Borland Educational - Math as Art

As the study of natural and abstract patterns, mathematics is an inherently creative and artistic pursuit.

Unfortunately, many students have been mistaught that math is dry, rote, and one-dimensional, devoid of creativity and inventiveness, and lacking any connection to the arts.

Advances in computing technology have finally made possible the widespread discovery and appreciation of the sublime beauty of mathematics.

The art of math can now be enjoyed by all!

I hope you'll like the following graphs of mathematical relations I created involving nothing more sophisticated than high school precalculus.

Pretty Precalculus Graphs

Designed in 2018 using the Google Search graphing calculator:

Egg Carton
z = sin(x/2)+cos(y/2)

Wrinkled Paper
z = x-y+tan(x)-cos(8x-sin(2y))

Fancy Lapels
z = arcsec(sin(1/x)-cos(1/y))

Corrugated V-Roof
z = abs(sin(2x)-y)

Bed o' Nails
z = sin(4y)+cos(4x)

Arizona Desert
z = (x-y)/(sin(x)+1/cos(y))

Celery Row
z = (sin(x)cos(y))/(x-y)

Three Fingers
z = log(sin(1.1x)-y)

Plastic Chair
z = -abs(x)+2x^4-abs(x^2*y^3)

Lettuce Garden
z = 1/(sin(x)-cos(y))

Designed in 2018 using Wolfram Alpha:

Mesa Group with Hills and Valleys - Contour Plot

Mesa Group with Hills and Valleys
z = sin(2x)cos(2y)exp(-(x^2+y^2)/6)

Slanting Waves
z = xsin(y)

Tartan Forest
z = tan(.01sin(x)cos(y))^2

Tech-Tonic Rift

Techtonic Rift
z = x^3+tan(y)+cot(x)+y^3

Spikes

Spikes
z = tan(sin(x)cos(y))^2

Gaping Maw
z = (x^2+1.5y^2)*e^(-x^2-y^2)

X Marks the Spot
z = (arccos(abs(x)+abs(y))/arcsin(abs(x)-abs(y))

Wavy Incline
z = -cos(xy)+y

Dimpled Vase
z^2 = x^6-x^2+y^6+y^2

Squarish Tube
x^2-y^3+z^4 = 1

Designed in 2018 using Apple's Grapher app:

Splotches
sin(sin(x)+cos(y)) = cos(sin(xy)+cos(x))

Ripples

Ripples
sin(xy) = cos(y)+sin(x)

Slinky

Slinky
x = sin(t^2)
y = cos(t)
t = {0..10}

The Wiggles
cos(2x)-sin(xy)-tan(2y) = .5

Controlled Chaos
x^3 = 1+7tan(xy^2)

Fender Butterfly
r = sec(x)
y = tan(x)+cot(x)+K
k = {-5..5}

Spiral

Spiral
x = (u/5)cos(u)
y = (u/5)sin(u)
u = {0..20}

Double-Back
r = t^2-t
𝜃 = tsin(t)
t = {0..40}

Line Dancing
cos(x^2)+sin(xy)+tan(y^2) = 1

Convergence
y = kx/3
k = {1..18}

$Basket Math$

Basket Math
x = cos(11t)
y = sin(13t)
t = {0..2π}

Designed in the late 1990's using the pioneering Mac app Graphing Calculator, later NuCalc:

rtan[sin(x)+cos(y)]^2 < xlog(y)

nsin(y)cos(x)/tan(x/y) < rcos[sin(x/y)]

sin(rx+y) > cos(ry+x)

sin{r[cosx(xy)]} > cos{rsin(y)]+x}

sin(x) < cos(r*ny)

tan(ry) < tan(rx)

tan(rx){cos[ry(sin(rx)]} < xcos(y)

sin(x/y) < cos[rn(y/x)]

tan[sin(x^2)-2cos(x)-y^2] < 1

tan{[sin(x^2)-2cos(x)-y^2]/(xy)} < 1

z = sin{[n(cos(x)+y)]/[tan(xy)]}

y < xtan[ycos(xy)]

rsin(y)cos(x) < rsin[xcos(y)]

sin(rx) < cos(ny)

sin(x/y) < cos(rny)

cos(y)+sin(y^x) < x/y

Articles

Mathematics and Art
Comprehensive article exploring the close relationship between math and art throughout the centuries.

Seven Times Mathematics Became Art and Blew Our Minds
Science Alert article highlights seven modern instances in which math and art imitate each other in beautiful ways.

Using Ancient Mathematics to Enrich Your Design Skills
Nowhere is the magical intersection of art and mathematics more clear than in areas comprising the broad field of design.

Making Mathematical Art
"Stunning symmetrical images created with just a few equations and a computer." From the venerable Scientific American.

Why the History of Maths is Also the History of Art
Article in The Guardian exploring how artists have utilized the inherent beauty of mathematical ideas in their work for thousands of years (at least).

Fifty Famous Curves
The title says it all. 50 fascinating general mathematical curves: Sinusoidal Spirals, Nephroids and Lissajous Curves, Fermat's Spiral, Epitrochoid/Epicycloid, more.

List of Curves
Wikipedia's accessible yet comprehensive index of articles discussing mathematical curves, their forms, and properties.

Wolfram Alpha Examples: Popular Curves
Yes, you can use mathematical relations to draw interesting, creative portraits of luminaries like Lady Gaga and Mew.

Other

GeoGebra: Math in Art
A gallery of beautiful but simple GeoGebra math art projects. Sliders enable interactivity, learning by exploration. A playground for budding math artists.

Orbiform
None other than Leonhard Euler proved the constant width of this intriguing solid. Acts like a sphere, looks like a ... what? Artsy artillery shell? Details here.

More!

Check out my Math-as-Art blog featuring additional original graphs like those above.

There's no better way to refute the notion of the lifeless sterility of mathematics than to visit the Wolfram Demonstrations Project. I urge you to take a moment to enjoy the intriguing mathematical objects and demonstrations you'll find there.

Reviews

In the words of my oldest: '... other teachers go over and over math concepts and it is still a blur. Chris explains it just once and it is "ah ha" crystal clear.'

Chris exemplifies all that we wish for in our children's teachers. He is calm, patient and always kind. He seems to have an immediate and intuitive understanding of his students ... a good role model for adolescents.

Both of my sons have had difficulties with math in the past. Currently, under Chris's tutelage, they are making excellent grades in honors math.

Chandra G. (Client: 1998)

I have spoken with [Chris's] students as well as their parents and have had nothing but enthusiastic, positive responses.

He is, apparently, a natural communicator and teacher, personable, bright and responsible.

Marilyn N. (Colleague: 1979)

Chris has taken this average boy and taught him not only several levels of math but also an appreciation for the subject. My son is now in the advanced program at his school, which is algebra in 8th grade.

Chris has also acted as a friend ... advising my son about education in general, organization, love of words as well as numbers.

I am extremely pleased.

Jeanne B. (Client: 1999)

Over this past year, I observed Chris tutoring the 8-year-old that I nanny. This child began working with Chris during 1st grade, and by now he is extremely confident in his mathematical abilities.

I was blown away as I overheard Chris teaching his student many of the same concepts that I had to review for the GRE - properties of exponents, square roots, fractions, negative integers, prime numbers, and even basic algebra. The boy was preparing for a graduate school entrance exam in 2nd grade!

Not only is Chris vastly knowledgeable in math, he is also infinitely patient and easygoing. He also takes the time to write thorough emails to update parents as his students progress.

I would encourage any parents that are on the fence to invest in a tutor. Math was a huge source of anxiety for me all throughout school (and most certainly on the GRE). From observing Chris with his student, I have no doubt that my grades and confidence in STEM subjects would have increased drastically had he been my tutor. I couldn't put a price tag on that!

Eve S. (Other: 2017)

My oldest son was struggling in math his freshman year. Chris was able to help my son raise his math grade. Today, my son is a junior, and is completing his math duties without any intervention.

Chris taught my son the importance of reading directions, slowing down and completing each process step by step.

Chris Borland is ... patient, kind, articulate and an excellent educator. WORTH EVERY PENNY!

Brigid K. (Client: 2007)

Chris has infinite patience. He never told Martha that she was 'wrong.' Instead, he gently questioned her and helped her to realize her mistake and come up with the correct answer herself. Her self-esteem always remained intact.

As I observed Chris work with Martha, math finally made sense.

Cornelia S. (Client: 1993)

Chris is a bright, positive man who relates easily to teenagers at the same time that he sets definite standards and goals.

Basil H. (Colleague: 1984)

My high school sophomore worked with Chris this past academic year. There are so many wonderful things to say. First, Chris is smart, thorough and takes his work with students very seriously. My son has a great relationship with Chris because he is a serious and respectful student, and Chris is the same as a tutor. These are not "social hour" tutoring sessions (which my son had with his freshman year math tutor in Greenbrae and what a waste of time and money that was). Chris' tutoring fees are not inexpensive but you WILL get your money's worth with Chris' sessions and your student will not be disappointed.

My son was worked so hard all year in his Honors Algebra 2 class yet he was disappointed he was only pulling "Bs" and "Cs" for first semester. He set his second semester intentions to raise his grades and get an "A" on the final and an overall "A" in the class. He set up a strategy with Chris and together they worked hard to reach my son's goal. I'm proud to report, my son reached his goal and he knows Chris' support was a pivotal part in getting him there. Thank you Chris!

L. A. C. (Client: 2019)

Claudio, as you know has been tested and labeled as a student with severe dyscalcula. I can assure you that you are not the first math tutor that has worked with Claudio.

Believe me to find someone that a fifteen year old looks forward to working with is a blessing.

Angela O. (Client: 2002)

Chris Borland is truly a phenomenal academic coach. He brings over 40 years of teaching experience to the table, and it shines in every aspect of his work.

Before working with Chris, I had been tackling the SAT like any other test - drilling on content, memorizing formulas and vocabulary, and practicing a few problems daily. I was consistently struggling with writer's block on the essay portion, and missing simple mistakes on the math section due to test fatigue. Chris revealed in his initial tutoring sessions how the SAT is a test of endurance and strategy, more than knowledge, and that developing the right test-taking skills could help me optimize my time and shift my focus to areas with a better payoff.

In our sessions, we'd alternate between individual practice problems and running through whole sections. This routine trained me to recognize the appropriate strategies for each problem, but beyond that, Chris coached me on managing test anxiety, and simple tricks to maintain focus throughout the test while keeping an eye on the clock... the things they don't teach you in school!

One skill that particularly sets Chris apart from other Bay Area tutors is his expertise with graphing calculators, which helped me shave off minutes of testing time that I could reinvest in the more difficult math problems.

Chris is patient, supportive, motivating, and genuinely fun to work with. He comes in with a high-energy that's infectious and really engages you to focus on the problem at hand. He gives a reasonable amount of homework, and frequently assesses your progress for transparency and accountability. Prepping for the SAT can take a lot of effort, but Chris ensures the effort you put in will be worth your while!

In the course of 7 months, Chris helped me raise my SAT score nearly 200 points to 2150, and reach a 710 score on the Mathematics Level 2 subject test, putting me squarely in the running for the top schools in my college list.

Chris has also tutored my niece, who recently finished her freshman year of high school with an A grade in her Algebra class thanks to his coaching! When shelter-in- place restrictions started mid-semester, she continued to have a very positive learning experience as Chris transitioned to support virtual tutoring sessions over Zoom. The extra 1:1 attention from hiring a coach like Chris during this time was particularly motivating for her to finish the year strong.

Lili D. (Student: 2010)

Chris is a phenomenal math, PSAT, and SAT tutor. I studied with him for over three years and am so grateful for all of his invaluable knowledge and the confidence that he instilled in me as a math student.

Now, years later, the same standardized test-taking skills that he taught me for the PSAT and SAT are helping me immensely in my LSAT prep.

Chris is patient and efficient in his teaching. I would highly recommend him to everyone!

Gary C. (Client: 2016)

Mr. Borland worked with our 8 year old daughter for 3 months. He introduced many teaching devices, including manipulatives which were very beneficial.

Mr. Borland made math fun, including recommending several 'fun' math books for her to do on her own. She now averages a 'B+' in Math and approaches the subject with enthusiasm!

Michele A. (Client: 1998)

Mr. Borland explains the materials very slowly so that it is easy to understand. After a few sessions with Mr. Borland my precalculus grade improved significantly. He definitely knows what he is doing.

Highly recommend!

Agung I. (Student: 2012)

I will preface this by saying that I have always been a humanities-oriented student - if Chris could help me succeed in math and standardized tests, he can truly teach anybody!

Chris was my tutor over the course of several years for integrated math (geometry + algebra 2), the PSAT, and the SAT. He helped me smoothly transition into a new advanced math curriculum when I switched schools, and I was very impressed with his knowledge of the local integrated math curriculum, which was completely foreign to me when I moved across the country. I was initially uncomfortable with geometry, but Chris helped me not only excel at it, but by the end of the year, even enjoy it! He explained new concepts very clearly and was skillful at coming up with new exercises tailored to the concepts I needed help with the most. When those concepts were solidified, he taught me more difficult challenges ahead of the rest of my class!

I really appreciated how well rounded and interdisciplinary Chris' approach to education was. Even now, over a decade later, I fondly remember at the end my first year studying with him the fictional book about geometry, Flatland, that he gave me, which opened my eyes to the aesthetic beauty and even the philosophy of math. I know for certain that without Chris, I never would have had the interest to later pursue calculus, purely for fun (and I'm very glad I did!).

Lastly, it's worth highlighting that Chris is immensely patient, compassionate, and accommodating, especially to students with disabilities. Throughout high school, I struggled with undiagnosed Lyme disease, and Chris really supported me in my math and standardized test studies, especially during the period before I received my academic accommodations, which allowed for extended time on exams. I also had a near death in the family, and Chris helped me get back up to speed after I missed a considerable amount of classes.

I will always be very grateful for everything I learned from Chris!

Jane D. (Client: 2020)

I have received many positive comments from Chris's students about his enthusiasm for the subject, clear explanations of concepts, and general knowledge of mathematics.

Chris is a responsible, capable, and compassionate person who cares that his students reach their potential.

Linda P. (Administrator: 1989)

My son attends junior college and has struggled with math for many years.

We wish we would have plugged into Chris many years ago. He has been instrumental in increasing skills as well as confidence and my son received his best grade ever …

I am so proud of him and so grateful for Chris.

Judy W. (Client: 2012)

Both Samantha and I have been extremely pleased with Chris. Samantha's grades immediately improved. Even more importantly, she really began to understand math and gained confidence in her ability to do well in math.

As an added bonus, Chris is always warm and friendly, and he is really a pleasure to see each week.

Carolyn C. (Client: 1999)

We have found Chris Borland to be a very effective communicator, and a personable, supportive, and talented academic coach. I wholeheartedly give Chris my highest personal recommendation.

Sheva F. (Client: 2013)

Chris Borland has been instrumental in my success as a business woman in the private education field. I met Chris a few years ago when I just started my private tutoring business teaching math for Elementary and Middle School children. Chris gave me lots of advice and encouragement which helped me to shape and grow my business.

I learned more in the two hour meeting with Chris than I did in a whole year of tutoring. His suggestion to focus on working only with girls lead to the successful specialization of my practice.

Thanks to Chris's help I have increased revenue and successfully filled a gap in the local market. Now I have around 50 girls that I tutor every year!

Vali G. (Colleague: 2017)

[Chris has] tremendous knowledge and enthusiasm for mathematics. He is conscientious, reliable and communicates well both with a teenager and his parents. I heartily recommend Chris Borland.

Paula M. (Client: 1988)

Chris Borland is a gifted tutor [who] understands that people learn differently. Chris figured out that I am a physical learner, applying teaching techniques that were applicable to my learning style.

Max P. (Student: 2013)

[Chris] did a splendid job. He was exceedingly conscientious, earnest, intelligent, and was lovely and patient with our son.

He was reliable, understanding, knew the [math] 'cold' ... I can recommend him highly.

Emily K. (Client: 1983)

Chris is an extremely skilled math tutor and helped me tremendously in improving my math score.

It is clear that he has spent a lot of time studying both the ACT and SAT because [his approach] is filled with tricks and test techniques that apply to both tests.

Thank you for all your work!

Olivia P. (Student: 2017)

Chris carefully guided her towards a new confidence about her capacity to achieve ... realized in her SAT scores which jumped hundreds of points, her new found excitement, her commitment to accept and conquer the challenge of a particular set of problems. He immediately instilled a sense of: 'no problem, you can do this!'

Improved confidence, renewed joy of learning, kind and thoughtful guidance, and improved scores – these are the fundamental and guiding principles of Chris Borland.

Margaret J. (Client: 1999)

Chris Borland has been a great tutor for our son in preparing for the ACT.

When we began the process of looking for a program to help our son study for the ACT, I was overwhelmed by the number of companies in our area that offered these services. As we were fairly new to the Bay Area, I turned to my friends and neighbors for advice on who to choose. Borland Educational was recommended to me by several people, all of whom had used Chris's tutoring services in the past and had excellent results. I decided to sign our son up for one meeting with Chris to see how they worked together. After the first 1 1/2 hour session, he felt more confident and calm about taking the ACT. He told me to sign him up for more sessions.

He has now been working with Chris continually since the beginning of his junior year. He tells me that Chris is in-depth and strategic in his teachings. Chris Borland has been a pleasure to work with and has provided our son with a lot of knowledge and confidence. I highly recommend Borland Educational.

Martha T. (Client: 2020)

I have been impressed with Chris’s high level of energy and enthusiasm for his work, his attentiveness to detail, his appropriateness and effectiveness.

Barbara C. (Client: 1983)

I started doing ACT prep with Chris at the beginning of my junior year. When I first took the ACT, I scored a 21. After a year of working with Chris, I scored a 32. His constant optimism and insightful feedback helped me develop the skills needed to succeed on the ACT. Through our test prep sessions, my confidence in math skyrocketed. In addition, my understanding of grammar and writing conventions grew. So, not only did my ACT score improve 11 points, but my understanding of math and English improved tremendously.

Now, I am working with Chris as I am in honors calculus. Chris's is the first teacher who has taught me calc in a way I understand it. This has enhanced my confidence in the classroom and improved my testing results. I would recommend Chris to anyone struggling with test prep or math.

No matter what math level you're in, go to Chris.

Emma P. (Student: 2019)

Chris did an excellent job of tutoring my daughter for the SAT. She walked away from her lessons feeling that she now knows how to approach the test in a more confident manner. With any of these tests just knowing the subject well isn't enough. She already loved math but Chris gave her the strategy needed to really succeed on the SAT.

Matt Q. (Client: 2020)

Did she tell you the amazing news about her report card? She got 4 A’s, 1 A- and one A+ ... her first time for straight A's. I think her A in math was to be expected considering how well you have brought her along, but I firmly believe that the sense of confidence and discipline you have instilled in her aided her in getting such great grades overall.

Thank you for helping her become such an accomplished student.

Laura L. (Client: 2002)

Our son, Bobby, was diagnosed last December with severe learning disabilities.

Bobby worked with you twice a week on his math. He began to understand the concepts, finish his homework, and excel in class. His self-esteem increased tremendously.

The most commendable accomplishment was the increase in Bobby’s SAT1 scores. Bobby met with you individually for two hours only and increased his scores from 860 to 1020, a total of 160 points! This has significantly increased his chances in several of his favorite college choices.

And this is the boy that was going to only look at schools that did not require standardized tests.

Sally N. (Client: 2003)

My son had been struggling with standardized testing for some time, and in a matter of one month, Chris Borland, who came to our home only once a week, was successful in helping [him] increase his Math Computation scores in the Iowa Tests of Basic Skills from 67%-tile to 81%-tile, [and] his overall composite score from 52%-tile to 80%-tile.

More importantly, our son gained self-confidence and a new passion for Mathematics. We are truly grateful for Chris Borland's dedication, patience and genuine concern.

Adriana P. (Client: 2006)

Chris is OUTSTANDING! He worked with our daughter to help prepare her for the SAT and the ACT. We started early as our daughter deals with a certain amount of test anxiety. Chris worked above and beyond to get our daughter as comfortable as possible with the test both emotionally and academically.

In the end, we are abundantly grateful to Chris for all that he has done. Our daughter was able to get her score to where it needed to be, and she came through the past many months with new skills on how to calm her mind as well as how to apply what she knows to these tests.

Aimee P. (Client: 2017)

Worth every penny! As the high school application process in SF is already stressful and time consuming, I was looking for an efficient strategy for my B/G twins to prepare for the SSAT. Chris has deep experience with test prep and presented a very clear, efficient, and effective strategy for the twins to follow. He supplied worksheets so that metrics could be tracked, and additional resources such as the most relevant word lists to practice and so on. He also is a wealth of information regarding test taking techniques and strategies. His approach resulted in a huge increase in test results. We would highly recommend him to anyone for standardized test prep!

Irene L. (client: 2018)

Chris is conscientious, prompt and very patient. Laurie actually enjoyed her tutorials, if such a thing is possible!

Jared E. (Client: 1983)

Chris Borland is an excellent tutor both in SAT and mathematics. He always answers all my questions thoroughly and walks me through math problems step by step so I understand exactly what I am doing and why I must do it that way.

As far as SAT tutoring, I could not ask for a better coach. Chris knows so many helpful techniques, hints and practices that helped me understand the SAT itself, study the right information and speed up my test taking ability.

I owe it to Chris for an SAT score that I am very proud of and that has definitely helped me get into the colleges that I applied to. And the practice and tips I learned from Chris for the SAT also has helped me so much on tests in all my other classes!

Chris is a very knowledgeable and skilled tutor and truly offers the very best tutoring service I can imagine.

Thank you so much for all you have done, Chris!

Lily H. (Student: 2012)

I have been working with Chris for the past six weeks and have already seen significant improvement in my practice ACT scores. I love working with Chris because he is extremely knowledgeable about the ACT and helps me with strategies to get through the test that you can't learn in a book.

Chris is always calm and encouraging even when my sessions are in the evening after a long day. He truly cares about his students and goes the extra mile. I like meeting at his Sausalito office which is conducive to focusing. I highly recommend Chris especially as a tutor for the ACT and SAT exams.

Will J. (Student: 2017)

Chris Borland is a wonderful, kind, patient and effective tutor in math and test prep.

My daughter wanted to jump up to honors algebra when she reached 8th grade. Chris worked with her the entire summer so that she could do just that. He came every week, worked quietly at the kitchen table and gave her lots of confidence.

Because of Chris's dedication and reassuring words, our daughter not only jumped up in math, but did it with straight A's!

Chris is efficient, generous with his time, and from our experience, an incredibly nice person! We couldn't have been more pleased.

Colleen H. (Client: 2011)

Marshall knew he had work to do on his vocabulary (synonyms and analogies) … I did some research with an educational consulting firm in San Francisco and got Chris Borland's name.

The systematic approach Chris employs with his students is extremely effective. Before he meets with a student, he sends parents an extensive questionnaire to help him get to know your child's needs. Next, he spends some quality time with your child asking questions and listening intently to their responses and concerns. Through conversation and observation, he assesses the situation and comes up with a strategic ‘game plan’’ for the student's success.

In Marshall's case, he watched Marshall take several SSAT practice tests and was able to see exactly what issues Marshall was having with the test. Together they addressed the types of mistakes he was making and how he needed to change certain behaviors.

Chris taught Marshall to strategically approach a standardized test. Everything from warm-up exercises to very specific techniques to assist him in ultimately getting to the right answer. Chris always left Marshall with a detailed outline and notes to keep him on track when he took practice tests at home.

The work Chris did with Marshall really paid off. Chris and Marshall only met four times for a total of 4.25 hours starting on 12/01/07 through 1/03/08. On the 1/7/08 SSAT, Marshall more than doubled his March 2007 verbal score to 71st percentile. Needless to say, Marshall felt very good about his verbal score improvement and credits Chris's strategies for his success.

Belinda L. (Client: 2008)

[Chris's] rapport with both of my sons has been very successful ... reflected in their improved grades during [Chris’s] guidance.

Lucilla B. (Client: 1983)

Chris took his time to walk each of our two sons through the cognitive process of learning how to learn. He crafted his approach to match the learning style of each of our sons personalities and academic capacities.

Today, they are both successful young men. One even graduated Summa Cum Laude from the university where he attended all four years on academic merit scholarship, while our other son is now excelling in that cryptic (to me) world of coding. Chris was instrumental as a catalyst in their intellectual awakenings at a crucial time in their lives.

Thank you again, Chris!!

G.U. (client: 2010)

Our daughter, a high school junior, met with Chris for a total of 8 sessions. She thoroughly enjoyed her visits with Chris, always coming away feeling confident and inspired. She vacillated until the 6th session about whether to take the ACT or SAT, so Chris couldn't tailor the majority of the tutoring to the specific test. She also had such a busy schedule and demanding level of AP homework that she hardly spent any time working at home with the practice materials that Chris gave her. Despite that, her score on the ACT (which she finally chose) increased from a 30 on her practice test to a 34 on her first real ACT.

Needless to say, we're thrilled with the result, and relieved that she doesn't need to take it again. One and done! We're also so grateful for how downright pleasurable Chris made the process for our daughter and us. We'll be back in a couple of years with our younger daughter!

William S. (client: 2018)

Chris really helped me prepare for the math section of the ACT. Math was one of my weakest subjects in testing but Chris gave me many strategies and helped me practice. He gave me so many practice tests and worked through all the problems with me to help me understand and improve. The practice helped especially to build my confidence and ease in completing these problems. In the end, Chris' help took me from a 28 to a 32 out of 36 in the math section of the ACT. I would definitely recommend going to Chris for tutoring!

Alison D. (student: 2016)

You did great with Kelsey. Her progress was reflected in her Stanford 9 testing. Her score for Total Math went from the 23rd percentile to the 66th percentile. Problem Solving went from 20th to the 48th percentile. Math procedures went from the 32nd percentile to the 80th. This was very important and significant progress between the second and third grade.

Fran E. (Client: 2002)

Mr. Borland was one of the best tutors I've ever had. His teaching style was highly individualized. A tutor like this is hard to find. He helped me study and prepare for the SSAT and thanks to him I was accepted to many of the private high schools I applied to.

He knew exactly what my problems areas were, and worked with me to correct them very efficiently. Thanks to Mr. Borland I got accepted to the school of my choice. He is very patient and knowledgable. He explained everything in a very clear and understandable manner. He goes above and beyond to make sure his students are successful.

His test taking tips and tricks were so useful. I felt confident in my test taking abilities after I worked with him. Mr. Borland is a brilliant tutor, one of a kind. He helped me so much! I highly recommend him to anyone preparing for the SSAT, or in general as a tutor.

Nicole R. (Student: 2017)

Thank you very much for your great help in preparing our daughter Sarah for her SAT tests. Her math went up 90 points to 650 and her verbal went up 50 to 780!

Linda W. (Client: 1998)

Before starting my [SAT] sessions with Chris, I received 450 on critical reading, 530 in math and a 490 in writing. After working with Chris and him teaching me all the strategies that he knows I was able to raise my critical reading score to a 540, math to a 610 and writing to a 580. I finally got a score that I could send to colleges.

As of right now I wish I never even had wasted my time with Ivy West and Kaplan and worked with Chris the entire time because I think that I could have raised my score even more had I worked with him for a longer period of time.

Sam A. (Student: 2009)

Chris was terrific in helping our child prepare for their SSAT for a few weeks before the test. Our child is a motivated and hardworking high academic performer but had never taken a test like this before, and needed guidance on strategy and review of materials. Chris is not just an excellent and ethical academic teacher and coach, but has a lovely calm demeanor, and works on instilling confidence as well as content. Our child's score was in the mid-90th on first attempt at the exam, so we were all really happy with the outcome!

As a parent, I appreciated that Chris was extremely personable and professional throughout. The costs were clearly explained up front, calendaring and paperwork (including a proper contract) was thorough and professionally handled, and he was always on time and on schedule. In-person meetings are run appropriately from a professional office space with an open door, and he has a full, high quality professional videoconference set-up if you need to meet remotely (no time wasting trying to log in).

I highly recommend Chris!

Sarah B. (Client: 2019)

Chris helped me improve my SAT math score by 150 points in a 2 month period! I had a great time working with him. I tried several other SAT tutors who were unable to provide me the same great results. I would recommend Chris to anyone looking to boost their SAT or ACT score.

Shoshana F. (Student: 2013)

My daughter did very well in her math classes and found Mr. Borland to be resourceful in his knowledge and in his methods of teaching.

His work with her clearly made a great deal of difference in her school performance and in her attitude about the subject matter.

I am able to recommend Mr. Borland enthusiastically ...

Clifford A. (Client: 1989)

From day one, Chris was a very out going person that I felt comfortable around. He makes jokes every once in while to keep a dry subject like the SAT's interesting. The most important thing for me was that he never moved on until he made sure where I was making mistakes and helped me so I would not make them again.

My favorite part about Chris’s program is that it's not all about content like the other SAT courses are. Chris didn't try to teach me new material but instead he taught me new strategies that I had never heard of before which helped me work at a faster pace and get more answers correct.

Overall, I am extremely happy that I worked with Chris and I truly believe that he is one of the best SAT instructors out there. I really enjoyed the sessions I had with him and I still use the strategies he taught me for multiple choice tests that I take today.

Sam A. (Client: 2009)

Chris is amazing. Both of my teenage sons have worked with him and saw tremendous success. My 14 year old was falling behind in math, very frustrated and losing his confidence. I was referred to Chris by a trusted friend whose son worked with Chris and received incredible results.

In just a few sessions with Chris my son was back on track and received an A on his next exam. He continues to work with Chris when he needs more instruction than his teacher is able to provide.

Kristin M. (Client: 2019)

Chris is an amazing math tutor and has helped my daughter immensely over the past year. He made math fun and interesting for her. She was not interested in math before starting her tutoring with Chris and now is really excited about math.

When he teaches he gets excited about thinking through a math problem which is infectious. He has a calm and patient demeanor but is also funny and cheerful with kids.

My daughter loves working with Chris and she has gotten a perfect score on all her math tests because he has taught her to enjoy math, the fundamentals of math, and the "why" behind specific math equations.

I highly recommend Chris Borland!

Kina I. (Client: 2022)

Chris is great. His advice on test taking techniques and his teaching style helped our daughter improve her overall SSAT score by 50 percentile points [to 83rd%-tile]. We will be coming back to Chris for SAT prep!

Brad B. (Client: 2014)

... when she was in seventh grade, our daughter qualified to take the SAT as the entrance test to the Johns Hopkins Youth Program.

Chris explained the structure of the test, helped her become familiar with the directions, and taught her strategies for answering certain types of questions.

Linden scored well on the SAT and gained admission to the program. The value of her sessions with Chris goes beyond this one test. She will continue to use the information and strategies she learned from Chris. I recommend him highly.

Lorraine B. (Client: 1996)

Our son is bright and capable but we felt that we needed to back away from closely monitoring his school work and put that process in the hands of a professional.

Chris was meticulous about monitoring Barth's assignments and tests. Chris had a talent for uncovering even those assignments that Barth might "forget" to mention. My husband and I wish schools still taught students how to do a detailed outline before they start writing, and we are so appreciative that Chris taught Barth that invaluable skill.

No detail, even what type of paper to use for a "standout" project, was too minor for Chris and he always presented his suggestions to Barth in a congenial, professional manner, taking the time to discuss why one approach might work better than another. He and Barth often brainstormed ideas, tactics, and approaches in a way that made Chris's tutoring so much more than just looking at homework.

Chris reminded not just Barth but his parents, by his approach to learning, that there can be great joy and satisfaction in developing ideas and opinions and in deepening one's knowledge of a subject. Barth became an editorial writer and cartoonist for his school's newspaper and I'm sure his growing confidence in his own ideas emerged from these collaborative discussions.

Finally, he was accepted by 9 of the 12 boarding schools to which he applied ...

Gordon and Janet W. (Client: 2001)

I’ve known Chris for a decade and have come to respect him more and more and more each year. He's a master with calculator programs for test takers! He is also a caring man and goes more than an extra mile for every one of his students. I have sent several students to Chris and each has reported significant score/grade gains and complete satisfaction.

Scott C. (Colleague: 2017)

Chris is an excellent ACT tutor.

I was able to go to a reach college that required a score higher than I thought I could get.

T J (student: 2018)

Chris is patient, knowledgeable and was helpful in identifying my weaknesses and addressing them. My work with Chris on test prep also helped me in the classroom resulting in improved grades both my junior and senior years. He is easy to work with and never made me feel stupid. I highly recommend Chris Borland for test prep and math tutoring.

T J (student: 2018)

Chris Borland tutored my l4 year old twins for several weeks before they took the SSAT test for entrance into private schools. He has a very good rapport with kids and makes the sessions fun and interesting.

Chris's strategies about standardized test taking proved to be excellent. Both boys improved their scores significantly and openly attributed their success to their work with Chris...

Anne C. (Client: 2003)

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