Parent perspective here 🙋🏻♀️. I endorse the long cut wholeheartedly. The struggle is good for them. Also, as someone who is a math enthusiast and has worked in technical fields, I’ve tried all sorts of math supplementation options for kids and found that I still need to “tutor” them myself to reinforce some concepts, identify and fill in gaps and offer the right kind of encouragement during those struggles with the long cut. What I do can’t be “measured”—but that doesn’t mean it’s not important. In fact, it may be one of the most important components.
Chip and Dan Heath talk about this type of thing in their book, The Power of Moments. There are some events that we remember much more strongly than others, and one of the factors in those events is to experience something that leads to an epiphany. I highly recommend the book, and it does have other education examples. https://heathbrothers.com/books/the-power-of-moments/
I love using the "long cut" for graphing linear equations. When working with the Slope-intercept Form, most resources I've used go straight for the short cut of plotting the y-intercept and using the slope to plot the points of the line, but this is usually meaningless to the students. Instead I require that they use a table, then identify the slope and intercepts. Usually after about 20 linear equations, they see it for themselves. Then they think they can trick me by "skipping the math" and just plotting one point and using the slope -- jokes on them :)
I agree. I come more from a computer science point of view, but I think that the experience of connecting the different ways that a problem can be represented, of using patterns and abstraction to make things easier, etc. are all important for giving students the flexibility to put the concepts into context so that they form a coherent system. Otherwise, it's just a disparate collection of facts and procedures. Seeing the connections and the ability to move fluidly within the broader system of mathematics is crucial for them to solve novel problems and use what they know to help them move to the next level.
I use this long-cut technique (or variations of it, perhaps) a lot when teaching programming!
As just one example, I use Processing often to teach CS1/beginners, where we draw things on the screen and therefore use x/y coordinates a lot. At one point in the learning progression, I have them draw objects using nicely named variables that store their coordinates. Then we start wanting to have many objects moving around on the screen, which starts to mean that we have to do a lot of manual work over and over to draw and adjust all their x/y values. This leads us to the shortcut of...arrays! :D So much easier to have an array of all the x values and another for the y values so we can just loop over the array and do what we need to do each frame of the animation.
There are plenty of opportunities for "long cuts" when using dynamic geometry software. Challenge students to construct the midpoint of a segment using the Compass and Line tools. So much work to construct one point! But then, with the construction complete, students can use the Midpoint tool in the future. Sketchpad makes the benefit of the long cut feel especially worthwhile because students can take the constructions they labored on mightily (the midpoint, a square, an angle bisector, etc.) and turn them into tools that carry out their construction steps instantly.
Great example. In a similar vein, I like the locus as a long cut for the construction itself. Okay everyone find a point that is roughly equidistant from these two points. We overlay them and hey check that out.
Love the discussion of "short cut" and "long cut." It comes up in Calculus, when teaching students about the slope of a tangent line and derivatives. The whole discussion of limits and the limit definition of the derivative is a "must do" in my opinion, but the students are always so relieved to eventually learn the "short cut." Also, quite a few of my students have been taught the "short cut" in Physics with no background discussion, and it's neat to see the proverbial light bulb go on in their heads as they develop a deeper understanding of the procedures they had previously memorized.
There is one "short cut" that I was taught as a kid that seems like it is no longer taught. Over the past 10 years or so, when faced with something like 84 divided by 4 in class, I will quickly answer "21" and a student will inevitably say "Wow! How'd you do that so fast?" At first I was impatient - "You can't do 4 goes into 8 and 4 goes into 4 in your head??" - but I've learned that so many students don't even know that this "short cut" is possible. It's painful to me to watch them do long division every time, even when the solution is, to me anyway, so evident on inspection. I once discussed this with our lower school teachers and the more experienced ones agreed that it was a topic once taught, but was left behind as their curriculum became so full.
Love "the long cut" and how it tries to give a name (and weight) to a part of learning that's easy to overlook and leave out. I think David Hawkins aims to do something similar when he describes "Messing about in science"--to make a Thing out of that crucial, evanescent part of the figuring-out process.
"I swear I do not know how to score this news" - a disturbance in the force, for sure. AoPS, BEAM, now National Math Stars and National Math Camps, and Beast Academy aiming to go into schools? The pace of changes in that whole ecosystem has certainly picked up!
This is good advice, but there's an important detail: this doesn't work well if too many students make a mistake on the long cut. Too many "wait I didn't get that"s and a teacher running around trying to fix those really kills the moment in lessons like this.
I might have students solve a few "evaluate expression for x = number" problems on the Do Now, go over them, maybe do a bit of practice if it feels rocky, and then launch into the long cut/short cut activity.
Teaching exponential equations, I always insist on solving by guess-n-check and then by graphing before I ever say the word "logarithm." Algebra students can learn a process for solving and then have no idea what their solution means (especially if what you do with the solution is nothing, except to check against an answer key and then move onto the next equation.)
When you solve by graphing, you're really seeing what the solution means, and can even tell a story about it: "Measles cases rise exponentially, they crash through this level here, n=10000, after ___ days." In Desmos, one of the first things I teach is how to set y=something to get a nice horizontal line that tells you when a given function reaches a given level, which is often the goal of equation-solving.
Dan reminds us to "Be Less Helpful", and I hope we're listening.
-I agree with Agasthya's point. It takes a trained eye to discern engagement from actual learning.
-SteveB is totally correct as well. When you've reviewed the handwritten work of hundreds of students you definitely notice patterns higher frequency mistakes. Sure, we may be get surprised by the occasional unique error or line of thinking, but those are usually more entertainment than a sign that my instruction needs to change.
-Kelly's concern is provoking. I can't imagine the frustration of a student asking for help with a strategy when you know there is a "better" strategy for this problem/student/class. Or imagine you're a newer teacher still working to develop your content knowledge and you need to recognize multiple strategies and be able to offer guidance on each one? And imagine the agony if you see a student being asked to use a short-cut like FOIL or CROSS-MULTIPLY to solve. Yikes.
-Alex's comment seems rational from an outsider's viewpoint, and I'm glad Dan can articulate a spot-on response. I joined the profession with the goal of being a great teacher by giving the BEST explanations. I quickly learned that I can't EXPLAIN my way to math achievement. I once agreed with Alex -- in 2012, I interrupted teacher's classes to show them Khan Academy. Such high hopes and dreams! 13 year later, and I'm still waiting on video link that I could just DM to a struggling student...
(BUT, if I taught calculus, I would totally include videos from the YouTube channel 3Blue1Brown because WHOA -- those visuals and animations are amazing. An exemplar of explanatory videos.
Finally, off course Dan's kid uses an AoPS book... That's some good stuff right there.
Common wrong answers almost always have some logic to them, that's what makes them common. 1/2+1/3=2/5 isn't correct, but it's not crazy, either. And if you try, it's hard to come up with things that are both: 1) incorrect and 2) logical which is why there's a limited set of common wrong answers.
In statistics there are lots of opportunities to calculate or derive formulas the long way first. One of my favorites is confidence interval calculations. Long cut: students use a Normal distribution to map out a 95% confidence interval, calculate the area under the curve and find the z critical values in the table. Short cut: most confidence intervals repeat the same percentages, so it’s easy to just memorize z=1.96 for 95% for example
How many permutation/combination questions do you require students to work through manually before you show the formulas? This is a thing I struggle with, so far all I have is "some number greater than zero."
I think that’s really dependent on how much time you have. One way I get around that is by choosing a small number of items to work with — usually four or five is enough to make the formulas come to life. I feel similarly about binomial probability. I usually do about two problems by hand before showing them how the calculator works.
Parent perspective here 🙋🏻♀️. I endorse the long cut wholeheartedly. The struggle is good for them. Also, as someone who is a math enthusiast and has worked in technical fields, I’ve tried all sorts of math supplementation options for kids and found that I still need to “tutor” them myself to reinforce some concepts, identify and fill in gaps and offer the right kind of encouragement during those struggles with the long cut. What I do can’t be “measured”—but that doesn’t mean it’s not important. In fact, it may be one of the most important components.
Chip and Dan Heath talk about this type of thing in their book, The Power of Moments. There are some events that we remember much more strongly than others, and one of the factors in those events is to experience something that leads to an epiphany. I highly recommend the book, and it does have other education examples. https://heathbrothers.com/books/the-power-of-moments/
I love using the "long cut" for graphing linear equations. When working with the Slope-intercept Form, most resources I've used go straight for the short cut of plotting the y-intercept and using the slope to plot the points of the line, but this is usually meaningless to the students. Instead I require that they use a table, then identify the slope and intercepts. Usually after about 20 linear equations, they see it for themselves. Then they think they can trick me by "skipping the math" and just plotting one point and using the slope -- jokes on them :)
NICE. You're like, "Noooo ... don't skip the math ... [softly] by doing more math!"
I agree. I come more from a computer science point of view, but I think that the experience of connecting the different ways that a problem can be represented, of using patterns and abstraction to make things easier, etc. are all important for giving students the flexibility to put the concepts into context so that they form a coherent system. Otherwise, it's just a disparate collection of facts and procedures. Seeing the connections and the ability to move fluidly within the broader system of mathematics is crucial for them to solve novel problems and use what they know to help them move to the next level.
I use this long-cut technique (or variations of it, perhaps) a lot when teaching programming!
As just one example, I use Processing often to teach CS1/beginners, where we draw things on the screen and therefore use x/y coordinates a lot. At one point in the learning progression, I have them draw objects using nicely named variables that store their coordinates. Then we start wanting to have many objects moving around on the screen, which starts to mean that we have to do a lot of manual work over and over to draw and adjust all their x/y values. This leads us to the shortcut of...arrays! :D So much easier to have an array of all the x values and another for the y values so we can just loop over the array and do what we need to do each frame of the animation.
Re Synthesis Tutor channeling Musk: we all know how much the Elites (ahem, Musk) pay attention to their children ;P
There are plenty of opportunities for "long cuts" when using dynamic geometry software. Challenge students to construct the midpoint of a segment using the Compass and Line tools. So much work to construct one point! But then, with the construction complete, students can use the Midpoint tool in the future. Sketchpad makes the benefit of the long cut feel especially worthwhile because students can take the constructions they labored on mightily (the midpoint, a square, an angle bisector, etc.) and turn them into tools that carry out their construction steps instantly.
Great example. In a similar vein, I like the locus as a long cut for the construction itself. Okay everyone find a point that is roughly equidistant from these two points. We overlay them and hey check that out.
Love the discussion of "short cut" and "long cut." It comes up in Calculus, when teaching students about the slope of a tangent line and derivatives. The whole discussion of limits and the limit definition of the derivative is a "must do" in my opinion, but the students are always so relieved to eventually learn the "short cut." Also, quite a few of my students have been taught the "short cut" in Physics with no background discussion, and it's neat to see the proverbial light bulb go on in their heads as they develop a deeper understanding of the procedures they had previously memorized.
There is one "short cut" that I was taught as a kid that seems like it is no longer taught. Over the past 10 years or so, when faced with something like 84 divided by 4 in class, I will quickly answer "21" and a student will inevitably say "Wow! How'd you do that so fast?" At first I was impatient - "You can't do 4 goes into 8 and 4 goes into 4 in your head??" - but I've learned that so many students don't even know that this "short cut" is possible. It's painful to me to watch them do long division every time, even when the solution is, to me anyway, so evident on inspection. I once discussed this with our lower school teachers and the more experienced ones agreed that it was a topic once taught, but was left behind as their curriculum became so full.
Love "the long cut" and how it tries to give a name (and weight) to a part of learning that's easy to overlook and leave out. I think David Hawkins aims to do something similar when he describes "Messing about in science"--to make a Thing out of that crucial, evanescent part of the figuring-out process.
"I swear I do not know how to score this news" - a disturbance in the force, for sure. AoPS, BEAM, now National Math Stars and National Math Camps, and Beast Academy aiming to go into schools? The pace of changes in that whole ecosystem has certainly picked up!
This is good advice, but there's an important detail: this doesn't work well if too many students make a mistake on the long cut. Too many "wait I didn't get that"s and a teacher running around trying to fix those really kills the moment in lessons like this.
I might have students solve a few "evaluate expression for x = number" problems on the Do Now, go over them, maybe do a bit of practice if it feels rocky, and then launch into the long cut/short cut activity.
Teaching exponential equations, I always insist on solving by guess-n-check and then by graphing before I ever say the word "logarithm." Algebra students can learn a process for solving and then have no idea what their solution means (especially if what you do with the solution is nothing, except to check against an answer key and then move onto the next equation.)
When you solve by graphing, you're really seeing what the solution means, and can even tell a story about it: "Measles cases rise exponentially, they crash through this level here, n=10000, after ___ days." In Desmos, one of the first things I teach is how to set y=something to get a nice horizontal line that tells you when a given function reaches a given level, which is often the goal of equation-solving.
Dan reminds us to "Be Less Helpful", and I hope we're listening.
-I agree with Agasthya's point. It takes a trained eye to discern engagement from actual learning.
-SteveB is totally correct as well. When you've reviewed the handwritten work of hundreds of students you definitely notice patterns higher frequency mistakes. Sure, we may be get surprised by the occasional unique error or line of thinking, but those are usually more entertainment than a sign that my instruction needs to change.
-Kelly's concern is provoking. I can't imagine the frustration of a student asking for help with a strategy when you know there is a "better" strategy for this problem/student/class. Or imagine you're a newer teacher still working to develop your content knowledge and you need to recognize multiple strategies and be able to offer guidance on each one? And imagine the agony if you see a student being asked to use a short-cut like FOIL or CROSS-MULTIPLY to solve. Yikes.
-Alex's comment seems rational from an outsider's viewpoint, and I'm glad Dan can articulate a spot-on response. I joined the profession with the goal of being a great teacher by giving the BEST explanations. I quickly learned that I can't EXPLAIN my way to math achievement. I once agreed with Alex -- in 2012, I interrupted teacher's classes to show them Khan Academy. Such high hopes and dreams! 13 year later, and I'm still waiting on video link that I could just DM to a struggling student...
(BUT, if I taught calculus, I would totally include videos from the YouTube channel 3Blue1Brown because WHOA -- those visuals and animations are amazing. An exemplar of explanatory videos.
Finally, off course Dan's kid uses an AoPS book... That's some good stuff right there.
Common wrong answers almost always have some logic to them, that's what makes them common. 1/2+1/3=2/5 isn't correct, but it's not crazy, either. And if you try, it's hard to come up with things that are both: 1) incorrect and 2) logical which is why there's a limited set of common wrong answers.
In statistics there are lots of opportunities to calculate or derive formulas the long way first. One of my favorites is confidence interval calculations. Long cut: students use a Normal distribution to map out a 95% confidence interval, calculate the area under the curve and find the z critical values in the table. Short cut: most confidence intervals repeat the same percentages, so it’s easy to just memorize z=1.96 for 95% for example
How many permutation/combination questions do you require students to work through manually before you show the formulas? This is a thing I struggle with, so far all I have is "some number greater than zero."
I think that’s really dependent on how much time you have. One way I get around that is by choosing a small number of items to work with — usually four or five is enough to make the formulas come to life. I feel similarly about binomial probability. I usually do about two problems by hand before showing them how the calculator works.