Christine Lenghaus

The third video in my series about understanding #mathshacks and #mathstricks is now out. This one features a lovely trick for approximating square roots of whole numbers that has beautiful connections to straight line graphs, calculus and even the Newton-Raphson method. Watch the full video here: youtu.be/IlMeZFe-m0M

@ihtesham2005 Is whether we process 'batch' or 'inline' relevant here?

A Soviet psychologist walked into a café in 1927 and watched a waiter do something impossible. He remembered every open order at every table. Perfectly. Without notes. Without effort. Then a table paid their bill. She asked him to repeat the order. He couldn't remember a single item. She spent the next two years figuring out why. What she found is now the operating system underneath every platform fighting for your attention. Her name was Bluma Zeigarnik, and she was a graduate student at the time, sitting with her professor Kurt Lewin, watching the waiters work the room. What caught her attention was something so ordinary that it had been happening in restaurants for centuries without anyone asking why. The waiters could remember every open order with perfect accuracy. Table four wanted the schnitzel with no sauce. Table seven had changed their wine twice. Table twelve owed for three coffees and a dessert. Every detail, held without effort, without notes, without any visible system at all. But the moment a table paid their bill, the information vanished. Completely. Lewin tested it on the spot. He called a waiter back minutes after a table had settled up and asked him to recite the order. The waiter could not do it. Not partially. Not approximately. The information was simply gone. Zeigarnik went back to her lab and spent the next two years turning that observation into one of the most replicated findings in the history of psychology. Here is what she proved, and why it changes how you think about attention, memory, and almost every piece of media you have ever consumed. She gave participants a series of tasks. Some tasks they were allowed to finish. Others were interrupted before completion. Then she tested recall across both groups. The unfinished tasks were remembered at nearly twice the rate of the completed ones. Not slightly better. Nearly twice. The brain was holding the incomplete work in a state of active tension, returning to it, keeping it warm, refusing to file it away. The finished tasks were closed, archived, released. The unfinished ones were still running. She called it the resumption goal. When the brain commits to a task and cannot complete it, it opens a file that stays open until resolution arrives. That open file consumes a portion of your cognitive bandwidth whether you are thinking about it consciously or not. It surfaces in idle moments. It pulls at the edge of your attention during other work. It is the thing you find yourself thinking about in the shower when you were not trying to think about anything at all. This is not a flaw in human cognition. It is a feature. The brain evolved to finish things. An open loop is a signal that something important is unresolved. Keeping that signal active increases the probability that you will return to it and complete it. In an environment where most tasks had real survival stakes, this was an extraordinarily useful mechanism. In the modern world, it is the most exploited vulnerability in human attention. Netflix did not invent the cliffhanger. But it industrialized it in a way no medium before it ever had. When a show ends on an unresolved question, it does not just create curiosity. It opens a file in your brain that stays active until the next episode closes it. The autoplay countdown that begins at 15 seconds is not a convenience feature. It is a precise calculation about how long the average person can tolerate an open loop before the discomfort of not knowing overrides every other intention they had for the evening. One more episode is not a choice. It is your brain doing exactly what it was designed to do: return to what is unfinished. The writers who built Lost, Breaking Bad, and Succession understood this intuitively without ever reading a psychology paper. Every episode ended on an open question. Every season finale answered three things and opened five more. The entire architecture of prestige television is a Zeigarnik machine running at industrial scale. But television is not where this gets dangerous. Every notification on your phone is an open loop. Every unread email is an open loop. Every task you wrote on a list and have not yet crossed off is an open loop. Each one is consuming a small but real portion of your available attention, pulling fractionally at your focus, degrading your capacity to be fully present in whatever you are actually doing right now. TikTok's algorithm does not just serve you content you like. It serves you content that ends one loop and immediately opens another, keeping the resumption system permanently activated so the cost of stopping always feels higher than the cost of continuing. The research on this accumulation effect is striking. Psychologists studying cognitive load have found that unfinished tasks do not sit passively in memory. They actively interrupt. They surface at the wrong moments. They are the reason you are reading something and suddenly remember an email you forgot to send. The brain is not malfunctioning. It is running its resumption system exactly as designed. It is just running it across forty open loops simultaneously, in an environment that generates new ones faster than any human nervous system was built to process. The most important practical implication Zeigarnik's research produced is one that most people use backwards. David Allen built his entire Getting Things Done system on the insight that the only way to close a cognitive open loop is to either complete the task or make a trusted commitment to complete it later. Writing something down in a system you actually trust has the same effect on the brain as finishing it. The file closes. The bandwidth is released. This is why writing a task down feels like relief even before you have done anything about it. You have not solved the problem. You have simply told your brain that the loop is registered and will be returned to, which is enough for the resumption system to stand down. The inverse is equally true and far more destructive. Every task that lives only in your head, unwritten and unscheduled, is an open loop burning cognitive resources around the clock. The mental cost is not proportional to the size of the task. A tiny nagging obligation consumes the same active tension as a major project. Your brain does not discriminate by importance. It discriminates by completion. Zeigarnik published her findings in 1927. The paper sat in academic literature for decades before anyone outside psychology paid attention to it. Then television got good. Then the smartphone arrived. Then the entire attention economy was engineered, largely by people who understood intuitively what she had proven scientifically: an open loop is the most powerful hook available to anyone who wants to hold human attention. Netflix knew it. Instagram knew it. Every designer who ever made a notification badge red instead of grey knew it. The café in Vienna is long gone. The mechanism she discovered there is now the operating system underneath every platform fighting for your time. Every "to be continued." Every unread notification. Every thread that ends with "part 2 tomorrow." All of it is the same waiter, the same unpaid bill, the same brain refusing to let go of what it has not yet finished. Zeigarnik noticed it over coffee in 1927. A century later, it is the most valuable insight in the history of media. And nobody taught it to you in school.

@howie_hua Days of the week = 7, eg simple to see % of 4 days of the week is approx 57%

The decimal representation for 1/7s is really cool

If you teach anyone at all, and you don't know about @helenrey's CogSci book summaries, you are missing out. It's a wonderful resource summarizing the most important books on cognitive science: bit.ly/3Q7dEnG

One of the big discussion points from our How Do We Learn? book study was active learning—learning by thinking. Students need to actively think about the learning object, seek meaning, and connect it to their prior knowledge to make learning meaningful. This has major implications for how we design learning activities: ✅Clearly define learning objectives ✅Ensure students spend more time thinking about the learning object than not ✅Avoid unnecessary details or context ✅Prioritize tasks that require thinking and meaning-making—explaining, summarizing, analyzing, comparing I also shared how Project Zero’s Visible Thinking Routines have long been one of my favorite ways to promote learning by thinking in this way. Here’s a reference document I created a few years ago that categorizes the routines based on the type of thinking you want to emphasize 👇 docs.google.com/document/d/1Gz…

Have you ever tried the amazing Painted Cube Task? This is me teaching my Stanford Undergraduates and a newsletter about it. Do share #youpaintcubes mailchi.mp/youcubed/newsl…

@pickover Thank you for sharing - I passed this on to my students in Myanmar, who I teach online and are interested in calculus.

Mathematics. Free calculus book, which even begins with "a brief review of selected topics from Algebra and Trig that are vital to surviving a Calculus course." The entire 1326-page book may be downloaded as a PDF here: tutorial.math.lamar.edu/Classes/CalcI/…

Place value is it +9, ×10 or both? youtu.be/cvuK9lTh3kQ?si…

@dylanwiliam youtu.be/wlfrNIvgUOo?si… Digging deeper into what is claimed.

Public service announcement: if you want to be taken seriously as a data scientist, please stop referring to Hattie's "hinge-point" of a 0.4sd effect size. Most effect sizes included in meta-analyses are net of business as usual, so it has zero value in interpreting effect sizes.

Focus isn’t forced. It’s trained. Small daily drills - tracing dots, steady lines, simple patterns - build stronger hand-eye-brain coordination. Patience today becomes confidence tomorrow. 🧠✨

If you want to help students develop conceptual understanding of points on a coordinate plane, then this lesson called Pomegraphit from can't be beet: classroom.amplify.com/activity/58cb0…

@joann_sandford Excellent learning and teaching!

Check out my latest blog post, Keeping it real jsandfordmath.ca/2026/02/06/kee… #iteachmath

How Long is Longitude? Granted, this took me a minute to understand. Source: pathfindergeospatial.ca/portfolio/how-…

Here's a cool math fact: If we can go up a set of stairs at one or two steps at a time, the number of different ways we can go up those stairs will always be a Fibonacci number! Here's why:

Seen a lot of teachers posting in past few days looking for ideas for games for their new classes. This one is called Imaparium and it's great. All you need to play is something write with. Enjoy! youtube.com/watch?v=9WPb3s…

Let's learn how to divide using rectangles

Here's a sample of some of my favorite mathy creations and why they work so well in the classroom. Check 'em out with this link: docs.google.com/presentation/d… #iteachmath #mtbos

Giving away our secret PD sauce :) My NEW post: Building a PD Program for Mathematics Teachers & Coaches fawnnguyen.substack.com/p/building-a-p…

Full video youtube.com/watch?v=Fd-_VD…

@fawnpnguyen Yes!

#unitchat When airlines use "minutes" to describe the delay.