You don't have to look too closely to see math at work in everyday life. From calculating transaction change and sale prices, to looking at the percentage of probability on a weather forecast, to setting timers so that your Christmas lights will turn on at a measured level of darkness, math is arguably irremovable from every experience on Earth. Even nature falls into patterns, routines and life cycles, down to the most micro ecological organizations.
But what about the Earth, high in the air? Birds and other flying creatures have long been making use of Mother Nature’s carefully attuned laws of physics to carry themselves from Point A to Point B. It turns out that the airline industry is also a great borrower of such universal mathematical and physical principles, and given the high economic and human stakes associated with flying, aviation is arguably one of the industries for which reliance on mathematics is the most critical.
It is easy to see how critical it is that a pilot is capable behind the controls. With thousands of pounds of metal and flammable fuel careening through the atmosphere at extremely high velocities, carrying human or economic cargo over invaluable infrastructure, civilizations, and natural environments, the stakes for safe airline operation are extremely high. Airline pilots are rigorously trained in countless formulas, models, and principles of physics. On the other side of airline training, students are equipped to see the world as a series of grids and graphs, and to understand how a whole host of calculable variables interact with each other in three-dimensional space.
Pilots must be proficient in formulaic calculation and must have a keen mind for deviation. They must be able to account for routine and circumstantial factors as well as safety buffers, to calculate fuel requirements and to know the mathematical limits of their resources in case they need to make emergency adjustments to their plans. Pilots have to know how different levels of cargo weight will affect their instruments and handling. Altitude, velocity, and pitch will have significant impacts on how the vehicle operates and where it ultimately goes. It’s the pilot’s responsibility to make sure that from a numerical standpoint, everything is in tip-top shape before taking off or landing.
From Creation to Transportation
Every single component of an airplane is optimized for maximum performance in the air. Decades of research and experience have gone into curating every bolt, every panel, every cavity, and every instrument to deliver the safest flying experience possible. Deviations of fractions can have detrimental consequences, which is why mathematicians and engineers are part of every stage of the development process.
Moreover, maintenance schedules need to be designed and followed meticulously for assurance that the vehicle is operating within its safety margins. These need to be mathematically optimized to calculate the exact amount of variation that can still provide a safe experience. Maintenance operators need to understand how to measure these variables and metrics in order to properly mark an aircraft as safe or unsafe.
And once in the air, safety is not just the responsibility of the cockpit, but it is also critically in the hands of air traffic control. Air traffic control operators manage a series of complex geometric representations of their air spaces. They must have a vivid understanding of flight trajectories, velocities, and geometry, and be able to extrapolate that information to prevent disasters in the air. Pilots and air traffic controllers operate on universal principles to measure and curate safety, and those principles are based on mathematics.
The Fabric of Space
Fixed mathematical principles define and explain how the world works. Math does not change, though our understanding of how to operate within its principles allows us to advance higher than was ever possible before. A constant reliance on math governs our travel in the air, allowing us to catapult forward in our understanding of how humans can move around this planet―and even unlock the potential we are currently exploring for traveling beyond it. Without an insatiable curiosity to explore physics and math when engineering rocket ships and space travel missions, international space travel efforts would simply fall flat.
These principles have helped us optimize all kinds of transportation on the ground, too. Cars and vehicles have gotten safer since we’re better able to mathematically measure risk and adjust accordingly. Roads have been optimized for traffic flow and stoplights have been timed for pedestrian safety. This is math at work, getting us from our everyday Point A to Point B. As we continue to dive into what math has to offer, and how we can creatively apply that to our insatiable desire to explore, perhaps math will one day get us all the way to Point Z, as well.
It’s a common complaint from math students all over the world, learning a new concept or practicing a particularly tricky one: “But when am I going to use this?”
While every student may not use every facet and formula from their math education on a daily basis when they reach the working world, we emphasize at Mentorhood that teaching math is about far more than simply teaching calculable numerical solutions.
Math and intuition are self-reinforcing: teaching math intuitively empowers students to smoothly advance to more complicated math topics―and the more mathematically a student can fluently think, the more intuition, critical thinking, pattern recognition, and analysis skills become second nature.
It’s those soft skills, paired with a mathematically logical framework for thinking, that fuel some of the highest paying careers in today’s modern economies. And globally, there’s no industry with a greater association to higher pay than the medical industry.
Math in Medicine?
From dentistry to surgery to anesthesiology, there is no doubt that the field of medicine boasts of highly skilled, highly specialized, and highly paid professionals. And up to a certain degree, the specialists of today’s medical sector share a similar educational background: a rigorous post-graduate medical degree, undergraduate studies that equip them to pass the competitive application process to medical school, and primary and secondary studies at a public or private institution, when they were just regular kids in regular classes.
While science is the primary academic thrust of medical training, math is inarguably linked to that science in a critical way. Many math and science studies are inseparable at the post-secondary level and are often packaged together on admission requirements. But why are they so connected when it comes to medical science? Don’t medical professionals need biology a lot more than binomials?
The Mathematical Foundations of Medical Science
The average emergency or family physician might draw on their years of studying, recognizing, and diagnosing biological anomalies a lot more than they draw on what they remember from high school calculus. But no branch of medicine can be separated from the math that helped it come to be.
Anesthesiology is the study of how anesthetics affect the human body. Working with something like anesthetics can have grave consequences if it is not administered properly. Every patient brings a unique set of variables to the operating table (pardon the bun), from basic metrics and vitals such as age, heart rate, oxygen levels, blood counts, and the like, to medical history, environmental factors, family medical risks, and countless more. The decision to administer anesthetics and how much must be fine tuned for each individual patient, and anesthesiologists must be able to predict and adjust for the resulting outcomes.
Continuous improvement of the study of anesthesiology cannot be done without complex empirical research and statistical analysis, measuring a plethora of variables that may or may not turn out to be relevant to the final conclusions. Moreover, practicing professionals must be critically aware deviations from the norm and will build up their own banks of pattern recognition experience over time.
In fact, any empirical justification of a medical intervention must be founded in reliable analytical studies of complex statistics before it can be considered both safe and recommended. Observable results must not only be non-harmful but must also be statistically significant. This cannot be understood without advanced knowledge in how the mathematics of statistics works.
Epidemiology is the study of viral spread in human populations, and its role has become increasingly more visible as the world faced the COVID-19 pandemic. Decision-makers continue need accurate data on case spread, symptom changes, mutations, effectiveness of policy measures, vaccine risk factors, treatment efficacy, and more in order to make informed policy decisions, and this is fundamentally rooted in statistics.
But statistics is just the start. With accurate statistics, medical scientists can develop formulas, charts, and data tables that enable radiologists to advise hospitals on radiation and magnetization levels for imaging technology. Oncologists can make decisions about how much chemotherapy to administer to a patient. Cardiologists can measure the variables in a proposed heart surgery and have a framework for maximizing the probability of success.
An Inseparable Pairing
Medical science is inseparably dependent on mathematics. Whether its the analysis of a revolutionary new technology, or the determination of correct doses of paediatric medicines, we would be far more behind in the medical capabilities of the modern world without the contributions that mathematics has made to the medical world. If you’ve had a major surgery or have seen a loved one recover from a serious illness thanks to the administering of a life-saving treatment, you’ve witnessed math-fuelled medicine at a personal level. In many ways, all of us, either directly or indirectly, owe our lives to the influences math has had on our advancements in medical science.
Future Career Math Career
We’ve spent a lot of time at Mentorhood Math creating a curriculum that teaches math through concrete, relatable methods, because we believe that’s the best strategy to building mathematical intuition. When it comes to building a solid foundation of understanding that gives our learners the ability and the confidence to advance to higher topics, we believe that teaching slowly, systematically, and intuitively is far superior than teaching through repetition, rote training, or spoon feeding.
Spoon-feeding and rote training give students mechanical answers to math problems, and repetition reinforces its memorization. Students have the option to copy and repeat the solutions without really understanding what they are doing, or without being able to extrapolate the principles to slightly different situations.
At Mentorhood, we want students to understand the problem, know how to arrive at the solution, and proceed to calculate it.
If repetition can take the thinking out of the equation, should it be done away with all together? Is it destroying our children’s ability to think through math for themselves?
Building Mechanical Understanding
As stated above, repetition reinforces the ability to repeat mechanical answers without spending too much time thinking about it. Repetition ingrains mathematical processes into a student’s mind and body so that they can repeat the mechanics with almost no hesitation.
This can be highly advantageous―so long as you don’t skip the thinking from the start.
The Path to Mastery
Mentorhood believes that understanding is the first step on the path to mastery, but that memorization falls on the path too. We believe that repetition should be something students work towards, not work from. We don’t train through memorization; we reinforce through it.
Our three-step approach to teaching mastery in a topic begins understanding and builds up in skillfulness:
Building their skillfulness from understanding to memorization not only makes them more efficient calculators, but gives them confidence that they have mastered the topic. They understand what they are doing so well, and how to apply it to different situations, that they don’t need to think about it anymore, and can focus on efficiency instead. They know that if required, they could pull out the knowledge underneath their memorized solution at the drop of a hat.
It’s All About the Numbers
Math really is a numbers game. It’s about speaking numbers, breathing numbers, and drawing inferences from how numbers operate and interact with each other in the real world. Math is the language of numbers, and like learning any new language, sometimes you have to memorize a few new words.
We believe in first helping students understand that mathematical questions come from tangible problems that they can relate to in their lives. We seek to help learners develop intuitive visualizations so that they can see how numbers relate to one another. And we believe that once understanding is achieved, the next step on the path to true mastery is expediency.
The more expedient students become at arriving at solutions that they have calculated over and over (rather than answers that have been fed to them), the more confident and quick they will be in integrating those calculations into more complex topics. They will be quicker to recognize patterns and relationships because their memory banks will be full of similar problem and solution sets that they have seen before. This can make them even more capable of drawing inferences and conclusions based on the numbers they see.
It Starts With Understanding
That’s the difference between our approach and the traditional methods of rote training. Memorizing through rote training fails to link these numbers into relationship with one another because questions-and-answers are memorized individually, and are stored in little memory banks as isolated instances.
Memorizing based on understanding allows students to draw on a well of related information that bubbles up into memorized answers. It allows students to tap into an interconnected network of mathematical relationships underneath the surface of what they can repeat without hesitation.
So, is there room for memorization at an institution that stresses learning through understanding and building intuition? Absolutely. Memorization is just one more step towards building a more thorough and robust mastery of mathematics. And with enough time and the right understanding-based approach, memorized math can feel just as reliable and comfortable as intuition, too.
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Throughout this series, we’ve been exploring how the teaching techniques we use at Mentorhood Math, inspired by the highly successful Singapore Math method, focus on learning through intuition.
We refer to our students as “learners,” subtly encouraging them to see themselves as in charge of their own learning, rather than subordinate to instruction. We take our time, solidifying basic concepts before moving on to more advanced ones. We utilize visual tools, concrete objects, and fun to make learning more accessible and relatable to young minds. All of this helps students internalize what they are learning.
But what really is the difference between CPA learning (working with Concrete, Pictorial, and eventually Abstract representations of math) and rote training, also known as learning through repetition or spoon feeding? Are the differences that major? Are the advantages truly significant?
Leaving More than Potential
In much of the world, math is told rather than taught. Instructors state a new type of problem that the class will be tackling, and then go on to explain a series of mechanical steps to solve the problem.
Often, the example problems are stated in abstract, mathematical terms and are hardly connected to the real world at all. Ever heard a child say, “But when am I going to use this?” Perhaps you’ve said it yourself.
Once the mechanical steps have been sufficiently memorized by a sufficient percentage of the class, the real keeners are given the opportunity to apply the mechanics to word problems. Students have the added challenge of picking out relevant details, determining which operation they should use, and proceeding with the calculations.
The rest of the class is left stumbling, puzzling over which number goes where in the formula and trying not to invert steps while asking themselves in frustration, “Why on earth is Joan wasting time calculating where her baseball will land at all? Just hit it!”
Rote learning leaves more than potential for understanding, connection to the real world, and appreciation for math in daily life on the plate. Rote learning often leaves a lot of its learners behind.
Lap It Up or Be Lapped
In much of the world, courses race through math concepts at top speed. This is understandable: regional, national, and global mathematical standards have ever-growing standardized lists of concepts that should be mastered in order to compare the performance of one particular group of students to another.
This causes an interesting dynamic of quantity over quality in the classroom: teachers must emphasize getting through a certain number of topics and hope the students keep up.
As a result, math is often taught abstractly, with mechanics and symbols and numbers and letters being the only experience with math many students have. Teachers are tasked with connecting math to the real world, but this is seen almost as superfluous to knowing the mechanics. Students learn for the sake of learning—or, more accurately, memorize for the sake of learning, without understanding why. Those that do not have a naturally abstract mind will have trouble following and executing, while those that can pick up mechanics are at an advantage. This is why we see students label themselves into two groups: “good at math” and “bad at math.”
The problem with rote or mechanical training is that it takes independent decision making out of mathematical discovery. Students are given a problem and a set of steps to solve it and taught to look for identical circumstances in which to put their memorized steps into practice.
This is different from the way that math was first developed. People started with real world problems that they needed solutions for. They discovered universal mathematical truths as they came up against dilemmas, and used abstract language—numbers and symbols—as a way to efficiently represent and extrapolate these truths.
Abstract math language evolved as an efficient way to represent real-world needs for math. In short, it was an eventual destination on the journey of mathematical inquiry. With rote learning, this “destination” is used as a starting point. Students rehearse answers without any real investment in the problem: they are spoon fed, when they don’t even know they are hungry.
The Link Between Independence and Discovery Math is about discovery. It’s about representing the world through awe-inspiring universal truths and finding information that we need to make decisions. It’s about solving problems one dilemma at a time, using arithmetic tools at our disposal to learn and create. Taking an intuitive approach to math enables us to make decisions along the way that we think will drive us closer to the answer we’re looking for.
At Mentorhood Math, time is on our side. We with little mathematicians to understand the simple problems they face in class. We make sure that they are involved every step of the way in determining how a problem should be solved, and we don’t leave students behind.
Students who have studied at Mentorhood Math are gifted with understanding on a very foundational level, so that they are not only set up to pick up more advanced concepts, but they also understand the why they are doing what they are doing. We are equipping them with tools and growing their confidence that they can apply the right tool for the job.
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A Review: VGCC
Last week, we looked at how one method we favour at Mentorhood Math can help students move from understanding concepts with concrete representation to advancing into the abstract world of symbols and numbers we call mathematics.
The method we highlighted was VGCC: Visualization, Grouping, Connection, Calculation.
Using this method, we might model two-digit addition and subtraction questions on the fingers and hands of a few student participants. The advantage of using hands is that students can immediately Visualize how larger numbers are broken up into smaller parts using base five or base ten. (For example, students can see visually on the hands of their colleagues that eight is five plus three, and thirteen is five plus five plus three.)
This helps students move to the next stage of the method, known as Grouping. By concretely understanding how larger numbers can be broken down, students can group together easier parts of the numbers, and finally make the Connection between all these number groups to arrive at a final answer.
We then ask students to repeat this representation using numbers and symbols. Little do they know that they are building a three-dimensional understanding behind every two-dimensional number they are penciling onto their page.
Highlighting IMMC: Visualizing Numerical Interactions
This type of approach is known as the CPA approach: walking students through advancing levels of understanding math problems Concretely (using objects and manipulatives), Pictorially (using visuals), and Abstractly (using numbers and symbols), developing a robust intuition behind the digits we see on the surface of math. This method was first developed in Singapore and made popular in a number of countries known for their high-performing math students.
IMMC is another CPA method developed by Mentorhood Math, which focuses on helping students visualize how numbers interact with and relate to one another using comparisons.
IMMC stands for Imagery, More or Fewer, Model Fill-in, and Calculation. Under IMMC, a teacher will present a word problem in subtraction or addition and model out the components of the problem through Imagery, using visual rectangles. The key component of the method is the next step: More or Fewer. Students are asked to analyze the text for the key words that indicate what kind of operation is being performed. By doing this analysis, students are already building a connection between the model they are seeing, the text in the problem question, and the math that will need to be carried out.
When students answer, the next step is to fill in the Model, or label the rectangles and their differences in size, with the numbers from the question. This type of modelling can help students tackle simple problems, such as adding up how many apples Sally and her friend have all together. But it possesses a ton of power with all kinds of comparisons, and can be used to solve problems such as figuring out how much money James would have spent if he started with twice as much money as Casey, bought a shirt that was half as expensive as the one she bought, made a few other purchasing decisions, and had three times as much money left over.
Students have now done the legwork of breaking down the problem, and have a head start in seeing the problem come together by identifying key interactions and relationships between the numbers. Students can now advance to analyzing what Calculations need to be done, and in which order if there are several.
Students are embracing a highly intuitive way to solve these kinds of problems, encouraging the mental visuals that many of us might eventually stumble on naturally. They will be able to build on that intuition to mentally model how numbers interact with each other when given more complex sets of instructions.
We’re Vocal About the Hidden Benefits
As hinted, the golden nugget of this method lies in the second step: More or Fewer. Students are asked to analyze the text for clues that indicate which operations are being performed. We’ve already talked about how models can help students understand relationships between numbers visually, but by starting off in a word based problem, we are helping students understand numerical operations linguistically.
We talk in comparisons all the time: ratios, portions, percentages, doubles, halves, triples. We don’t always take the leap in our minds from vague comparisons to concrete math. Using IMMC and methods like it, we are helping students recognize math in everyday language.
Not only this, but especially in English, we represent a lot of operational concepts in many different ways. Just look at all the ways we can say “subtract” in English: minus, less than, take away, fewer…not to mention less-than-obvious ways that only make sense in context. (Three penguins walked away. How many are left?)
For students for whom English is already a fluent tongue, we are teaching them to think about their world mathematically: to take an instant leap between conversation and calculation, dialogue and operations. For students who are learning English, we are giving them an expanded, flexible vocabulary, enabling them to grasp the nuances of math spoken about in English.
Intuition, Intuition, Intuition
For a lot of the world, math is learned through repetition, repetition, repetition. At Mentorhood Canada, we believe in intuition, intuition, intuition. We believe in building a solid understanding of basic mathematical concepts that more advanced concepts can be built on later, saving students from having to be re-taught earlier concepts.
At Mentorhood, learning math can really be as simple as 1, 2, 3…so long as it’s spelled C, P, A.
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