AI companies know that children are the future - of their business model. The industry doesn't hide their attempts to hook the youth on their products through well-timed promotional offers, discounts, and referral programs. "Here to help you through finals," OpenAI said during a giveaway of ChatGPT Plus to college students. Students get free yearlong access to Google's and Perplexity's pricey AI products. Perplexity even pays referrers $20 for each US student that it gets to download its AI browser Comet.
Popularity of AI tools amongteens is astronomical. Once the product makes its way through the education system, it's the teachers and s …
Unmesh Joshi finds LLMs to be a useful tool, but
explains why their help becomes illusory if we use them to shortcut the
learning loop that's an essential part of our professional practice.
I’ve always felt like the math curriculum moves too fast. I teach topic A, and I know students could use some more practice to consolidate their understanding. But we have to move on to topic B. And worse, topic B builds on topic A, so that shaky understanding of topic A starts a domino effect of confusion.
Practice is important. But the best way to practice is in multiple short chunks, spaced out over time, with plenty of chances to check for understanding and give feedback. With the rush to get through everything, I often feel like I don’t have a choice but to give students lots of practice all at once and then move on to the next thing.
Spending a full lesson on a topic can feel like a slog. Sometimes I realize students don’t have the foundation I hoped they did, or I sequence examples poorly, but I have to push through to stay on track.
Math is sequential. New topics build on earlier ones. But math also isn’t one giant staircase. Two-step equations build on earlier ideas of one-step equations and inverse operations, but they don’t have much to do with geometry or proportions. Percent increases build on a few different ideas, but they aren’t connected to operations with negative numbers. Still, I end up in this quagmire pushing through my equations unit so there’s enough time to teach geometry.
The Multi-Stranded Math Curriculum
Here’s a rough outline of a typical 7th grade math curriculum.
This isn’t totally accurate. The units aren’t all the same size, and you might break things up differently, but this is a reasonable first approximation.
There are six big ideas, and I teach these big ideas in separate units, one after the other. That basic approach has led to a lot of the problems I described above.
Here’s what I’m doing this year instead:
Again, the sizes aren’t precise, but you get the basic idea.
Four of the six units are sequenced a bit like before, though now there’s more time for each unit. The other two units stretch in parallel through the entire year. I call these strands, so in this case there are three main strands: the four sequential units, rational numbers, and expressions and equations.
Those bottom two units are the most important and challenging topics I teach. Rational numbers involves adding, subtracting, multiplying, and dividing positive and negative whole numbers, fractions, and decimals. The heart of this unit is working with negative numbers, but students need plenty of review with fractions and decimals as well.
Expressions and equations refers to both finding equivalent expressions, for instance by combining like terms or expanding brackets, and solving two-step equations and inequalities, building on work with one-step equations the previous year.
These are both tough topics. Stretching them across the entire year helps me break the concepts down into small, manageable chunks, and give students time to practice each chunk before moving on.
Here’s an example. The key representation of negative numbers is a number line. We use other representations as well, but number lines are the linchpin. So the very start of the rational numbers strand is just putting positive and negative numbers on the number line. We start with integers, and gradually get into decimals, fractions, and mixed numbers. And that goes pretty slowly, with lots of quick rounds of practice so students get fluent thinking about and moving around on the negative part of the number line. We never spend an hour doing number line problems, but a few minutes a day adds up and gives students a strong foundation — while we’re also working on other topics. All that pays off in a big way when we get to adding and subtracting with negative numbers.
Here’s another example. The biggest challenge with solving two-step equations in 7th grade is that students are supposed to solve two-step equations with positive and negative fractions and decimals. But we start with simple, straightforward equations like 2x + 1 = 11 and 10x + 3 = 43. Get students comfortable with those. Then bring in some subtraction. 2x - 1 = 11, 10x - 3 = 37, stuff like that. Then let’s spend some time practicing one-step equations with negatives, problems like 2x = -10, -10x = 40. Then let’s bring in two-step equations with negative numbers and gradually make those harder. Eventually we work in larger numbers, fractions, decimals. I’m oversimplifying a huge part of my of teaching into a paragraph but hopefully you get the idea. There’s no time in a normal curriculum to teach one small chunk, check for understanding, practice, and consolidate before moving on because it’s all go, go, go. But with multiple strands, there’s more time — I can spend a few days doing a few problems of practice and checking for understanding on the equations strand while spending most of the class on the other strands.
One key of the multi-stranded approach is that not every class is divided 1/3, 1/3, 1/3 between the three strands. Sometimes we spend a large majority of a class on one topic. Other classes are split evenly three ways. Most often there’s one “new idea” each class, and then a few quick chunks of practice from the other strands. I didn’t add it to the image above but we also do a bunch of fact fluency practice, so that’s like an extra parallel strand that runs through most of the year.
The Advantages
Here are a few things I like about this approach:
I have more time to review prerequisite skills. Sure, students were supposed to learn them last year, but students forget. Stretching out each unit creates more time to review and practice tough skills from previous years before they’re necessary.
Students have time to consolidate their learning. If I teach skill A, and then skill B which builds on skill A, I can take my time, give plenty of practice with skill A, and check for understanding before moving on. I just prioritize the other strands for a few days while mixing in short chunks of practice with a skill until students are ready for what’s next.
I can do a much better job breaking skills down into small steps. Having a single objective for each class creates artificial divisions in the curriculum. Some ideas take a class to get through, but in other places one class actually represents a few different smaller ideas. With this approach, it’s much easier to help students learn one step at a time.
Teaching multiple strands at once lends itself to spaced and interleaved practice. Practice is important, and the best way to practice is in lots of small chunks, spaced out, interleaving different skills as students become more confident. A typical curriculum doesn’t do this very well, but the multiple strands approach spaces and interleaves by default.
I teach lots of lessons that don’t go well. In a typical lesson, I push through and try my best. If that activity is only one strand of several in a lesson, it’s easy to punt. I can say hey, let’s come back to this tomorrow, and focus on the other strands for the rest of class. Then, when students are gone, I can regroup and plan a new approach for the next day.
Here’s a comparison I think is helpful. An English teacher would never create artificial divisions where unit one focuses only on vocabulary, and then doesn’t do any work on vocabulary the rest of the year, and then unit two focuses on close reading, and then no close reading for the rest of the year, and then unit three on fluency and prosody and then moves on again. Instead, those key skills are woven throughout the year. That doesn’t mean everything gets stretched out for an entire year. If you’re going to read The Crucible it doesn’t need to take the whole year, and it’s reasonable to have a unit that focuses on argumentative writing or short stories. But the most important and most challenging skills come up over and over again throughout the entire year.
Some Practical Tips
Let’s say you want to try this. Here are a few pieces of advice.
This approach to curriculum does not work if you are set on having a single objective for each lesson. If you have a principal who loves to walk in and nag you about writing your daily objective on the board...well, I’m sorry you have to deal with that. In this approach there are still objectives. I think really hard about what I want students to learn each class. But most classes will have multiple objectives, and they look a bit different. Some days the objective is to learn something new, but others it’s a quick check for understanding to inform class tomorrow, or some practice to help solidify an idea before moving on to the next thing.
Find ways to break single lessons down into smaller pieces. Take an objective and turn it into two or three smaller objectives, or introduce students to a topic and then break the practice up into a few chunks to do over multiple days. There are lots of ways to do this, but it takes practice and effort and will need to be adapted to your curriculum.
It’s helpful to have a few teaching techniques that don’t require much prep to give students some quick practice and check for understanding. I wrote about this last week, and using mini whiteboards and five-question “stop and jots” make it easier for me to give students quick practice on one strand without spending forever prepping materials.
You don’t need to go all in all at once. I started an early version of this approach because I struggled teaching two-step equations when students were still shaky with one-step equations. I would try and do some quick one-step equations review at the start of the equations unit but it always felt rushed and students weren’t solid with those skills. So I started stretching that time out, and would spend a month or two gradually working through different types of one-step equations in parallel with the previous unit. Then I started to stretch out two-step equations, focusing first on equations with whole numbers, then getting into negatives, fractions, and decimals in parallel with the next unit on geometry. Over time, I moved closer to what I’m doing now with multiple strands running through the entire year, but I didn’t go for it all at once.
Start with one topic, and pick something that’s either especially important for future years or especially hard to teach. If I were teaching 5th grade I might make a year-long strand for multi-digit multiplication and division. We would take our time working on multiplication facts at the start of the year, then gradually work into simple multiplications without carrying like 34×2 and 12×3, then bring in carrying and slowly increase the number of digits. Then we’d work on division facts, and gradually introduce multi-digit division with simple questions like 84÷2 before getting into tougher and tougher problems. If I were teaching 8th grade I might make a year-long strand for systems of equations, starting with equation solving and very simple substitutions and gradually getting harder and working through the different strategies for systems.
Direct Instruction
I learned about this idea from the book Direct Instruction: A Practitioner’s Handbook. I’ve written about the general idea previously, though that post describes a very different version of teaching multiple strands than what I’m doing this year.
A lot of people have an allergy to Direct Instruction. I would challenge you: if you don’t think of yourself as a direct instruction teacher but this idea appeals to you, you should read the book. I’m not an advocate for Direct Instruction, but I’ve still learned a ton from the program. It is very different from anything in the contemporary curriculum world, and whether or not you become a true believer, I think all teachers should understand Direct Instruction and see what they can learn from it. If you’re curious for a bit more, I wrote about what I like and don’t like about Direct Instruction here.
In the US, in the last decade, there has been a flood of new curricula that all feel the same. They’re all vaguely constructivist. They don’t emphasize practice. They are obsessed with rigor, so there isn’t much review and the difficulty of each topic increases quickly. They focus on big ideas and don’t break content into small chunks.
What I’m describing in this post feels like the opposite of the current trend in math curricula. I have felt incredibly frustrated with the two curricula my school has used in the last five years. This multi-stranded approach has felt like the antidote to all the things I hate about the curricula that are popular in the US today. If you have similar frustrations, maybe this approach is for you.
Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat. The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in…
Biology can be hard to intuit, in part because it operates across vastly different scales, from single atoms all the way up to entire ecosystems. Students of biology therefore often first meet its agents and mechanisms through metaphors: molecules are charged balls connected by sticks! evolution designs organisms to maximize their fitness! mitochondria are the powerhouses of the cell! While metaphors give us qualitative handles to grasp, they often oversimplify complex ideas.
This is because most metaphors fail to address specifics — especially regarding numbers. Consider another common bio-metaphor: DNA is the blueprint of the cell. That’s useful for conceptual understanding, but how big is this blueprint? Is it as big as a novel or an encyclopedia? How much space does it take up? It’s possible to look up or calculate the answers to these questions; the human genome is 6.2 billion base pairs, which takes up about 10 cubic microns.1 But how big is that compared to the total volume of a cell? Is it most of it or just a tiny fraction?
To answer questions like these, you need more quantitative metaphors. Whereas a standard metaphor says that mitochondria are the powerhouses of the cell, a quantitative metaphor says how big a battery a mitochondrion would be, say, if a cell were a toaster. If qualitative metaphors are like containers, then quantitative metaphors are closer to yardsticks.
And these yardsticks are exceptionally useful down at the scales where biology operates. We all know cells are small. But so are proteins, nucleic acids, and water molecules. We often think of everything “small” as equally small, but that is not the case. Proteins are titanic, hulking machines compared to water molecules, and the mRNA that encodes a protein is orders of magnitude larger still!
To make the sizes and shapes of various biomolecules concrete, let’s imagine that each water molecule within a cell has been blown up to the size of a grain of sand.2 If this were the case, then …
Proteins come in a variety of shapes and sizes, but a typical protein would now be a knobby ball the size of a blueberry.3
The largest human protein, titin, would be a floppy spring 3-4 mm in diameter. Coiled loosely, it would be about as long as a golden retriever end-to-end but could stretch out to the height of a double-garage door.
An antibody would consist of three blobby arms, each roughly the dimension of a grain of basmati rice, connected to a common center by a short chain.
Proteins come in all kinds of shapes and sizes. ATP synthase is more than an order-of-magnitude larger than insulin! Credit: David Goodsell
DNA
A nucleotide would be a paper-thin oblong disk about the width of a poppy seed (1mm, also the width of three grains of sand).
To make double-stranded DNA, glue two stacks of nucleotides together side-to-side, then twist them so that they make a complete turn every ten bases. You should end up with a 2-stranded braid about as thick as a #2 pencil lead. Real nucleotides are extremely thin but have some space between them; for simplicity, imagine them as tightly-packed stacks of card stock. Short chains are stiff, but they start to get floppy around 5cm (2 inches).
A typical human chromosome would be a thread of double-stranded DNA about 100,000,000 bases long, which is just long enough to span the English Channel (34 km).4 In its natural context, it would usually be a snarled hairball about as long as a minivan (4-5 meters long and oblong), though thinner. During mitosis it would curl up to a length similar to that of a compact car.
A typical bacterial chromosome is a thread or circle of double-stranded DNA a couple million bases long, which would have a length comparable to the height of the tallest buildings. It is normally loosely coiled into lots of twisty bundles, like a headphone cable left loose in a bag; in this form, it would be a meter or two across.5
A messenger RNA is vastly larger than the protein it encodes. Credit: David Goodsell
Single-Celled Organisms
Human viruses come in a variety of shapes and sizes, but most are balls of protein covered in a lipid membrane studded in protruding proteins, between ping-pong ball and soccer ball in size.
E. coli (a fairly typical bacteria) is a capped cylinder one meter across and two meters long — about the size of a cow and weighing about two metric tons.
Single-celled eukaryotes would come in a variety of shapes, and range between cow and large town in size, with most between whale-size and city-block-size. Yeasts would tend toward the smaller end, with Saccharomyces cerevisiae being a sphere the size of a medium-weight moving van. The largest amoebae, on the other hand, would be several kilometers across.
Exceptionally deep writing about biology. Always free. Sign up today.
Human Cells and Organelles
A “typical” human cell is a fibroblast, which is the cell that makes up most of skin and connective tissue. This cell likes to attach itself to a surface and spread itself into an irregular shape, which would be about the dimensions of an 8x8 square block of king-bed, standard-sized hotel rooms.
The nucleus of such a fibroblast, which contains its DNA and transcriptional machinery, would be an oblong disk about the size of a 2x2 square block of hotel rooms. Remember those minivan-length human chromosomes? We need to pack 23 of these into the nucleus, or 46 if it’s about to divide. The chromosomes take up only a few percent of the nucleus’s total volume, loosely packed most of the time and, therefore, tending to expand to fill the space.
The mitochondria are coral-like webs of branching tubes reaching throughout the cell, with widths between that of a human and a cow.
Various organelles are attached to a dense spiderweb of fibers called the cytoskeleton. These come in a variety of thicknesses, lengths, and consistencies, but most are either:
actin fibers, which are tough threads that would be about as thick as a fat nail, or
microtubules — stiff, hollow, finger-width tubes.
The cell contains hundreds of lysosomes (organelles specialized for breaking down food, waste products, and damaged organelles), which would be blobby bags the size of a rat at the smaller end or house cat at the larger one.
The cell is encased by a lipid membrane about the thickness of a thin porcelain tile (~5mm). It is rough with proteins, and probably6 looks patchy, with many temporary “rafts” of different densities of lipids and proteins. A 2x2cm square of membrane (about the dimensions of a key from a desktop keyboard) contains around 2,000 lipids and 10-15 proteins.
Cells with nuclei stained in blue, mitochondria in green, and actin cytoskeleton in red. Credit: D. Burnette and J. Lippincott-Schwartz, NICHD.
Organs and Body Parts
A human hair is a towering cylinder of tightly-packed dead cells. Shape and thickness will vary by hair color and texture, but a fairly typical hair would be about as thick as a soccer pitch is long. At shoulder-length, the hair would extend about 250 kilometers, or long enough to cross California (or England) east to west.
Human blood vessels vary massively in scale. The smallest capillaries would be as thin as five meters (smaller in cross-section than a fibroblast!), just large enough to pass one or two inflatable-pool-sized red blood cells at a time. In contrast, the aorta, where the heart pumps into, would almost extend across Washington D.C., with surfaces a couple of kilometers thick.
A human eye would be an orb about 25 kilometers (15.5 miles) across, with a wall 0.5-1 kilometers thick. The cornea (the clear part at the front of the eye encompassing the pupil and iris) would be some 11 kilometers across. The retina (the “camera sensor” at the back of the eye) covers about three-quarters of the interior surface of the eye and would be made up of ten distinct layers of cells, each about 7-8 building stories tall.
A human brain, removed from the body and placed on the ground, would span half the surface area of Belgium and almost reach space.
Animals
Perhaps the smallest commonly-studied animal isC. elegans, a near-microscopic nematode worm consisting of exactly 959 cells.7 According to our metaphoric yardstick, this nematode would be a tube a kilometer long (about as long as the Golden Gate Bridge) and 40 meters (about 12 stories) high and wide.
A fruit fly would be a behemoth some 2-3 kilometers from mouthparts to abdomen, and 1-2 kilometers across.8 Each eye would be a rough hemisphere made up of thousands of whale-sized facets, and every other vertex between facets would sport a giant guard-spike some twenty meters long (as long as a semi truck).
The average human would be about 1,700 kilometers tall. The International Space Station would orbit somewhere around the knee.9 This human could cover the U.S. west coast with their stretched-out arms.
Manmade Things
Just to hammer home how compact biological entities are, let’s look at some human-made artifacts at the same scale:
A US penny laying flat on the ground would now be a disk a kilometer and a half tall (a bit shy of twice the height of the Burj Khalifa) and nineteen kilometers across (a reasonable day’s walk for an experienced hiker).
The ball from a fine-tipped ballpoint pen would form a sphere 400 meters tall — as tall as the Empire State Building. This 400-meter height is precise to within 250 millimeters (about 10 inches). It is not a perfect sphere, but it is surprisingly close — its manufacturing imperfections have an average height (or depth) of less than three centimeters (about an inch).
A transistor would form a rectangular block about two centimeters (¾ inches) across.10 An iPhone processor has 15-20 billion of these blocks, and is roughly 10 kilometers on a side.
Room Scales and Larger
Unfortunately, the sand-as-water-molecule metaphor begins to lose its usefulness when we get to the scale of the spaces in which humans and other animals live. Arguably, it has already lost concreteness by the time you get to whole humans (do you really grasp the length of the U.S. west coast or the orbital height of the International Space Station?).
If you wanted to visualize, say, a hotel room at sand-as-water-molecule scale, you’d be talking about carpet fibers as tall as mountains and areas comparable to continents. And real organisms can operate over much larger distances than the width of a hotel room! Scaled up, the migrations of the monarch butterfly or the arctic tern would be literally astronomical, best measured in units of solar system radii. Traditionally, astronomical distance is where we start using quantitativemetaphors to scale down.
This shift in metaphoric type is also useful — the difference between the scales of a cell and the scales of the everyday are like the difference between everyday earthly lengths and the smallest “astronomical” ones. In our next essay on metaphors, we’ll tackle the other dimension of biology: time.
Samuel Clamons is a bioinformatics scientist at Illumina, Inc. with a PhD in Bioengineering and training in applied mathematics and computer science. Outside of his day job, he writes science fiction and researches theoretical questions in biology at Asimov Press.
Cite: Clamons, S. “Metaphors for Biology: Sizes.” Asimov Press (2025). https://doi.org/10.62211/27he-92qw
Sand comes in a surprisingly wide range of sizes, with average beach sand ranging from 0.1 mm to 0.5mm. For convenience, I’ve assumed sand with a diameter of 0.28 mm, which a) is well within “typical” range for beaches and b) conveniently makes sand exactly one million times larger than a water molecule.
If this seems low to you, that’s because the International Space station is a fairly low-flying satellite. In contrast, a GPS satellite orbits about 10x the height of the scaled-up human.
Tech-savvy readers may note that this number doesn’t line up with modern “3nm architecture” and “5nm architecture” standards. That’s because the “nm” ratings on transistors stopped following the actual size of transistors sometime around 2008, and now refer to a kind of “performance equivalent.”
A puzzle by Paul Hoffman, from Science Digest. Could this game ever have resulted from a strict adherence to the rules of tic-tac-toe (noughts and crosses)?
No. The player with the Xs has won and so must have made the last move. If that player moved first, then there ought to be one more X than there are Os in the diagram. If the player making Os went first, then there should be an equal number of Os and Xs. The diagram contains more Os than Xs, so the position is impossible.
Log in
