. Many students do not even know what they want to do with their life next week let alone once they graduate from high school or even college. Even though much of mathematics may not be directly applicable to everyday life, mathematics can teach you to … [7]
- Persevere
- Problem solve
- Apply prior knowledge
- Think critically
- Justify your answer
- Plan ahead
- Construct viable arguments
- Create models
- Attend to precision
- Use appropriate tools
- Analyze information
- Critique other’s reasoning
- Explain your thinking
- Make connections
- Discover patterns
- Find solutions
- Describe relationships
- Represent ideas
- Challenge your thinking
- Experience an “Aha” moment
- Think logically
I wholeheartedly agree with the above list! Mathematics has served me well to accomplish each item above in all my endeavors.
| Take Away |
|---|
| The language of mathematics or mathematical language is an extension of the natural language (e.g., English) that is used in mathematics and in science for expressing results (scientific laws, theorems, proofs, logical deductions, etc.) with concision, precision and unambiguity. – Wikipedia |
Another reason I see for student’s apathy toward mathematics is the memorization of math trivia, i.e., information that can easily be looked up when needed. This is memorization for the wrong reason.
Memorization
In school, and even on the job after graduation, sometimes requires a person to memorize information, e.g., definitions of words, formulas, how to use a piece of software, and processes. Those things used day-to-day. There is no need to memorize everything but there is a need to understand concepts and ideas so you can explain and use them.
The same is true when learning a foreign language. When I was taking a foreign language, my teachers had us memorize the translation of English words into the foreign language and how to conjugate verbs. I struggled with verb tenses. What I thought was not taught very clearly was putting the two together because I did not know how to speak in complete sentences but I could rattle off the individual words and conjugate a verb.
Here is a good example of improper memorization in mathematics (IMHO):
“What’s the sine of π/2?” I asked my first-ever trigonometry class.
“One!” they replied in unison. “We learned that last year.”
So I skipped ahead, later to realize that they didn’t really know what “sine” even meant. They’d simply memorized that fact. To them, math wasn’t a process of logical discovery and thoughtful exploration. It was a call-and-response game. [4]
I wholeheartedly embrace what Paul Nowak states:
While memorization is essential to learning, over memorization does much harm and little good. It makes learning take longer and more stressful, makes learning superficial, does not help comprehension, makes you forget faster, and does not encourage practical application. Instead of over memorizing, embrace more meaningful learning by linking to previous knowledge, visualizing, and repeatedly using what you learn. [3]
“The principle of sleep-teaching, or hypnopædia [2], had been discovered.” The D.H.C. made an impressive pause.
Brave New World by Aldous Huxley: Chapter Two
…
(A small boy asleep on his right side, the right arm stuck out, the right hand hanging limp over the edge of the bed. Through a round grating in the side of a box a voice speaks softly.
“The Nile is the longest river in Africa and the second in length of all the rivers of the globe. Although falling short of the length of the Mississippi-Missouri, the Nile is at the head of all rivers as regards the length of its basin, which extends through 35 degrees of latitude …”
At breakfast the next morning, “Tommy,” some one says, “do you know which is the longest river in Africa?” A shaking of the head. “But don’t you remember something that begins: The Nile is the …”
“The – Nile – is – the – longest – river – in – Africa – and – the – second – in – length – of – all – the – rivers – of – the – globe …” The words come rushing out. “Although – falling – short – of …”
“Well now, which is the longest river in Africa?”
The eyes are blank. “I don’t know.”
“But the Nile, Tommy.”
“The – Nile – is – the – longest – river – in – Africa – and – second …”
“Then which river is the longest, Tommy?”
Tommy burst into tears. “I don’t know,” he howls.)
It seems that the three R’s taught in schools – Reading, wRiting and aRithmetic have been replaced with the three new R’s: Read, Remember, and Regurgitate. In my day, we used the phrase cram, memorize, regurgitate and forget. Students today seem to apply the same philosophy when they take a test of fail to answer a question about the subject the next day.
To quote Ryan McCoy [6] ‘… similar experience with nearly everyone learning math — pressure to perform. If we slow down and realize that this is a way of thinking, not a series of equations to memorize, there are infinitely many possibilities on how we can create useful math without the fear of receiving a bad grade.’
| Take Away |
|---|
| “Understanding will often lead to memorization, but memorization alone often does not lead to understanding.” |
Many teachers and substitute teachers also have concerns surrounding student distractions, i.e., the student’s minds are not on school (boring!) but on what game they can play, or what they can view and listen to, on their laptops (exciting!). These concerns are valid.
“I’m so confused! I mean, I’m not sure if I’m confused or not.”
The above quote is from Vinnie Barbarino, a character on the TV show Welcome Back, Kotter. He answers most questions with a simple “what?” or “when?” but it’s no joke – it doesn’t take much for him to become confused.
Are students really confused? Based on my substitute teaching of grades 4-12, the answer is a resounding no! They may say they are confused but in reality they do not understand the material for the following reasons.
- Not paying attention to the information being presented in class.
- Not accomplishing the assigned work and are not familiar enough with the material.
- More interested in music and videos they can access on their phones or computers.
- Some students would rather just ask for the answer than go looking for it.
Note what some have stated in their response to the question “Is the American School System Failing American Children?” [5]
“I have taught students in several grade levels from elementary grades to college and most of them have had some interest in something, whether it was cars, sports, fashion, or music. Unlike former students, my current students have very little interest in and almost no desire to learn about anything. Whenever I ask them to engage in activities that are designed to broaden their horizons, they perform only the most perfunctory work and research, and they are willing to believe anything that they read online without checking the validity of what they are reading or the credibility of the source.“
“I’m in high school right now. 98th-99th percentile, straight A’s every quarter, etc. etc. All that good stuff. Yet I go home, and I think to myself every single day: what did I LEARN? I find myself unable to answer. I will take a test, score 100, and not know the answer to the most basic questions about the concepts; I only knew the study guide.”
The thoughts and information students absorb depends on how it is presented and is it relevant to them. Marketing and entertainment are good at presenting information that students will absorb. Is it the same with how math is presented in schools? If math is not presented in a relevant way, then students can become disengaged from the class and become bored, confused or frustrated.
PEW RESEARCH CENTER
In addition to asking public K-12 teachers about issues they see at their school, we asked how much each of the following is a problem among students in their classroom: [8]
- Showing little to no interest in learning (47% say this is a major problem)
- Being distracted by their cellphones (33%)
- Getting up and walking around when they’re not supposed to (21%)
- Being disrespectful toward the teacher (21%)
Some challenges are more common among high school teachers, while others are more common among those who teach elementary or middle school.
Cellphones: 72% of high school teachers say students being distracted by their cellphones in the classroom is a major problem. A third of middle school teachers and just 6% of elementary school teachers say the same.
Little to no interest in learning: A majority of high school teachers (58%) say students showing little to no interest in learning is a major problem. This compares with half of middle school teachers and 40% of elementary school teachers.
Getting up and walking around: 23% of elementary school teachers and 24% of middle school teachers see students getting up and walking around when they’re not supposed to as a major problem. A smaller share of high school teachers (16%) say the same.
Being disrespectful: 23% of elementary school teachers and 27% of middle school teachers say students being disrespectful toward them is a major problem. Just 14% of high school teachers say this.
| Take Away |
|---|
| “The first half of my life I went to school, the second half of my life I got an education.” – Mark Twain |
Conclusion
I encounter many of the problems addressed above as a substitute teacher, and some are the same problems I had as a student when in school. So what then does Cool Hand Luke have to do with teaching and learning mathematics?
Like the Captain of Road Prison 36, sometimes schools forge chains that limit the freedom of the students as far as learning mathematics. The chains consist of at least the following links: language, unnecessary memorization, and not presenting the information so the student will learn and apply the mathematical concepts. Unless things change, I believe today’s students and educators may have the same conversation as the Captain and Cool Hand Luke.
Captain
You gonna get used to wearing them chains after a while, Luke.
But you never stop listening to them clinking.
Because they gonna remind you of what I been saying …
… for your own good.
Luke
Wish you’d stop bein’ so good to me, Cap’n.
Caveat
I work with and know a number of public educators, and they do their best to teach their students. The problem, as I see it, lies in the bureaucracy of the system that puts chains on their ability to teach, e.g., government laws, school boards, parents, book vendors, and budgets. Note the following statement.
“The question of whether the American school system is failing American children is a complex and debated issue. There are various factors to consider, such as disparities in funding, the quality of education, standardized testing, and the impact of socioeconomic factors on educational outcomes. It’s important to recognize that the education system is constantly evolving and efforts are being made to address its shortcomings. Different stakeholders have different perspectives on this issue, and ongoing discussions and reforms are aimed at improving the educational experience for all children.” [5]
Most of the “ongoing discussions and reforms are aimed at improving the educational experience for all children” address the symptoms and not the problems facing education today. Educators are the doing the best they can with what they have. Kudos to them!
References
[1] Singh, Sunil. 2023. “Mathematical Fluency Has A Constrained And Dangerous Definition in Education.” Medium. Medium. December 21. https://sunilsingh-42118.medium.com/mathematical-fluency-has-a-constrained-and-dangerous-definition-in-education-ecc4000450b4.
[2] Hypnopaedia or sleep-teaching is a way that the governing bodies in Brave New World teach children about morality and class distinctions. To impart information onto a child, speakers placed next to the child repeat slogans and messages while they sleep in order to ingrain those messages in the child’s memory. These messages promote societal ideals regarding class roles and proper behavior regarding sex and conformity. “Brave New World.” 2023. enotes.com. enotes.com. Accessed December 31. https://www.enotes.com/topics/brave-new-world/questions/what-is-hypnopaedia-teaching-in-brave-new-world-71323.
[3] Nowak Paul. 2022. “6 Consequences of Over Memorization (And What To Do Instead).” Iris Reading. May 4. https://irisreading.com/6-consequences-of-over-memorization-what-to-do-instead/.
[4] Orlin, Ben. 2018. “When Memorization Gets in the Way of Learning.” The Atlantic. Atlantic Media Company. June 12. https://www.theatlantic.com/education/archive/2013/09/when-memorization-gets-in-the-way-of-learning/279425/.
[5] “Is the American School System Failing American Children?” 2024. Quora. Accessed January 19. https://www.quora.com/Is-the-American-School-System-failing-American-children.
[6] McCoy, Ryan. 2024. “Math: When Am I Ever Going to Use This?” Medium. Math: When Am I going to use this? https://medium.com/math-when-am-i-going-to-use-this/math-when-am-i-ever-going-to-use-this-0e373207f1ad.
They say math is the language of the world. It can be seen in what is expected, like engineering, science, programming, and finance. However, what’s not often seen is that math exists in great literature, deep sea scuba diving, making the perfect cup of coffee, or the brush strokes in a Van Gogh painting. In a way, this makes math kind of beautiful.
[7] I Love Math. “When will I ever use this? Math Bulletin Board”, TPT. https://www.teacherspayteachers.com/Product/When-will-I-ever-use-this-Math-Bulletin-Board-3530651.
[8] Lin, Luona. 2024. “What’s It Like To Be a Teacher in America Today?” Pew Research Center’s Social & Demographic Trends Project. Pew Research Center. April 4. https://www.pewresearch.org/social-trends/2024/04/04/whats-it-like-to-be-a-teacher-in-america-today/.
Additional Reading
Language of Mathematics
💭, Drop of. 2024. “MATHEMATICS FOR ELECTRICIANS.” Medium. Medium. March 17. https://medium.com/@kjolzen/mathematics-for-electricians-13ae49f9e24b.
A language is based on words, phrases, expressions, and usages that have been brought together through the ages in haphazard fashion according to the customs of the times. Mathematics is built upon the firm foundation of sound logic and orderly reasoning, and progresses smoothly, step by step, from the simplest numerical processes to the most complicated and advanced applications, each step along the way resting squarely upon those which have gone before.
This makes mathematics the fascinating subject that it is.
Morgan, Eliot. 2024. “Mathematics Language: Developing Student Vocabulary.” Third Space Learning. March 4. https://thirdspacelearning.com/us/blog/math-language-skills/.
Mathematics language is an aspect of math teaching we may not often consciously consider when planning and delivering lessons, because we’ve already been through the process of absorbing and becoming fluent in it.
But explicit teaching of mathematical language is one of the most important and impactful aspects of developing our students into able mathematicians.
With that in mind, let’s take a detailed look into why mathematics language is important, how it might be presented, and how we can teach it effectively.
Memorization
Arden, Melanie, “Rote Memorization vs. Meaningful Learning – Is There a Place for Both?” 2024. prodigygame.com. Accessed January 16. https://www.prodigygame.com/main-en/blog/rote-memorization/.
“Cram, Memorize, Forget: Why TESTS Are The Problem With Higher Education.” 2017. Everything Braden. February 7. https://bradensthompson.wordpress.com/2011/12/08/cram-memorize-forget-why-tests-are-the-problem-with-higher-education-2/.
Ferlazzo, Larry. 2021. “The Roles of Memorization in Teaching & Learning (Opinion).” Education Week. Education Week. March 5. https://www.edweek.org/teaching-learning/opinion-the-roles-of-memorization-in-teaching-learning/2020/07.
Ferlazzo, Larry. 2021. “‘Memorization Often Comes Without Understanding’ (Opinion).” Education Week. Education Week. March 5. https://www.edweek.org/teaching-learning/opinion-memorization-often-comes-without-understanding/2020/07.
Students flipping to the very last page of their essays to see the grade rather than the comments. Parents emailing about their students’ scores rather than their growth. Government initiatives focusing on test scores and test scores only. Do any of these situations sound familiar?
For many teachers and principals across America, the emphasis on grades in education is irritating at best and damaging at worst. Have students always been obsessed with grades? Why? Is there anything educators can do about it?
Jamshaid, Dr. Faisal. 2020. “Use Your Brain for Thinking, Not Memorizing.” Medium. Medium. December 21. https://faisaljamshaid.medium.com/use-your-brain-for-thinking-not-memorizing-116423ea0a8c.
Kaufman, Peter. “Cram. Memorize. Regurgitate. Forget.” 2012. Everyday Sociology Blog. April 26. https://www.everydaysociologyblog.com/2012/04/cram-memorize-regurgitate-forget.html.
Marco Learning. 2023. “Grades vs Learning – Shifting Attention to What’s Important.” Marco Learning. April 6. https://marcolearning.com/retiring-the-red-pen-shifting-attention-from-grades-to-learning/.
Patterson, Ransom. 2021. “Why Cramming Is Hurting Your Learning (and What to Do Instead).” College Info Geek. September 14. https://collegeinfogeek.com/cramming/.
Whether because of procrastination or a full schedule, you’ve probably stayed up all night trying to cram information into your brain so you can pass an exam the next day.
Despite its prevalence on college campuses, however, cramming isn’t an optimal way to study. Not only is it stressful, but most of the information you cram into your brain will be gone soon after you’re tested on it.
If your only goal is to pass the exam, then that’s not a huge issue. But if your goal is to learn material for the long term, to deeply understand the concepts you’re studying, then cramming is a terrible technique.
Why is cramming so bad, exactly? And what should you do instead? Let’s find out.
Open Colleges. 2015. “Understanding the Limitations of Academic Success.” Open Colleges. July 20. https://www.opencolleges.edu.au/blogs/articles/why-straight-a-students-havent-learned-as-much-as-you-think.
A famous study on forgetting textbook materials compared the percentage of material remembered after different intervals of time. The results were as follows:
After 1 day: 54% was remembered.
After 7 days: 35% was remembered.
After 14 days: 21% was remembered.
After 21 days: 18% was remembered.
After 28 days: 19% was remembered.
After 63 days: 17% was remembered.
“I’m so confused! I mean, I’m not sure if I’m confused or not.”
Seelig, Tina. 2016. “Teaching – It’s about Inspiration, Not Information.” Medium. Medium. December 14. https://tseelig.medium.com/teaching-its-about-inspiration-not-information-1f64ddf019e7.
Teaching is really about inspiration, not information. Effective teaching focuses on why and how, not what. The goal should be to spark each student’s imagination, to find a hook in their heart and mind so that they feel a need to learn the material. The rest is easy, because the student then drives his or her learning. My role as a teacher is to ask provocative questions, and to help the students make a path toward the answers. If they are motivated to find the path, they will carve it themselves. If I have to pull out a mental machete to expose the path, then I haven’t done my job.
Questions Young People Ask—Answers That Work. “How Can I Get along with My Teacher?” JW.ORG. Accessed January 29, 2024. https://www.jw.org/en/library/books/Questions-Young-People-Ask-Answers-That-Work-Volume-1/In-and-Out-of-Class/How-Can-I-Get-Along-With-My-Teacher/.
Barbara Mayer, a teacher herself, adds: “Teachers, who have probably repeated these same lessons more times than they care to remember, tend to fall into a routine.” What can you do to liven things up? “Raise your hand for a change and ask for more information . . . Make him really tell you all he knows.” Will the teacher resent this? Not if you do so respectfully. Says Mayer: “You’ll discover that your teacher is coming to class a bit more prepared, and with more than just surface information.”
“Why Is School so Boring? 10 Reasons Why You Might Struggle with Boredom at School🥱.” 2024. The Grown-Up School. March 30. https://thegrownupschool.com/why-is-school-so-boring/.
Have you found yourself feeling completely fed up of school, wondering, “Why is school so boring?” It’s normal to feel bored with things that you do regularly, and you’re definitely not the first person to feel that way. Today we’re explaining 10 reasons why you might feel bored at school, and how you can make it feel more fun!
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Mathematical Mysteries 