Shri Khalpada, for PerThirtySix, breaks down musical sounds into math to illustrate how synthesizers work. Interactive graphics let you play with the parts and hear how the sounds change.
This would’ve been useful in that Fourier transform course I took in college, twice a week for two hours after lunch in a dark room using static PowerPoint slides.
Tags: instrument, music, PerThirtySix, Shri Khalpada
I just taught a couple of summer school classes here in Oakland, CA. The students were rising ninth graders and attending the class because some combination of parents and teachers insisted they needed some extra math for success in ninth grade. The students were not there for course credit (unlike the “intensive” classes down the hall) nor were they there, let’s say, for the love of the game.
“I have some lessons that fit the syllabus,” I told the teachers who were willing to let me get in their classes and make a mess of things. “How do you feel about digital lessons?”
These teachers told me they had put the laptop carts away for the summer. Their students would be off screens. Not my preference, but a worthy challenge. I planned for a) a teacher screen at the front of the room, b) paper on student desks.
Here is what I won and lost going screen-free, and how I’m now thinking about my work developing math edtech afterwards.
In the lesson, students watch a 15-second video of a thing happening—water filling a bowl, a student climbing up a play structure and sliding down. Then they fit a graph to that thing. We talk about the implications of the graph, pulling out mathematical ideas like a) the nature of a graph, b) the meaning of a point, c) the meaning of slope, d) the value of precision. This is Graphing Stories.
I had students create their graph of Tyler’s height as he climbed up and slid down the play structure. I walked around the class looking for a particular graph, a graph with a particular level of precision, one that caught some details but not all. I told that student that I loved their graph and thought it would be great for conversation. I asked them to go up to the front of the room and draw it on the whiteboard on top of a PDF of the handout that I had projected.
Then we talked about the graph. What happened in this section? What do you like about the graph? What should we change? Why? Okay, given everything we have all learned here, make your next and final draft.
Our start time is faster on paper than screens. The difference between …
passing out a handout
… and asking students to …
get laptops,
log into the laptop,
enter the activity URL,
correct their misspelling of the activity URL,
log into the activity
… is minutes. Precious minutes.
Our range of start times is smaller on paper. This is subtle but important. Worse than the overall time cost of screens is how screens distribute that cost across students. With paper, the time difference between the first ready student and the last is measured in seconds. Screens, by contrast, involve more complexity and failure points, so the first ready student might wait several minutes for the last ready student to get ready. This is frequently experienced by the first students as time wasted.
Fewer barriers separate us with paper. Even when students are working on their laptops without distraction, I do not, personally, appreciate the glass and metal between us. I appreciate how paper lays flat on a desk and lets everyone see everyone else above it.
Student screens give me more visibility into student thinking. In the digital version of the activity, I can access, present, and save the state of every student’s graph in real-time. Instead of walking around the classroom, I can quickly scan my screen for a particular graph, then show that graph in an anonymized mode for discussion. This is pedagogical power.
Student screens let me make better use of board space. With student screens, I can put the student’s graph in front of the class. I can then swap back to the video. I can put two student graphs side by side. I can bring the original student’s graph back. All at will, all without losing any work. Meanwhile, with paper, I have to ask the student to draw their graph on the board, and that’s basically all we can talk about until I erase it, and then it’s gone.
Students screens create dynamic connections between representations. As the video plays, a vertical line traces across the coordinate plane, helping students develop the connection between the graph and the world. I missed that.
On balance—wins minus losses—I wish I had had devices here. We would have used them for a total of ten minutes and they would have helped me invite and develop student thinking in ways I couldn’t with paper. Overall, this experience makes me want new ways to mix media, to add a student’s print work to our digital space, for example.
It’s a bit outside the edtech industry’s power to make faster hardware, but all of us nerds should get into classes as often as it takes to understand that what you experience when you pop open a new tab on your professional-grade laptop bears little resemblance to the dragging sensation experienced by students when teachers say “get out your laptops.”
¶ I’m under no illusion that this use of technology in math classes is common. It should be. But so much of math edtech loses all the benefits of paper and gains none of the benefits of screens as I’ve outlined them here. With most math edtech, teachers gain quiet kids and, like, a graph of how many questions students did over time. I don’t blame parents for wondering whether that kind of technology is worth their kids’ one childhood.
¶ “What opportunities did I miss?” I asked Victoria Sepe, the teacher whose classes I had borrowed. She mentioned asking the group of four students to come up with a group graph versus four solo graphs, both to generate more discussion and to share some of the risks of presenting work. Smart. Writing that down for next time.
I got to kick off a summer institute for Gwinnett County math teachers earlier this month. I arrived early enough to join grade-level sessions and was excited to see a lot of coherence across middle grades—both in content and pedagogy.
Next up:
Florida’s annual math teacher & supervisor conferences this week.
Come say hi!
Megan Schmidt gets the value of what we’re up to at Amplify and the support teachers need to do it well:
As someone who is very familiar with various curricular resources and mathematics instructional practices, Amplify is impressive (and unique) in the ways it has built technology tools that are a conduit for student thinking and exploration. I remember when Function Carnival was new and students could *see* the actual effect their graph had on the man shooting out of the cannon, and then the teacher could display the entire class’s graphs on one set of axes.
And then as a district math coordinator, I’m concerned about what support math teachers will need in order to facilitate (orchestrate?) this kind of math lesson, where students need to toggle between a screen on their desk, a screen in the front, along with various verbal inputs from others in the room. At the same time, Amplify Desmos Math is really quite different from most everything else I’ve seen (in a lot of good ways), and I’m excited for our teachers to get to try it.
¶ Okay, can we please be adults here? When it comes to the effect of generative AI on student learning, the evidence is clearly pointing in one particular direction:
And they’re using it to cheat.
Or if you prefer more nuance: they’re using it to disintermediate the artifacts of learning (essays, answers, grades, etc) from learning itself.
Since we last chatted, the studies keep dropping. The hits keep coming.
Nature retracted the big study showing positive learning gains from AI. (Hundreds of citations previously.)
Researchers found that AI gives different feedback based on a student’s race and gender.
A giant study of 30k Chinese students found that their AI usage increased homework scores, decreased completion time, and lowered exam scores by 20%.
Failing grades are way up in UC Berkeley’s computer science classes.
A study of 3.2M math practice interactions showed kids spending significantly less time on problems that are easier to send into an LLM (pure text-based) and a corresponding 25% decrease in exam performance. (The design of that study is so crisp you could layer it onto a sandwich.)
We need to be adults here. A thing got invented and its existence in the world—without denying any of its affordances—has not only failed to produce the learning gains promised by so many and instead produces evidence of negative transformation weekly.
All the usual suspects are placing their faith in pedagogical AI—the kind that won’t just burp an answer in a student’s direction—and that sounds potentially helpful during class. But why would kids use it outside of class where they have unrestricted access to the answer-burping AI?
I’m still working on some AI stuff that interests me, but now that this technology has arrived, the exciting education innovations are IMO almost entirely social and political:
Curriculum that gives kids more reasons to do the hard work of learning.
Stronger social ties in learning.
More accountability between parents, students, teachers, and schools.
Tighter connection between the artifacts and the learning. Blue books, public presentation of learning, etc, etc, e.g., these interesting pedagogical design patterns.
A political economy that is quite a lot less cutthroat, one where grades and credentials aren’t perceived as one of a vanishing few exit tickets from poverty.
¶ Congrats to the Knicks on winning the NBA championship. I loved this clip of coach Mike Brown describing a pedagogy of relationship with center Karl Anthony-Towns.
He’s a great player. I come in with a plan. Maybe the plan doesn’t work. Who adjusts? Him or me? Me. I adjust. The adjustment’s not enough. Every once in a while, we aren’t on the same page. We talk about it. I adjust again. He’s feeling better. We talk about it. It’s my job as a coach to fit whatever scheme we have on both sides of the floor to all of our players.
¶ Jill Barshay won an award for outstanding reporting. Way to go, Jill.
LLMs have broken legibility of effort - our ability to tell, at a glance, whether something took a human real work. What happens next?
Read the full post on my blog!Here's a raw link, if you need it: https://eieio.games/blog/legibility-of-effort
And how lugworms tweak the rules by pooping upside down
The post The Physics Behind the Poo Emoji’s Shape appeared first on Nautilus.
In James Bell’s math class at Chapman High School, sophomores are trying to pinpoint exactly where two lines cross.
The students in this rural Kansas high school already solved for that meeting point in previous lessons, using graphs and other techniques. But this recent lesson shows them how to use a matrix — a box made up of rows and columns representing a system of equations. Matrices are often used in engineering or video game design to calculate the locations of two objects moving through space.
Traditionally, this kind of complex linear algebra would be taught in the third year of high school math, when many students would take Algebra II. But a decade ago, the Chapman Unified School District, a rural district about 80 miles west of Topeka, Kansas, decided to drop the traditional high school math pathway — Algebra I for a year, followed by a year of geometry and then a year of Algebra II — in favor of what is called “integrated math.”
Integrated math takes concepts from both algebras and geometry, plus a little from trigonometry — the study of triangles and angles — and blends them over multiple years instead of teaching them separately, a year at a time. That means students can move from lessons in geometry to algebra and back again within the same year.
Bell, who helped write the math curriculum that Chapman High School uses, said he has seen how practicing both algebra and geometry throughout the year keeps them fresh in students’ minds.
“You’re going to have the opportunity to change course and change direction and see different things,” Bell said. “Students even do a little trig — which is a scary word for kids, but when it’s integrated into every year of math, it doesn’t sound as scary. It doesn’t make it as overwhelming.”
He added: “This is better for students. This is the best of both worlds.”
Chapman is part of a small group of districts and states moving to integrated math — all part of a larger movement to reimagine secondary math, give high schoolers more choice in courses, and modernize what some say are outdated ideas about what constitutes rigorous, college-level math by expanding course options beyond just calculus to include data science and statistics.
Some experts say blending algebra and geometry gives more students a shot at learning high-level math later on in high school and college, touting high-achieving European and Asian countries that have taught integrated math for decades.
But others say the integrated approach creates new problems. Some teachers have objected, saying they prefer to focus solely on algebra or geometry: Georgia mandated an integrated math approach in 2008 but made it optional in 2015 after teachers and parents complained. Others worry that integrated courses sometimes have to drop concepts that would be taught in the traditional math progression of Algebra I, geometry and Algebra II. That might leave some students unprepared for college calculus, because the integrated approach doesn’t stick with one subject for long enough for students to fully digest it.
In Chapman, where integrated math was installed along with other reforms, students went from 11 percent proficient to 41 percent proficient on the state math test the first year it was introduced, in 2015. In 2025, 67 percent scored proficient.
Kate Thornton, Chapman High’s principal, said the decision to move to integrated math had a lot to do with the Kansas state high school math test, which is administered in 10th grade. Many of the test questions were based on concepts from both Algebra I and II.
Related: This state tried to overhaul math instruction. It didn’t go as planned
Advanced students who took Algebra I in 8th grade and geometry the following year did well on the test, she said, but “regular students were not getting those concepts because they were taking geometry as sophomores. So they were having a whole year with no algebraic involvement.”
Integrated math still makes up only a fraction of high school math courses nationwide. A 2023 report from the Center for Education Market Dynamics noted that about 16 percent of districts offer integrated math either alone or as an option alongside the traditional Algebra I-geometry-Algebra II progression, often referred to as “AGA.” California and other Western states were seeing the most growth at the time of the report, based on the organization’s sample of over 900 districts around the country.
But more districts have adopted integrated math since the time of that report, said Lora Kaiser, the executive director of the center. “From 22-23 to 25-26, we’ve seen growth in every region outside of the Midwest, which showed a very modest decline. The West region, and California specifically, show most growth,” said Lora Kaiser, the executive director of the center. She also noted that fewer districts are offering a choice between the AGA and integrated models — instead, they are only offering integrated math to their students.
One factor driving the move to integrated math is the nationwide push to update high school mathematics for postsecondary education and the modern workforce. The Launch Years Initiative at the Charles A. Dana Center at the University of Texas at Austin, for example, is helping 27 states rework their transition from high school to postsecondary mathematics, mainly through offering more math options to students than the algebra, geometry, precalculus and calculus courses that have dominated high school math for decades.
Blending algebra and geometry courses can give students more room in their schedules to take other courses like data science or statistics, concepts that are very present in people’s everyday lives. Students take integrated math in ninth and 10th grade and then have the option of pursuing different math paths in 11th and 12th grades.
Lya Snell, director of building capacity for innovation at the Dana Center, said the AGA sequence is familiar to many and people are used to that framework. However, “where we are right now with tech and innovation is a lot different than where we’ve been before. We have to look at how we are preparing students for life today and in the future, and it requires us to create a more relevant experience.”
Maryland, one of the states working with the Dana Center, is rolling out a two-year course of integrated algebra and geometry that will be required statewide beginning in the fall of 2027. State leaders hope integrated math bolsters student achievement — only 30 percent of high schoolers scored proficient in math on the most recent state exam.
Related: A lot goes on in classrooms from kindergarten to high school. Keep up with our free weekly newsletter on K-12 education.
Lyndsey Brightful, the director of mathematics in the division of instructional programs at the Maryland State Department of Education, said the previous math progression didn’t align with the Blueprint for Maryland’s Future, the 2021 state law providing extra funding to accelerate flagging student achievement.
Maryland now expects students to be ready for college-level math by 11th grade, which means condensing “the most essential content and concepts” from Algebra I, geometry and Algebra II into the first two years of high school, she said.
“We had sort of a mismatch in our math progression,” Brightful said. “If we wanted students to be ready for an entry-level college math course by grade 11, then we needed students to be finished with their foundational math skills by the end of 10th grade. With the traditional Algebra I-geometry-Algebra I progression, that was not true for on-grade-level students. It required acceleration.”
After students have completed two years of integrated math, Maryland will offer them multiple pathways of what it calls rigorous, college-level math courses beginning in 11th grade, including calculus, college algebra, data science and statistics. Students will also have access to high school data science, discrete mathematics, and AP and IB math courses. For decades, the calculus pathway was considered the only choice for advanced and STEM students, but even top universities are now saying it’s “not the only advanced math,” she said.
Related: One state tried algebra for all eighth graders. It hasn’t gone well
“Higher-level statistics is advanced math, data science is advanced math,” Brightful said. “For students who are going into humanities majors, or even in some cases students who are going into nursing or business majors, they need to have strong math application skills or statistics skills, so pursuing other math pathways is still advanced.”
Brightful said the state is currently working with community colleges and the University of Maryland system to align admission requirements with the revised pathways. Currently, the state system requires four years of math for admissions, including Algebra I, geometry and Algebra II. Ideally, the state’s university system would accept the two-year integrated math course in place of the three years of AGA, Brightful said.
Because integrated math can cover a range of math topics over different time frames — states like Maryland are only offering two years, while districts like Chapman offer three or four years, depending on whether students choose to take calculus — it’s hard to determine whether the approach makes a difference in student achievement or helps more students with other goals, like succeeding at college-level math.
Related: Proof Points: Talk nerdy to me: Teachers who use math vocabulary help students do better in math
Some states have seen gains they believe are tied to the integrated approach. Mike Spencer, secondary math specialist at the Utah State Board of Education, cites the state’s move to three years of integrated math a decade ago as one factor among many contributing to Utah students’ consistently strong math performance on both the National Assessment of Educational Progress and the ACT.
“Some of the value in integrated math is you see things come up each year, versus having gaps in some of that content knowledge,” Spencer said. “And when it’s done well, you make connections” between topics.
California, on the other hand, hasn’t seen the same testing improvement. Even after more than a decade of districts using integrated math, only 37 percent of California students are proficient in math, lagging behind most states.
Research on implementing integrated math — at least in the complicated U.S. context — is relatively limited, said University of Florida education policy researcher Elizabeth Huffaker. A 2015 study, for example, showed that teacher effectiveness may improve while teaching integrated math.
Huffaker recently examined California districts to compare students who took integrated math to students in the traditional math pathway. Her just-published study showed a “small and positive” effect on 11th grade test scores among students who took integrated math, but she said this has to be taken with a grain of salt: Districts were also implementing Common Core math at the same time. The gains were equivalent to two to three months of high school math learning, about the same effect as Common Core implementation, Huffaker’s study showed.
“I would not make this change expecting giant, transformative, high-impact-tutoring-type impacts,” Huffaker said. The costs of switching to integrated math may well outweigh the benefits, she said, “but I do think that the idea of creating more coherence and more opportunities to revisit key topics is sensible.”
But many educators find aspects of integrated math frustrating. Students may not be getting the message that higher-level traditional math like trigonometry and calculus is still essential for college STEM majors, especially at selective schools. David Merryman, a professor of biomedical engineering at Vanderbilt University in Nashville, Tennessee, said the handful of students he’s advised over the years who arrived at Vanderbilt without high school calculus were completely lost in their first-year math courses.
And even if integrated math courses promise they are covering trigonometry woven in with geometry and algebra concepts, Merryman is not sure it’s enough to really prepare students for the rigors of college engineering courses.
“The kids who come to my class and are weak in trig, they struggle,” Merryman said. “For the non-STEM students, integrated math is probably great, you get a better understanding of how everything relates to each other. I don’t think it serves kids who’re going into STEM.”
Ben Kitchl, a first-year global studies major at Loyola University in Chicago, briefly thought about being a STEM major early in high school. But his eighth grade Algebra I credit wouldn’t transfer to his new high school that only offered integrated math, and he had to start over, putting him behind.
“I had to take Integrated Math II and III, even though I already knew a lot of it,” he said. He would have had to take two math courses in one year in order to make it to calculus his senior year.
It’s some of these tensions that are causing some states, including high-performing Utah, to revisit their integrated math requirement. Even some teachers are questioning it. “A lot of our veteran teachers, they’re the ones who say, ‘Oh, I miss teaching the AGA format, of being compartmentalized a little bit,” Spencer said. The state education agency plans to survey secondary teachers for their thoughts on the integrated model compared to the traditional math sequence and report back to the board of education on its findings.
In North Carolina, which like Utah has a three-year progression, the state board of education is also revisiting integrated math, looking into a possible two years of choice like Maryland to make more time for statistics and data science. Emily Hare, the director of pre-K-12 math at the state board of education, said attitudes are slowly shifting away from what she called the “race to calculus.”
“I think that we have proven that the integrated pathway can work. I also think that sometimes folks think that it’s a little more different than what it is,” Hare said. “Sometimes, yes, you have the opportunity to mix algebra and geometry, but it’s not like we’re teaching different math. It’s just teaching it in a different order.”
This story about integrated math was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Sign up for the Hechinger newsletter.
The post Blending algebra and geometry: An approach to high school math slowly gains favor appeared first on The Hechinger Report.
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