Stephen Hawking is quoted as saying, “Intelligence is the ability to adapt to change.” And we live in a rapidly changing world, don’t we? The technological advances of the past fifty years have far surpassed those from the previous five hundred. From industry and machinery, to computing and electronics, to artificial intelligence, humans continuously carve out for themselves new spaces for societies to flourish.
One thriving group of these new spaces are the STEM industries―science, technology, engineering, and mathematics. They’ve been getting a lot of attention, especially over the past couple of decades, as both markers of human progress and opportunities to further flourish.
The industrial revolution allowed blend tools that we were familiar with―gears, wheels, and the like―and blend them with sources of power and energy, such as coal and oil, allowing us to tackle problems mechanically, much more efficiently than ever before possible. As electricity became more readily available, this enabled mechanics to become embrace automation, and the internet and computing revolution opened up a whole new set of technologically-assisted opportunities. Now, hardly a daily task is accomplished without technological support of some capacity. As AI becomes more sophisticated, we have the increasing ability to automate solutions, further closing the gap between human thought and human accomplishment.
This automation, however, is far from autonomous, and is a far cry from the creative problem solving happening in the minds of the people who use it. Behind every process that occurs at the touch of a button, a complex stream of rational code guides the automation through a multitude of variables and decisions. Replacing traditional gears and wheels, this code represents the metaphorical inner workings of the machine―and all of this has to be developed by highly educated, skilled, and specialized individuals. Coding, therefore, the hidden in the “T” in STEM, underpins all three other areas, and indeed, all kinds of aspects of modern life.
Coding is the process of predicting and accounting for any scenario that you wish to design the machine to handle, and engineering the response that the machine should have. This complex set of instructions needs to be written in a language that the machine can process, or understand. Smart machines require smarter code.
But these skills do not come at a one-stop shop. Technology continues to evolve in sophistication, driven by the discontent of our race with the status quo. Humans imagine and expect more from their devices, seeking a seamless relationship between their thoughts and their tools. This requires skill to navigate, alongside a hungry mind with a strong ability to adapt.
S - Coding in Science
The sciences rely on the computational power of technology for far more complex problems than we ever would have been able to tackle before. Modern scientific research in anything from geological study to extra-planetary analysis to pharmaceutical development to climate mapping and prediction requires AI support to organizing vast amounts of data into a format that enables drawing inferences and conclusions. Sciences that run physical experiments, such as those in chemical or food safety, require precision robotics to manipulate minuscule variables. Beyond the lab, science-centric practices such as conducting eye surgeries and manufacturing rely on precisely programmed robotics to carry out exacting results.
Behind all these databases and calculators and mechanics, computer code has to be developed to allow for the predictable and critical results these practices depend on. This requires minds that are not only specialized in the various niches of scientific study, but ones that are prepared to translate their requirements into computer language.
E - Coding in Engineering
Engineering is the practice of taking complex scientific knowledge and turning it into actionable solutions to real-world problems. This could take the form of anything from civil engineering and infrastructure construction, to machine design, to electrical mapping, to computer network management and more. Engineering requires precision; imagine the devastation caused by a miscalculation in the load bearing abilities of 14,000 tons of steel. Or imagine a key variable in municipal electrical power being overlooked, resulting in outages, shortages, or dangerous electrical overloads.
Engineering requires the wealth of lessons we have collectively learned in these fields be accounted for. And just what kind of system could handle such a wealth of information? You guessed it: modern engineering is absolutely dependent on computing. Engineering management requires algorithmic technology to process past history and output exacting calculations with the goal of optimizing both efficiency and safety.
The Unofficial “a” - Coding in Artistry
Did you know that AI and computing is making an appearance on the world stage of creative arts and design? Computing made its debuts in the art space through skill-support and time-saving tools. Examples of these include graphic editing programs with built in templates and settings, and cameras that automatically optimized for photographing specific environments. These freed amateurs and semi-professionals from the burden of understanding all the nuances of ISOs, exposure, colour balancing, and layering, while still making decent results possible. As this kind of support gets more sophisticated, it can be hard sometimes to even tell the difference between professional-grade and computer-assisted designs. Technology is eroding the technical skills gap between amateurs and professionals in many forms of art.
Now, there is even experimentation with voice-dictated created art creation, which would in theory even eliminate the hurdle of software mastery. Some groups are even experimenting with not just AI-assisted art, but AI created art, in which computers (attempt to) write stories, paint paintings, and generate graphic designs to rival the creations of humans.
Accomplishing believable results requires feeding massive collections of information to the AI, along with sophisticated coding that allows software to sift through and choose what bits of information are relevant. Combine this with the need to refine such algorithms for biases, taste, and continued evolution in human imagination and, to be frank, you create a need for computer programmers who are really good at their jobs.
M – Coding and Mathematics
So, what does all this have to do with mathematics? Mentorhood is a math-focused organization, is it not?
Well, for one thing, pure mathematics as dependent on computing support as any other modern scientific industry―crunching complicated calculations takes much more computational power than a human can muster. But even more generally, the practice of coding itself requires a mathematically-wired mind that is capable of handling detail, consistency, logic, and structure, as well as an advanced level of comfort with algorithmic calculations and statistics. Coding is not just about writing down thoughts in “computer-speak”. It’s about developing complicated formulas that account for myriad variables and create logical outcomes. Moreover, programmers are often required to operate different kinds of software to create and test code, and a brain comfortable with logic will have an easier time picking up new applications by thinking the way a computer does.
All of our teaching at Mentorhood emphasizes this sort of logical framework that, really, underpins our whole world. It’s important to us to instruct our students in visualizing mathematical and logical concepts because this develops their logical intuition. Strong roots in logical consistency and mathematical thinking sets an individual up for success regardless of the industry they choose to pursue a career in, and especially if they find themselves interested in one of the STEM professions.
Dealing with coding specifically, however, this year we are launching a new initiative. We’ll be starting camps specifically geared towards coding and computer science. We’re excited to be able to show students how their studies of math can translate into real, in-demand skills that will continue to evolve as human imagination continues to push the boundaries of technology. We’ll be announcing more details on these specific educational opportunities soon.
Coding Our Future
Society used to function purely on muscle―we worked with what we could build with our hands. Our limits were blasted open through the invention of machinery, expanding our world to include what we could engineer. Now, technology and computing is the muscle of society. We create the parameters of our world based on what we can program.
Coding and computer science continue to increase in importance in our world. We’re preparing our students to take advantage of these opportunities, and we wouldn’t be surprised to see many them coding not only their own futures, but the world around all of us one day.
We say, go for it. Full STEaM ahead.
Will Singapore Math be out of reach for my child, who’s only ever participated in mainstream instruction before? How does it compare?
At Mentorhood, the Singapore math techniques complement lessons that follow standard curriculum expectations for each grade level. While there is a learning curve associated with visualizing concepts in new ways, the goal of Singapore math is not to teach advanced mathematics. Rather, it is designed to equip students with advanced tools in understanding and problem solving, enabling them to face more advanced mathematical concepts later with less of a steep curve.
As with anything new, student new to Mentorhood and the way we employ Singapore math may experience a bit of a learning curve. While new models for representing and manipulating mathematical information may need to be explained, this is no different than learning a new concept in any setting. Most of our techniques are relatively intuitive, and can be a virtually non-existent hurdle depending on the child’s experience.
The real learning curve will come in in how students understand why certain methods are being used, and learning to apply this understanding to new problems. And this is where the very power of Singapore math comes into play. While students are learning concepts on pace with counterparts at their grade level, they are also learning to approach them in a much deeper way. They may not be aware that in learning to solve one type of problem, they are building a multi-faceted understanding of how to solve problems with similar qualities. They are not so much climbing faster up the staircase in conceptual advancement, but rather are taking their intuitive foundations and digging them down deeper.
We celebrate the learning curve, because the learning curve is just that―it’s learning. Without realizing it, our students at Mentorhood are bending that curve to support deeper understanding and to build more capacity for understanding later. We’re not teaching them new math; we’re training their brains in a new way to look at math, which will serve them later when more complexity is demanded of them.
Singapore math also emphasizes individual mastery. It’s less about diffusing information, like scattering seeds far above a field and hoping that at least some of it will be taken in and germinated. Mentorhood’s model is more like gardening: we thrive on small classes because they give us the chance to plant and fertilize each concept and provide special attention to the flower beds that may need it. We recognize how understanding creates more understanding, and our passionate teachers strive to care for each step along the way. And for those families that desire even more intimate attention, we offer plenty of one-on-one tutoring, providing a space for flexible practice.
And not the least importantly, we make math fun! We certainly have a blast working with our students via accessible, engaging materials, and we pair this with games, visual problem solving methods, and plenty of encouragement that makes our environment rich with satisfying “Aha!” moments.
While the Singapore math methods and emphases might be new for you, we think this is a great thing! Any learning curve our students experience is there because their brains are working at deeper levels than they’ve been used to. This will prepare them for even smoother learning experiences later, as they grow accustomed to reaching into the deeper levels of their intuitions.
And as students strengthen their conceptual abilities working intimately with instructors, we believe that this provides them with a special opportunity to grow in a way many of their counterparts don’t have the chance to.
Singapore math is for all. Students who are mathematically inclined and already delight in math’s logic love our teaching methods. They can be challenged to grow and apply their knowledge broadly. Students who have previously struggled in math often comment on their newfound understanding and comfort level with figures. And teachers can attest to the universal sense of joy in their classrooms.
So, have you missed the boat? Is it too late to start? Never. There’s always time to cultivate a bright understanding of math, to build the foundation blocks that make the mathematical realities of our world more intuitively relatable. We’ve got a spot for you, and we hope you’ll join us.
When was the last time your instinct answered something before your brain did? Maybe you tossed in the right pinch of spices to a recipe without looking. Maybe you instinctively knew how to spell something without looking it up. Or maybe you brought your groceries to the checkout, and your mind estimated within a few dollars how much your bill was going to be (actually…maybe we’re all under estimating a little bit these days).
Intuition can be a powerful thing. We might know people who seem to be naturally gifted to be more intuitive than others. That can be intimidating. And while to some degree intuition may be a gift, intuition is also both teachable and learnable. Can intuition be taught in a subject as laden with rules and memorization as math?
After all, how can math be intuitive? You can’t instinctively know that the diameter of a circle is equivalent to a ratio slightly greater than 3.14 without calculating it, and once you’ve calculated it, you won’t know next time without committing it to memory. It would be difficult for the first person to come up with the long division technique to simply know how to do it intuitively. It requires figuring. And following figuring, it requires memorization.
Let’s take a look at the definition of intuition. Merriam-Websters defines it as:
a. the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference
b. immediate apprehension or cognition
The first definition seems like it’s talking about knowledge pulled out of thin air. You may not be able to determine the intercepts of a parabola this way, but perhaps this type of intuition helped the great mathematicians of old bank on an intuitive belief that there was a mathematical order to the world.
But take a look at the second definition: “immediate apprehension or cognition.” This is the ability for a new truth to be easily plugged in to your existing framework of understanding. It’s the ability to pick up new knowledge quickly, because to some degree, it already fits with your expectations—even if you were previously unaware of them.
One way that we start to teach this is through problem solving strategies on basic questions that can be used for more advanced questions later. We rehearse visual modelling techniques so that when students approach a type of question they have never seen before, their brains already have the tools to start mentally mapping the scenario. Familiar techniques can also be used by instructors when explaining how to solve a question in a way that students don’t have to struggle to follow.
We also emphasize the use of reason, which is very closely related to intuition. We practice breaking down similar questions in several different ways rather than sticking exclusively to one “correct” approach. This trains students not only explore a problem from different angles, but to look for the most efficient way to solve a question based on the variable parameters of the question.
We encourage estimation when approaching a question, because making a reasonable guess sharpens a student’s eye for picking out unreasonable answers. If their mind is already calculating an approximate answer, this can act as a bit of a self-guide as they determine if they are on the right track with a question. They won’t need to wait until they get a final answer to check if they were correct; the better they are at estimating reasonableness, the more they can check in with their methods along the way.
Our strategies develop not just an understanding of a concept, but a three-dimensional understanding; rather than the memorization of problem solving techniques, we prioritize transferable application. All this can help build towards that first definition of intuition:
a. the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference
In mathematics, new knowledge is never created―only discovered. Math is numerical archaeology, digging deeper into the truths that are buried in the universe to discover the architecture of the way things are. New knowledge cannot be intuited out of thin air, but the most advanced mathematical minds can use this kind of intuition to deduce mathematical truths that were previously untaught.
While the powers of mathematical deduction may be a rare gifting of the mathematicians among us, we can steer our learners towards the same patterns of thinking that turn mathematics into more low-hanging fruit―ideas that aren’t too much of a leap from where they are currently standing. The more our students learn reasonableness, problem solving, and flexibility in their approaches, the stronger their framework of understanding becomes―the stronger their framework, the stronger their powers of cognition, intuition, and deduction.
And who knows―the more confident they become in their intuitions, the more that thinking might just spill over into other areas of their lives. They may even be looking over your shoulder one day and telling you how much spice to put in dinner.
Do you know your learning style?
Chances are you can harken back to a time in grade school when someone handed you a quiz: Which Learning Style Are You? You may have found out you’re a visual learner, or a kinesthetic one. You might be prefer to intake information auditorily, or recite through writing and reading.
Maybe you took the test and you were convinced that you were through and through a visual learner. You understood math best with diagrams and history best with pictures. But then, the next year, you stepped into your first science lab, and all of the sudden, your comprehension shot through the roof as you began to engage your hands in duplicating the experiments. The diagrams on the board seemed confusing at best, compared to the kinesthetic experience you were getting with the equipment. Does that make you a kinesthetic learner, then? And if that’s the case, why did you find it so easy to listen to the sports game from the other room and formulate a mental arena that mimicked exactly what you were hearing auditorily?
One thing you might remember from your test is that you probably didn’t score zero for any of the learning styles. Chances are you had a style, or perhaps two, that seemed to be dominant, along with points in all the other categories. It’s even possible that you experienced receiving different results for your dominant style over different test-taking instances.
What a mess! Is there no hope for a clear-cut answer as to whether it would be best for me to watch through my YouTube list with no sound, or to listen behind a wall?
It’s important to point out that with self-diagnostic tests always carry more inherent bias and inaccuracy than blind scientific trials. Even with that aside, however, it turns out that the criticism of the “messy science” of learning styles is shared by some in the science community as well.
The criticism comes from the lack of standardized definitions and testing for learning styles. A quick Google search will reveal a few the discrepancies. There are tests for four learning styles, tests for seven, tests for categories unique to adults, and just about every other categorization you can think of. No wonder scientists don’t agree, and can’t agree to disagree.
Perhaps, however, the untidiness of learning style sorting is actually, in some senses, a true reflection of the learning style phenomenon. Perhaps in some senses, we all experience learning in such uniquely individual ways that it’s hard to categorize. At the same time, maybe we learn more similarly than we originally thought when we first looked at our results in grade school: “You’re a Visual Learner!”
“But,” you say, “I promise! I am a visual learner! I learn much better with pictures and diagrams, or even better, with a 3D model. In fact, if I’m watching a video and seeing the pictures and objects move, that’s even better!”
And here’s what we run into the real magic about learning styles. It turns out that human brains learn relatively similarly to one another, in that we take aspects of all kinds of learning methods into account when we intake and process information. Therefore, the best kind of learning integrates them all.
If you think of a video, it might start with a narrator, explaining a general concept. Maybe this is proceeded by a few interviews with experts as they shed a bit more light on the idea. Then a diagram is shown on the screen, and as different parts move and interact, the narrator explains what’s going on. We talk to another expert, and then eventually, we get to see a genuine example of the situation in the real world. The video is concluded by the narrator drawing implications from what we just witnessed.
Think of all the different learning styles that are at play in that one video. You have a consistent auditory output. The animated diagrams would probably be much more difficult to understand if they were instead a series of still images with no one to explain them. You have different experts talking, so you’re employing a bit of social intelligence, watching for body language cues and inflections. Simulations and real-world visuals give your brain the sensation of actually being there, much like how dreams, movies, and novels can make us feel kinesthetically activated.
You might find you get a different quality of understanding from certain parts of a diverse learning experience like this, which could depend on your personal cognitive inclination, the quality of the content, the environment you’re in, or how you’re used to taking in information. There is something to be said for learning preference. The reality is, most of us are engaging all parts of our brain at all times and it’s not always easy to create clear-cut distinctions between learning styles. Perhaps this lends itself to the “creative” nature of learning style categorization. It’s not a perfect science, but an experiential one.
Should math cater to learning styles? For sure it should. But that’s not to say that incorporating visual elements and auditory explanations are going to specifically cater to different students. Layered approaches, such as our video example, offer a well-rounded approach to a concept that consistently engages each student as they use all kinds of strategies to internalize and understand the information. Tackling multiple learning styles at once also allows teachers to cast wide nets and continuously capture the attention of several unique brains in one room.
This is the approach we take at Mentorhood. Our materials are chock full of visuals, text explanations, loads of practice, and a caring teacher to animate them all. Our mixed approaches to learning are powerful because they enable students to develop a more thorough and nuanced intuition and understanding, enabling a three-dimensional appreciation for how mathematics flows into every aspect of daily life. We want our students to see mathematical concepts from all angles―whether acute or obtuse, which not only heightens their understanding but multiplies its depth as well.
Romanelli, Frank et al. “Learning styles: a review of theory, application, and best practices.” American journal of pharmaceutical education vol. 73,1 (2009): 9. doi:10.5688/aj730109
Fournier, Mikelya. “7 Major Learning Styles and the 1 Big Mistake Everyone Makes.” LearnDash, 14 Jan. 2020, https://www.learndash.com/7-major-learning-styles-which-one-is-you/.
Do you remember the students in your class who seemed to live for math? They just seemed to be gifted. They did extra problems, and seemed to be genuinely excited for a new lesson or by figuring out a solution or a connection to an old one.
And then there were many of the rest of us for whom math was a required subject that didn’t feel all that relevant to our regular lives. “But honestly, Mrs. Webster. When am I going to need to know the difference between a rhombus and a parallelogram and a trapezoid?”
But if you take a look at the people for whom math seems the most natural, typically, they genuinely seem to be enjoying themselves. Perhaps they enjoy the practice of fitting a new math concept into an evolving database of mathematical truths that frame their world. Perhaps they like the challenge of memory recall. Maybe they like to solve puzzles and find it extremely satisfying to getting to the right answer.
Sometimes, it’s the application of the math that seems the most compelling to them. Engineers or physicists-to-be may enjoy thinking of the world geometrically or algorithmically because it allows them to exploit the science that makes their investigations possible.
There is at least one thing that all of these perspectives have in common: there is a thread of true enjoyment. This likely comes from having a task with a clear reward proposition at the end. This makes enjoyment possible and allows the challenge to be fun and/or meaningful.
Kids who do well in math are usually the ones for whom math is fun. The right balance of challenge and possibility, mixed with a dash of meaningfulness can make math rewarding, and therefore engaging. Think of the things you most enjoy doing―not just when you’re relaxing, but when you really feel engaged, in your element, the most satisfied. Chances are you have a similar balance of these three: challenge, possibility, and meaning.
At Mentorhood, one way we foster this is by engaging students in math through games. Games have built in reward systems that learners can care about. Solve a question, get a point. Roll the dice, hope for a good result. When there is a reward at hand, such as a bit of friendly competition, a race, or a personal challenge, it gives students the opportunity to invest. Investment gives the activity meaning, and this in turn makes it engaging.
We have full-year clubs that are dedicated to playing through the mathematical concepts of each grade level through games―and certainly, these are always popular and a blast. We also keep in mind the concept of gamification when we are designing our full-year instructional programs. Our class materials are varied, sometimes including mazes, diagrams, games, matching, and pictures. Moreover, it’s easy to spot the increase in challenge from the beginning of a lesson to the end of it, giving students the feeling of “leveling up.” And our small classes facilitate an environment where the teacher is engaged with each student, allowing each student to feel like the material is possible to engage with.
Math should be fun. For the people who are the best in math, it’s usually the people who find the most fun in it. They may find a level of fascination with the immutability and the complexity of mathematics, or they may find a valuable end in it. To put it simply: math matters to them.
We make our classes as engaging and fun as possible to make it as easy as possible for our students to find something that matters to them in the math. This can bridge the gap between math that feels distant or intimidating to something that is a part of their world, something that they can take on and master little by little.
It’s rewarding for us as educators to see students really get something out of their math lessons, whether it’s the pure joy of winning a game, the thrill of a revelatory moment, or simply the sheer pride we see in their smiles when they get a question they didn’t think they could.
For some of them, we may be opening the door for them to start to love the beauty of math itself. For others, we may be one step in a journey of self-discovery and a trust in themselves and their own abilities, and a willingness to engage with their world. At Mentorhood, we believe that math matters for our students, and that fun matters, too, because ultimately, it’s them―our students―that matter the most.