Mathematical Innocence

As the father of two school-age boys, and as a second-year, second-career high school mathematics teacher, I am now more attuned to their attitude towards math, and how it changes with time.  My vantage point is bittersweet, however, as one son distances himself from mathematics, the other engages in conversation with me about various aspects of it.

Innocence Lost

My eldest, at fourteen and a half, and a rising high school freshman, dislikes mathematics.  While his mathematical intuition is strong, his self-doubt is stronger.  In a traditional mathematics classroom, where grades of A through D range from 100% to 60%, his end of course grades range from the mid 80’s to the low 90’s, albeit a test score recently touched 6x%.  As he attends a very competitive school, many of his classmates score higher than him in mathematics, so he feels inferior comparatively; hence, his negative self-perception about his mathematical ability.  In every other subject, he scores well, to include science, which requires a fair amount of mathematical skill.  In English Language Arts (ELA), he scores at or near the top of his class every year.

My hypothesis is that he, and likely many others with similar backgrounds, believes he can apply the same skills used in reading to mathematics, where skipping a word or two minimally impacts understanding.  Whereas, in mathematics, missing any digit, operation, or sequence typically results in an incorrect solution.  This likely leads to a feeling of helplessness where one understands the concept, but makes procedural, or careless mistakes, and scores lower than desired.  Repeating this pattern can often lead to self-deprecating thoughts, comments, and behaviors, ultimately distancing a potentially successful, conceptually sound student from further pursuit of mathematics.

I believe this scenario plays itself out in classroom after classroom throughout the world, but definitely in the US with our obsession with mathematical procedure, as shown by the over emphasis on an exercise’s answer and not the thinking that went into sizing up the problem and developing a path to a solution that makes sense.  This is not to say that procedure, precision, or accuracy in a solution is unimportant.  However, if the plan is to increase the number of citizens in the science, technology, engineering, and mathematics (“STEM”) fields, I believe we are focusing incorrectly on their development.

Intermittently over the past three years, in trying to dispel his negative self talk about mathematics, I have worked to help him reframe his self-view with some success, however, a large residual remains.  This summer, as I teach him, and a friend, geometry, I have more opportunities to help him put his skills and capabilities in a better perspective.  And beneficially, working with his friend allows him to see how others succeed, and struggle, in their own ways; this benefits both since they overlap in some areas, yet differ on others so they are mutually supportive.  Sadly, though, my son’s disposition towards mathematics has existed in negative territory for so long, it is taking  some time for it to hit neutral, much less positive.  As an example, I was so proud of his sustained, increased efforts last semester where he much improved his understanding of mixture and rate problems as well as rational expressions.  Those two topic areas baffle many students of algebra.  He persevered at working to understand how to succeed in all the second semester topics, then scored lower than he expected on the end of course exam, as it was cumulative and included some topics with which he still had some misunderstanding from first semester.  Unfortunately, he was unable to receive a copy of his test so that I could decide which areas to focus on with him over the summer.  I plan to have him take a diagnostic test to help me there soon anyways.

Innocence Remains

My youngest son who is a ten-year old entering fifth grade this fall, however, seems genuinely excited about mathematics, most of the time.  I caveat this statement since he does complain a tad when asked to work on the summer mathematics work book my wife bought for him. [1]  However, when engaged with working mathematics, he rarely exhibits any sign that he thinks he is incapable, or less capable than others.  In fact, I write this post since he was so excited about a small discussion about mathematics we had yesterday.

I cannot recall how our conversation started.  At some point though, he started asking me about percentages for certain fractions.  He eventually asked me what was the percentage for one-third (“1/3”).  As a teacher, I should have asked him to figure it out himself using long division.  I did not.  I said it was thirty-three percent.  He then mentioned that it seemed wrong to him since adding up three amounts of thirty-three percent each did not equal one-hundred percent.  He even giggled a little since he identified the apparent inconsistency before me.  Since I did not study mathematics formally, as a focus of study, but instead as prerequisites for my electrical engineering degree, I rarely thought about the basis of mathematics for this type of situation, so was ill-prepared for a rejoinder.  My mind quickly fell into the common trap that 3(0.3333)=0.9999 which was obviously less than one, even though I clearly knew that representing the product instead as an integer and a rational number yielded one (e.g. 3(1/3)=1), and even that the limit of the series expansion of the repeating decimal 0.9999 equaled 1.

Quickly following his repeating decimal rational number inquiry, he asked me something related to infinity.  I mentioned that you never reached infinity, which led to his asking what was the largest number, presumably the number before infinity in his mind, although that is just speculation on my part.  I tried to explain that since you could never reach infinity, you could never arrive at one less than infinity, or even a largest number close to infinity.  My explanation left him dissatisfied and laughing again as he felt that surely a largest number existed.  My later research revealed that while there is no number mathematically speaking, there is a largest defined number, humanly speaking, which lent aspects of its name to the renowned leader of internet search engines, namely Google.  The basis for the largest named number is ‘googol,” or 1.0 x 10^100, a one with one hundred zeros to the right of the one, before the decimal place.  The largest named number is the “googolplex,” or 1×10^(1×10^100).  In words, a googolplex is one to the googol, which has googol zeros between the leading one and the decimal place; not that bad of a conversation for a ten-year old.

I only hope my eldest will find this as interesting!

[1] She teaches high school mathematics as well so while I work with my eldest and his friend, she plans out work for my youngest.

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About Dave aka Mr. Math Teacher

Secondary math teacher teaching math intervention, algebra 1, honors precalculus, and AP Calculus AB. I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.
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7 Responses to Mathematical Innocence

  1. eo says:

    Next you should throw the continuum at him. Then take away all of the integers.

    Like

  2. Pingback: Timed Tests and the Development of Math Anxiety | Reflections of a Second-career Math Teacher

  3. MRM says:

    “My hypothesis is that he, and likely many others with similar backgrounds, believes he can apply the same skills used in reading to mathematics, where skipping a word or two minimally impacts understanding.”

    That is SO insightful! My older daughter was born reading–clearly an exaggeration, but not by a whole lot. By 18 months she knew all the letters by sight and was doing phonics at 2-1/2. (“Mom, “fun” starts with “f”!) She could sight-read any word she knew the meaning of before she was 3. All this language ability brought an unexpected liability to other things–most notably math. Because reading was so effortless, she thought, as a tiny kid, that anything she didn’t already know–like the math facts–were an indication that she was stupid in that area. She just couldn’t conceive of the idea that nobody was born with the math facts already programmed and that she needed to practice them to learn them.

    In fact, this was so painful to contemplate that she just avoided the issue. In first grade, the teacher started administering timed tests; and my daughter was wounded day after day when she didn’t know the answers. It took me five years–about the amount of time necessary for her to develop her sense of who she was and what she was capable of–to get her to understand the idea of practicing basic things to develop fluency and to convince her that just because she hadn’t YET learned the math facts did not mean she was stupid in math.

    Like

  4. Dave ( also a career changer Math teacher ),a few years ahead of you. says:

    Now you see why we have google; never ending enslavement to the internet

    Like

  5. Pingback: Math Teachers at Play 52 « Let's Play Math!

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