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Maslow’s hierarchy of needs is a theory in psychology proposed by Abraham Maslow. His first discussion of this idea was in his 1943 paper “A Theory of Human Motivation” in Psychological Review. It was developed further in his 1954 book Motivation and Personality.
Contrary to popular belief, Maslow never created a pyramid to represent these needs. Nor did he conclude that in order for motivation to arise at the next stage, each stage must be satisfied. Much that teachers have heard about Maslow’s hierarchy of needs isn’t what he taught.
How his ideas were changed, incorrectly claimed as scientifically proven, and then became the basis of profitable seminars in business and education, is the subject of these papers:
Who Built Maslow’s Pyramid? A History of the Creation of Management Studies’ Most Famous Symbol and Its Implications for Management Education, by Todd Bridgman, Stephen Cummings and John Ballard, Academy of Management Learning & EducationVol. 18, No. 1, 3/1/2019
“Who Created Maslow’s Iconic Pyramid?” by Scott Barry Kaufman Scientific American, 4/23/2019
Even today the popular packaging of Maslow’s work is popular in management training and secondary and higher psychology and education instruction.
Saul McLeod points out that
Maslow continued to refine his theory based on the concept of a hierarchy of needs over several decades. Regarding the structure of his hierarchy, Maslow proposed that the order in the hierarchy “is not nearly as rigid” as he may have implied in his earlier description. Maslow noted that the order of needs might be flexible based on external circumstances or individual differences. For example, he notes that for some individuals, the need for self-esteem is more important than the need for love. For others, the need for creative fulfillment may supersede even the most basic needs.
Maslow’s Hierarchy of Needs, 3/20/2020, Saul McLeod, Simply Psychology
In Scientific American, Scott Barry Kaufman writes
Abraham Maslow’s iconic pyramid of needs is one of the most famous images in the history of management studies. At the base of the pyramid are physiological needs, and at the top is self-actualization, the full realization of one’s unique potential. Along the way are the needs for safety, belonging, love, and esteem.
However, many people may not realize that during the last few years of his life Maslow believed self-transcendence, not self-actualization, was the pinnacle of human needs. What’s more, it’s difficult to find any evidence that he ever actually represented his theory as a pyramid.
On the contrary, it’s clear from his writings that he did not view his hierarchy of needs like a video game– as though you reach one level and then unlock the next level, never again returning to the “lower” levels. He made it quite clear that we are always going back and forth in the hierarchy, and we can target multiple needs at the same time.
If Maslow never built his iconic pyramid, who did? In a recent paper, Todd Bridgman, Stephen Cummings, and John Ballard trace the true origins of the pyramid in management textbooks, and lay out the implications for the amplification of Maslow’s theory, and for management studies in general. In the following Q & A, I chat with the authors of that paper about their detective work.
Question: Why did you set out to answer the question: Who built “Maslow’s Pyramid”?
My colleague Stephen Cummings and I have long been interested in how foundational ideas of our field, management studies, are represented in textbooks. Textbooks often present ideas very differently than in the original writings. We’re interested in understanding how and why this happens. We’ve taught Maslow’s hierarchy of needs for many years and were aware the pyramid did not appear in his most well-known works, so were interested in delving deeper. We contacted John Ballard, who knew Maslow’s work better than we did and who shared our concern about Maslow’s theory being misrepresented. Thankfully, he agreed to join us on the project.
Question: Do you think the popularity of Maslow’s hierarchy of needs is due in part to the iconic appeal of the pyramid that became associated with it?
Yes, absolutely. Maslow wasn’t the first psychologist to develop a theory of human needs. Walter Langer presented a theory with physical, social and egoistic needs that appeared alongside Maslow’s in an early management textbook. And Maslow’s theory generally hasn’t performed well in empirical studies (although I’m aware of your recent research which challenges this).
In fact, this lack of empirical support is one of the main criticisms of the theory made by textbook authors. So why do they continue to include it? The pyramid. We know from having taught management courses for 20 years that if there’s one thing that students remember from an introductory course in management, it’s the pyramid. It’s intuitively appealing, easy to remember and looks great in PowerPoint. Students love it and because of that, so do textbooks authors, teachers, and publishers.
Question: So what’s your problem with the pyramid?
It’s described as ‘Maslow’s pyramid’ when he did not create it and it’s just not a good representation of Maslow’s hierarchy of needs. It perpetuates unfair criticisms of the theory. For example, that people are only motivated to satisfy one need at a time, that a need must be 100% satisfied before a higher-level need kicks in, and that a satisfied need no longer affects behavior.
Another is the view that everyone has the same needs arranged and activated in the same order. In his 1943 article in Psychological Review Maslow anticipates these criticisms and says they would give a false impression of his theory. Maslow believed that people have partially satisfied needs and partially unsatisfied needs at the same time, that a lower level need may be only partially met before a higher-level need emerges, and that the order in which needs emerge is not fixed.
Question: How did this inaccurate interpretation of the hierarchy of needs become established in management textbooks?
It’s a complicated story and one we address fully in the paper. Douglas McGregor is a key figure, because he popularized Maslow within the business community. McGregor saw the potential for the hierarchy of needs to be applied by managers, but for ease of translation he deliberately ignored many of the nuances and qualifications that Maslow had articulated. To cut a long story short, McGregor’s simplified version is the theory that appears in management textbooks today, and most criticisms of Maslow’s theory are critiques of McGregor’s interpretation of Maslow.
Question: Did McGregor create the pyramid? Or if not, who did?
No pyramid appears in McGregor’s writing. Keith Davis wrote a widely-used management textbook in 1957 that illustrated the theory in the form of a series of steps in a right-angled triangle leading to a peak. The top level shows a suited executive raising a flag, reminiscent of the flag-raising at Iwo Jima. But this representation of the theory did not catch on.
We traced the pyramid that we associate with the hierarchy of needs today to Charles McDermid, a consulting psychologist. It appeared in his 1960 article in Business Horizons ‘How money motivates men’ in which he argued the pyramid can be applied to generate “maximum motivation at the lowest cost”. We think McDermid’s pyramid was inspired by Davis’ representation, but it was McDermid’s image that took off. If there is an earlier pyramid, we did not find it.
Question: Is it right that you actually found no trace of Maslow framing his ideas in pyramid form? Where did you look, and how comprehensive was your search?
That’s correct. It was a comprehensive search. Maslow was a prolific writer. We examined all of his published books and articles that we could identify, as well as his personal diaries, which are published. John immersed himself in the Maslow archives at the Centre for the History of Psychology at the University of Akron in Ohio and examined many boxes of papers, letters, memos, and so forth. We found no trace of the pyramid in any of Maslow’s writings. Additionally, John went through pre1960 psychology textbooks for any discussions of Maslow. Most psych books in those times did not even mention Maslow.
Question: Why didn’t Maslow argue against the Pyramid once he saw it? He could have criticized it, right? I heard from someone who knew Maslow that he actually thought the pyramid on the back of the $1 bill was a fair representation of his theory.
Also, one of his students who took his course at Brooklyn College told me he would include a slide of the pyramid when he described his theory in class. So perhaps he was pleased with the iconic pyramid even if he didn’t invent the depiction himself?
Answer: Those are interesting questions. Maslow lived for 10 years after McDermid presented the pyramid. We found no evidence of Maslow challenging the pyramid at any time. We don’t think that’s because he regarded pyramid as an accurate representation. A more plausible explanation, which comes from our analysis of his personal diaries, is that aspects of his professional life were unravelling.
He felt underappreciated in psychology. The major research journals in psychology had been taken over by experimental studies, which depressed Maslow for their lack of creativity and insight. He also had more pragmatic concerns, suffering periods of ill health and financial difficulties. Key figures in the management community saw him as a guru and rolled out the red carpet. They gave him the recognition he felt he deserved. Furthermore, through speaking engagements and consulting, he could generate additional income. Seen in that light, it’s not surprising he went along with it.
Question: You wrote: “Inspiring the study of management and its relationship to creativity and the pursuit of the common good would be a much more empowering legacy to Maslow than a simplistic, 5-step, one-way pyramid.” I agree! It seems like Maslow’s original thinking about self-actualization is at odds with how business leaders treated the concept, right?
Definitely. Following the publication of Motivation and Personality in 1954, Maslow emerged as one of the few established psychologists to challenge the prevailing conformism of the 1950s. He spoke out on how large organizations and social conformity stifled individual self-expression. At times he was frustrated that the business community treated his theory of human nature as a means to a financial end–short-term profits–rather than the end which he saw, a more enlightened citizenry and society.
It would be great if students were encouraged to read what Maslow in the original. Students would better understand that motivating employees to be more productive at work was not the end that Maslow desired for the hierarchy of needs. He was concerned with creativity, freedom of expression, personal growth and fulfillment – issues that remain as relevant today in thinking about work, organizations, and our lives as they were in Maslow’s time. We think there’s an opportunity to create a new Maslow for management studies by returning to Maslow’s original ideas.
State of the theory today
William Kremer and Claudia Hammond write
There is a further problem with Maslow’s work. Margie Lachman, a psychologist who works in the same office as Maslow at his old university, Brandeis in Massachusetts, admits that her predecessor offered no empirical evidence for his theory. “He wanted to have the grand theory, the grand ideas – and he wanted someone else to put it to the hardcore scientific test,” she says. “It never quite materialised.”
However, after Maslow’s death in 1970, researchers did undertake a more detailed investigation, with attitude-based surveys and field studies testing out the Hierarchy of Needs.
“When you analyse them, the five needs just don’t drop out,” says Hodgkinson. “The actual structure of motivation doesn’t fit the theory. And that led to a lot of discussion and debate, and new theories evolved as a consequence.”
In 1972, Clayton Alderfer whittled Maslow’s five groups of needs down to three, labelled Existence, Relatedness and Growth. Although elements of a hierarchy remain, “ERG theory” held that human beings need to be satisfied in all three areas – if that’s not possible then their energies are redoubled in a lower category. So for example, if it is impossible to get a promotion, an employee might talk more to colleagues and get more out of the social side of work.
More sophisticated theories followed. Maslow’s triangle was chopped up, flipped on its head and pulled apart into flow diagrams. Hodgkinson says that one business textbook has just been published which doesn’t mention Maslow, and there is a campaign afoot to have him removed from the next editions of others.
The absence of solid evidence has tarnished Maslow’s status within psychology too. But as a result, Lachman says, people miss seeing that he was responsible for a major shift of focus within the discipline.
“He really was ground-breaking in his thinking,” Lachman says. “He was saying that you weren’t acting on the basis of these uncontrollable, unconscious desires. Your behaviour was not just influenced by external rewards and reinforcement, but there were these internal needs and motivations.”
Abraham Maslow and the pyramid that beguiled business, William Kremer and Claudia Hammond, BBC World Service 9/1/2013
Easy labs and manipulatives
Creating the periodic table
Exploratorium Science Snacks
(San Francisco, California)
How do viruses spread?
Not by individual virus particles
An individual virus particle is unbelievably tiny.
Since they are so lightweight they can float in the air for relatively long distances. So that makes them airborne, right?
Yet these airborne individual virus particles are almost never a problem. Studies show that people are not at risk of being infected by single viral particle.
Why not? We’re likely always inhaling single viral particles here and there. But they quickly break down, or if they persist then our immune system quickly wipes them out.
So if that ain’t the problem then what is? The problem is when we encounter a drop of fluid, or a solid surface, which may have many hundreds or thousands of such viral particles.
Try not to touch people who may be infected! If you do touch someone then wash your hands first.
When it comes to this novel coronavirus (its formal name is SARS-CoV-2) we have to be very careful: An infected person can leave viral particles behind on anything they touch or breathe on.
Infectious material could be left behind on a table, supermarket cart, keypad on an ATM machine, a computer keyboard, on a phone, etc.
A healthy person might touch one of those surfaces, and then touch their face, which then lets those virus particles get in to your airway. That’s a problem, but we can avoid danger: Be careful of what you touch and wash your hands!
Viruses spread exponentially
How does the likelihood of death from any common cause compare to the likelihood of death from something that spreads exponentially? The important difference is that for any other cause of death, that cause is (a) usually not transmissible, and (b) the rate of death stays (more or less) the same over time.
But for deaths caused by a virus the situation is different – (c) it is transmissible from one person to another, and (d) the number of people infected grows exponentially over time.
Droplets from sneezing and coughing
Sneezing or coughing sends out lots of tiny, snotty water droplets. Each droplet could hold thousands of viral particles. If we inhaled some of these drops then that is enough to make us sick.
Most droplets are short range. The larger ones only go about six feet before they fall to the ground. That’s why it is important to practice social distancing. Stay at least six feet away from people outside of your home.
But read on – with this novel coronavirus (SARS-CoV-2) there is a bit more danger:
Smaller droplets remain in the air longer
The big particles fall quickly, but the small particles float in the air longer – and then they dehydrate (they lose water molecules.) That leaves an even tinier, lighter particle.
These super tiny particles are almost like a gel. Some call these droplet nuclei, or an aerosol, or a bioaerosol. The danger is that these very tiny globs remain airborne much longer, and can travel a further distance. They can float over 20 feet!
That isn’t quite far enough for a virus to technically be called “airborne,” but it still is super dangerous. So if you are indoors – like in a supermarket – the air could become saturated with lots of these tiny droplet nuclei, making the location unsafe.
So when people were saying “this is new coronavirus is bad, but at least it isn’t airborne,” we now know that they were partially incorrect. When indoors this virus is somewhat airborne (6 to 20 feet), and that’s why one needs to avoid supermarkets unless necessary.
Health authorities suggest wearing a mask if you have to do so. Even an imperfect mask is better than none at all.
Truly airborne viruses
An airborne virus is one that can float in very tiny aerosol drops, less than 5 microns across, for hours and still remain infectious.
A micron is 0.001 millimeters , or 0.000039 inch.
Its symbol is μm
We now have evidence that this novel coronavirus, SARS-CoV-2, is an airborne virus.
The National Academy of Sciences (NAS) has given a boost to an unsettling idea: that the novel coronavirus can spread through the air—not just through the large droplets emitted in a cough or sneeze. Though current studies aren’t conclusive, “the results of available studies are consistent with aerosolization of virus from normal breathing,”
researchers reported earlier this year in The New England Journal of Medicine that SARS-CoV-2 can float in aerosol droplets—less than 5 microns across—for up to 3 hours, and remain infectious
You may be able to spread coronavirus just by breathing, new report finds, Science, AAAS, Robert F. Service, 4/2/2020
Yes, wearing cloth face masks works!
Cloth masks can help stop the spread of COVID-19, save lives and restore jobs. About 95% of the world lives in countries where the government and leading disease experts both agree that masks are effective at reducing the spread of COVID-19.
Anyone not wearing a cloth mask in public puts everyone at risk of getting infected and they hurt our economy by increasing the chances of a second lockdown.
Why? The U.S. CDC and most experts agree that many infected and contagious people don’t know they’re sick because they don’t have symptoms. Wearing a mask significantly reduces the chances of spreading COVID-19 from you to others.
“Some people have said that covering their faces infringes on their rights, but…it’s about protecting your neighbors…Spreading this disease infringes on your neighbors’ rights.” –Larry Hogan, Governor of Maryland (Republican)
“If everybody’s wearing a mask, it will dramatically reduce the opportunity and possibility of spread.” –Charlie Baker, Governor of Massachusetts (Republican)
Countries that have contained major COVID-19 outbreaks have close to 100% mask usage. An international review of the scientific research on masks by 19 experts (from Stanford, MIT, Oxford, UPenn, Brown, UNC, UCLA, and USF) concluded that:
Near-universal adoption of non-medical masks in public (in conjunction with other measures like test & trace) can reduce effective-R below 1.0 and stop the community spread of the virus.
Laws appear to be highly effective at increasing compliance and slowing or stopping the spread of COVID-19.
There are “34 scientific papers indicating basic masks can be effective in reducing virus transmission in public — and not a single paper that shows clear evidence that they cannot.” –The Washington Post
Flight of the aerosol, Ian M Mackay et al. Virology Down Under, 2/9/2020
Due to the likely imminent arrival of coronavirus outbreaks in America, many school districts are preparing to fight the pandemic with the most effective tool: social distancing. This may include cancelling school, and other public events, for up to 2 weeks in affected areas.
As such, teachers may be asked to prepare 2 weeks worth of lessons for students to do at home as stand alone work. How do we efficiently prepare for this? I have been uploading resources to my website and creating worksheets based on them. In a pinch I could email our students 2 weeks worth of worksheets in PDF format, and print out packets for students with little or no internet access.
Teachers can supplement the work they assign by creating a YouTube channel with daily mini-lectures. Students could listen & ask questions from their homes.
Teachers of course don’t need their own website. They can use any of their favorite science websites, such as The Physics Classroom.
However, we shouldn’t need something like the coronavirus to begin preparing. It is just good practice for teachers to create a library of ready-to-go, self-contained lesson plans for each topic. Instead of waiting for an emergency, make this part of your weekly schedule: Each week pick your best lesson, and write a self-contained worksheet for it.
As you create handouts (whether destined for PDF or paper) please think about readability, and about students who are slow readers or who have IEPs:
Have a brief, clear introduction so the student knows what we are learning and why. See below for details.
Use a large enough font.
Have some space between each section.
Break long paragraphs into a smaller paragraphs.
Add a color graphic to help explain the concept in each section.
A packed page is a poorly-designed page. Trying to shove a class onto just a page or two is especially irrelevant since we are no longer limited by paper. Students can view as many pages as we need on their PCs, tablets, or phones.
Writing a brief, clear introduction
* Content objective:
Briefly discuss what we are learning, and/or why we are learning this. This may involve teaching new ideas, procedures, or skills.
* Vocabulary objective – What are the critical words in this lesson?
These include not only new terms that you introduce, but supposedly “common” words that one assumes the students “already know.” (The problem is that many students don’t always know what these terms means. Please click the link for more information.)
– Tier II vocabulary words: High frequency words used across content areas. They are key to understanding directions, understanding relationships, and for making inferences.
– Tier III vocabulary words: Low frequency, domain specific terms.
* Build on what we already know.
Very few lessons start completely from scratch. Most will include some vocabulary & scientific concepts that were learned in earlier grades. So in this section of your introduction, briefly make connections to prior concepts.
How are you preparing for this?
Reliable sources of information
An essay on the role of a teacher today, and the wisdom of the standardized social and emotional learning phenomenon.
By Nick Parsons, Chemistry Teacher. 12/18/2019
In wrapping up and reflecting on the 2019 year, I’ve been thinking about school a lot… too much. Overall, teaching has been mostly good to me.
I liked school as a kid for the common, superficial reasons, like socializing and doing extracurriculars (by extracurriculars I mean sports, not all the other stuff I “did” to pad up a resume).
I REALLY liked school for deeper, private, and more personal reasons, such as being a place where I had the opportunity to challenge myself and either: (1) fail, reflect, and grow to do something more challenging or (2) succeed, reflect, and grow to do something more challenging.
For reasons I don’t understand, satisfying the incessant, burdensome need to challenge myself and be better always, and probably forever will, feel better than almost anything else.
On a day to day basis, I slept through most of my school day because it was usually boring and school/ practice/ homework was a 7:20am-9pm job, with 2 hours somewhere in there to eat, shower, and take a break to collect myself to get through the current day, in preparation for the next day. The majority of days I would feel stressed, tired, and mostly, just content – not overly happy, not overly sad. I certainly was not a daily ray of sunshine.
I LOVED all of the teachers I had in high school, but I can’t remember any going out of there way to help me sort out, identify, regulate, reflect, or otherwise support my social-emotional state… and that was totally fine. Honestly, if they did it would be weird. Plus, I trusted they were all conscionable, decent humans and if there was something seriously wrong with me, I bet they’d notice and reach out.
Not only that, I knew who the adults were who would support me as a young person. They were the same people who held me to high expectations, saw me at my highs and lows, would call me out if I was lacking focus or being a jerk. It was all love and it was real.
I liked school because it was satisfying, and I got through it because I was hooked on that feeling and I had genuine, supporting people that would afford me opportunities to challenge myself, hold me accountable to engage with these opportunities, and support me, if necessary, in adapting to these challenges so I could be better equipped to face them in the future.
I liked school (and coaching) SO much, I decided to do it, not as a job, but as a vocation (there is a difference). My average work day is essentially the same as in high school. During the hours of 7am-9pm I go to school, then coach, then come home to lesson plan or grade. Somewhere in there, I get two hours to eat, shower, and collect myself to get through the current day, in preparation for the next.
The majority of days I feel stressed, tired, and mostly, just content – not overly happy, not overly sad. I’m a little more outwardly positive and “sunshiny” because I don’t want to be that weird, mopey coworker, and students are more productive if I, at least, give the outward appearance of being happy and enthusiastic. Thankfully, I usually am. Somedays, I have to pretend, But mostly, my students can read my by the end of a semester, understand how I’m feeling, and they’re usually cool and respectful of it.
I’m a firm believer in creating fair, well thought out policies and adhering to them with fidelity. I teach chemistry – not because I have some deep, inherent passion for it – but because it’s “hard”, demanding, and doesn’t require much background knowledge – almost every student starts on an even playing field of knowing pretty much nothing. It’s the kind of subject where if you work hard, challenge yourself, fail, and reflect, you do REALLY well in. And nothing feels better to me when students do really well.
We definitely do not take content time aside to share our feelings, but I for sure will ask students how their days are going, what their other classes are like, how their game went last night, if they were able to get sleep after their rehearsal last night. BUT only if they got the work asked of them first.
Once in a while I remind the children that they have my unconditional support. Far more often, though, I’m reminding them that school is like a job, effort doesn’t matter much to me because it isn’t a material thing I can grade, and in my room, to do well, they need to be producing the best work they can. If they don’t, they’ll know when it’s graded. And no, I don’t allow revisions or retests. My favorite teacher’ism might be, “Hey now, don’t get mad about it, just get better.”
In general, I have good relationships with students. I hear through the grapevine and through survey data that they like my class, that they know I care about what I’m doing, I put in a lot of effort into my job, and they admit that they learn some chemistry by the end of it.
I’d be omitting the truth if I didn’t say I also constantly badger kids to put their phones away, cold-call the kids sleeping in class, cold-call the kids who are otherwise distracted, or raise my voice to talk over students talking over me. And I definitely piss off a kid once in a while with my daily urgency for order, focus, and productivity above having “fun”.
If a kid gets disrespectful toward me after I hold them accountable for violating clearly laid out classroom policies that are fair, constantly revised, and regularly communicated, I don’t ask them how they’re feeling or worry about how I can help them regulate their emotions. I got 20+ other kids that need to learn! I don’t have the time or capacity for that. Plus, I don’t need to: I’m a teacher, they’re students. On a professional level, in no way are we equals.
They get another chance, in that moment, to do better. If they can’t do that, well they can talk about it with an administrator, or at home with their family after I notify whoever is taking care of them that I’m not tolerating that kind of nonsense. Usually the kid gets better after that, sometimes they don’t. While I would like better outcomes in these situations, I’m not going to take on that burden of altering their psychology and modifying their behaviors. I’m a teacher, not a psychologist or a therapist. That’s not my place.
Since I got into teaching, all I’ve been listening to is this SEL (social emotional learning) thing and how I’m supposed to teach students how to recognize and regulate their emotions to feel good all the time. But again, I teach chemistry, no one taught me or trained me how to be a therapist.
Most schools push for this SEL thing, but there are only 6’ish hours in a school day. I’ve observed, substituted, and taught in 5 schools in the past 5 years. I’ve noticed that SEL pushes out other time-consuming tasks like content and discipline.
I’ve also noticed and have crossed paths with plenty of literature saying rampant depression is ailing teens, in-person communication is bizarre for them, and that they are encouraged to fight for their right to feel good all of the time. Plus, I’ve had a lot of students who are far more interested in me being their friend rather than their teacher. Like, no… that’s super weird, not to mention unprofessional.
I’ve also witnessed, experienced, heard about, and read about early career teachers leaving the profession ENTIRELY from burnout, veteran teachers saying “school wasn’t always like this”, and almost unanimously, by all kinds of teachers, some level of regret for even entering the profession. I was warned by SO many teachers to not enter the profession when I made the decision to. That’s not hard, qualifiable, or quantifiable data, but it was a lot – trust me!
I think I was better off in school as a student than my current students that I teach. I went to school to learn, because that was the whole purpose of school. I definitely didn’t go to feel good, and it wouldn’t matter! Going to school wasn’t a “choice” thing for me – I went because it was my job. I got the sense that it was my job and I would work hard during it because that was prioritized more by my teachers and parents than them worrying about how my social-emotional state was.
It feels backwards now. Every year, I get the sense students think my role in the room is to be some entertaining, funny guy that is supposed to make them feel good and never make them feel guilty, ashamed, embarrassed, or like they’re doing a bad job. It’s not my goal as a teacher for kids to feel that way, and it’s not something I particularly enjoy, but yah, that’s going to happen. School is hard, and if it isn’t I don’t know if we could define whatever “it” is as school.
And I know why they feel this way. Social-emotional learning is the new “in” thing. I suspect they’re interpreting the message from SEL as, “Yes, getting good grades is important, but definitely not as important as feeling good.”
This is sad, because I think we’re robbing kids of opportunities to intrinsically “feel”. Instead, unqualified, therapist like, social-emotional focused teachers (like myself) are unintentionally pushing these half-baked, extrinsically sourced emotions onto kids. Then, when they find themselves in situations where intrinsic, genuine, and powerful emotions rise up, they’re not equipped with dealing with them because they haven’t been put in organic situations to deal with them independently.
I also worry that this constant expectation about how to feel and in what situations is making students (and plenty of adults) abhorrently intolerant to events, people, or actions that make them “feel” bad. I’m not saying people should never feel offended, mad, or hurt in their dealings with people and events, but there needs to be less pressure on people to HAVE to feel certain ways.
People make mistakes and upset other people. It happens, and I don’t think it’s going to stop happening while humans roam the Earth. But if we’re encouraged to reject anything that upsets or offends us in lieu of experiencing the myriad of emotions that naturally emerge in these situations, the opportunity to develop self-regulation, empathy, and perspective taking is in danger of being lost. Further, the inclination and need to reflect upon, think critically about, and consider future actions in regards to whatever it was that is upsetting becomes unnecessary and unimportant. Absentmindedly rejecting sources of negative emotions bring too hasty of closure.
And I think that is happening everywhere, especially in classrooms. Because emotions have become so embedded in the teaching profession, the line between what is personal and professional and how to feel about a person or event are becoming blurred. My biggest fear is that demanding expectations, conduct policies, and rigor, the stuff that makes school “school”, become villainous ideas that can be rejected and attacked.
I dunno. This SEL thing is complicated. But I liked how I went through school as a student, and I know I’m better for it. I’m really proud of who I’ve become and what I do and school was a big part of that.
I also know that whatever I do as a teacher, it’s going to be focused on:
1. Enabling my students to be equipped to be successful in the real world, and…
2. Providing them reasonably challenging obstacles, a focused environment to work hard in, and several opportunities to achieve something that they thought was beyond their current capacities, and…
3. Most importantly, to get them to FEEL, GENUINE satisfaction and joy at being successful, in SOMETHING. It only takes once to get hooked to that feeling.
Long story short, I like going to my teacher job to just teach because it is simple and rewarding. I think my students benefit the most from that as well.
The idea behind “blizzard bags” and similar programs is to provide an alternative to making up school days missed due to weather disruptions or other unplanned school closures. The MTA Board has some serious concerns about blizzard bags.
In February, we asked MTA members for their thoughts on what the Department of Elementary and Secondary Education refers to as “alternative structured learning day programs” — otherwise known as “blizzard bags.” Your input will help guide our activism on this matter.
The “Blizzard Bags” program that allowed Massachusetts students to do class work at home during a winter storm and not have to make up the day in the summer comes to an end with this academic year.
The Massachusetts Department of Elementary and Secondary Education announced in June that it was discontinuing the Alternative Structured Learning Day Program, commonly known as “Blizzard Bags,” in fall 2020. It based its decision on a review of the “development and implementation of these programs.”
Some parents argued that “Blizzard Bags” could not take the place of a full day of school with face-to-face instruction or adequately address the needs of students on Individualized Education Programs.
Also, the Massachusetts Teachers Association voiced “serious concerns” about “Blizzard Bags” as a means of making up for a lost day of classroom instruction.
In the fall of 2018, the state Department of Elementary and Secondary Education established a working group to review the policy. Representatives from the Massachusetts Teachers Association participated, along with representatives of administrators from 10 Massachusetts school districts.
“The decision to discontinue the use of Alternative Structured Learning Day Programs is based upon a variety of factors, including concerns about equitable access for all students,” Jeffrey C. Riley, commissioner of Elementary and Secondary Education, stated on the state DESE website on June 27. “In addition to making every attempt to reschedule school days lost due to inclement weather, leaders should consider holding the first day of school prior to Labor Day. Other possibilities include scheduling a one-week vacation in March instead of week-long vacations in February and April.”
But here’s a question that almost no one seemed to even ask: Do snow days actually affect a student’s learning? This study claims that they don’t:
“Snow days don’t subtract from learning”
School administrators may want to be even more aggressive in calling for weather-related closures. A new study conducted by Harvard Kennedy School Assistant Professor Joshua Goodman finds that snow days do not impact student learning. In fact, he finds, keeping schools open during a storm is more detrimental to learning than a closure.
The findings are “consistent with a model in which the central challenge of teaching is coordination of students,” Goodman writes. “With slack time in the schedule, the time lost to closure can be regained. Student absences, however, force teachers to expend time getting students on the same page as their classmates.”
Goodman, a former school teacher, began his study at the behest of the Massachusetts Department of Education, which wanted to know more about the impact of snow days on student achievement. He examined reams of data in grades three through 10 from 2003 to 2010. One conclusion — that snow days are less detrimental to student performance than other absences — can be explained by the fact that school districts typically plan for weather-related disruptions and tack on extra days in the schedule to compensate. They do not, however, typically schedule make-up days for other student absences.
The lesson for administrators might be considered somewhat counterintuitive. “They need to consider the downside when deciding not to declare a snow day during a storm — the fact that many kids will miss school regardless, either because of transportation issues or parental discretion. And because those absences typically aren’t made up in the school calendar, those kids can fall behind.”
Goodman, an assistant professor of public policy, teaches empirical methods and the economics of education. His research interests include labor and public economics, with a particular focus on education policy.
Flaking Out: Snowfall, Disruptions of Instructional Time, and Student Achievement, by Joshua Goodman, Harvard Kennedy School of Government, April 30, 2012
Why learn math? When are going to use this in real life?
When students ask “when will we ever need this in real life?” they often aren’t actually being curious about their future. They are actually just unhappy with being assigned work in the present. But some students truly do want to learn the answers to this question – and teachers, one would hope – should know answers as well. And there are several answers to this question, not just one.
I. First we should recognize that this is an unfair question. Douglas Corey, at Brigham Young University, writes:
In truth, the when-will-I-use-this question is unfair for the teacher. She doesn’t know when you will (or even might) use it (except on the exam and in the next course in the sequence). She might explain how other people have used it, but, as we saw above, that response is not convincing. The difficulty in answering this question lies with an implicit assumption hidden beneath the question. The student has an idea of the kinds of situations that she will encounter in her life, and when the response from the teacher doesn’t apply to any of these situations, the mathematics seems useless. But it is fraudulent to assume that we know at a moment of reflection the kinds of situations in which we might use something. Why? Because we typically don’t know what we don’t know.
II. Does a football team go onto the field and lift weights? Of course the team doesn’t do that. However if they didn’t practice lifting weights then they certainly wouldn’t have a chance to win.
III. We’re actually not learning hard math that mathematicians study in university. For every subject that you think you are studying – algebra, trigonometry, calculus, etc. – you’re really just learning the introduction to these subjects! Yes, even after a year in high school calculus all you have done is scratch the surface of that field of math.
So why learn any of these math topics at all in grades K-12? Because children don’t know what they are going to be 10 or 20 years from now. So consider: If we don’t teach students how to be fluent and literate in English, then how can they read and learn anything? How can they communicate using the written word? They literally would be unable to even consider a career, right?
Now realize that the same is true for math. If we don’t teach students how to be fluent and literate in mathematics and logical thinking, then how could they ever even have a chance to consider a career in medicine, engineering, coding, chemistry, artificial intelligence, astronomy, physics, or math? No one ever would even be able to consider such a career.
IV. Here I’m excerpting some thoughts from Al Sweigart.
A math teacher is giving a lesson on logarithms or the quadratic equation or whatever and is asked by a student, “When will I ever need to know this?”
“Most likely never,” replied the teacher without hesitation. “Most jobs and even a lot of professions won’t require you to know any math beyond basic arithmetic or a little algebra.”
“But,” the teacher continued, “let me ask you this. Why do people go to the gym and lift weights? Do they all plan on becoming Olympic weight lifters, or professional body builders? Do they think they’ll one day find an old lady trapped under a 200 pound bar bell and say, ‘This is what I’ve been training for.’”
“No, they lift weights because it makes them stronger. Learning math is important because because it makes you smarter. It forces your brain to think in a way that normally it wouldn’t think: a way that requires precision, discipline, and abstract thought. It’s more than rote memorization, or making beautiful things, or figuring out someone’s expectations and how to appease them.”
“Doing your math homework is practice for the kind of disciplined thinking where there are objective right and wrong answers. And math is ubiquitous: it comes up in a lot of other subjects and is universal across cultures. And all this is practice for thinking in a new way. And being able to think in new ways, more than anything, is what will prepare you for an unpredictable, even dangerous, future.”
IV. We learn mathematics without realizing its ramifications and applications
This attitude comes partly from ignorance and partly from our faulty education system. We learn mathematics without realizing its ramifications and applications. You have been led to believe that it is useless but it is not. Look around the world in which you live. Almost everything that you experience and enjoy is possible because of mathematics.
You drive a car. A car company uses CAD software which lets it design and model components with absurd ease. Do you know how a CAD software works? It uses rigorous mathematics from geometry to matrices.
That’s one part of it. The calculation part. To display a model on your computer screen is yet another story. Processes are set, algorithms are developed and executed. But merely developing an algorithm is not sufficient. You have to optimize it. To develop and optimize an algorithm you need mathematics. Somebody has to develop the optimization algorithm. Know that the optimization algorithm is an algorithm to optimize a different algorithm. To develop such feat you would probably need to master functions, graphs and calculus. To perform stress analysis on such a component you would need yet another algorithms. To develop them you would probably need to study finite element analysis and matrices. This is true for any industry and not just for car industry.
Consider a security firm. It need to be able to identify a person’s face. They need a face recognition algorithm. Now some geeks have developed many such algorithms. Some of them are simple and less accurate while some are highly accurate but difficult to employ.
Development of each such algorithm requires extensive knowledge of matrices, probabilities, and other 100 things but do you know what is beautiful? The security firm may audit itself and using yet another mathematical process, assess exactly what type of algorithm it would need. Mathematics. Again. This is true for any forensic analysis. Fingerprint matching, face matching, pattern recognition and what not. Many private and public security firms, law enforcement agencies and spy agencies are using and developing such specialized tools thanks to mathematics.
Let’s come to gaming. You will be thrilled to know that while playing combat games, you are actually fighting with an algorithm which can ‘learn’ you. Genetic algorithm, neural networks and such things. Google it. Imagine yourself at a scene in a game. You can’t see what’s behind you in a scene, but as you look at it, the scene develops. There are special compression algorithms who use the information of the scene in compressed format when nobody is looking at it. I guess I don’t have to repeat now but still, I will. Mathematics.
Investment funds, hedge funds and other financial institutions predict the market and make decisions using mathematical software. Again, it require, number crunching, statistics, pattern recognition (which itself requires a lot of mathematics), optimization, functions and graphs and calculus (for effective predictions). Insurance companies need to use probabilistic models of customers to come up with new policies. They invest money in stock market. Now again read this paragraph just put insurance companies in place of investment funds.
V. Kalid Azad, of BetterExplained, writes in How to Develop a Mindset for Math
Math uses made-up rules to create models and relationships. When learning, I ask:
What relationship does this model represent?
What real-world items share this relationship?
Does that relationship make sense to me?
They’re simple questions, but they help me understand new topics. If you liked my math posts, this article covers my approach to this oft-maligned subject. Many people have left insightful comments about their struggles with math and resources that helped them.
Textbooks rarely focus on understanding; it’s mostly solving problems with “plug and chug” formulas. It saddens me that beautiful ideas get such a rote treatment:
The Pythagorean Theorem is not just about triangles. It is about the relationship between similar shapes, the distance between any set of numbers, and much more.
e is not just a number. It is about the fundamental relationships between all growth rates.
The natural log is not just an inverse function. It is about the amount of time things need to grow.
Elegant, “a ha!” insights should be our focus, but we leave that for students to randomly stumble upon themselves. I hit an “a ha” moment after a hellish cram session in college; since then, I’ve wanted to find and share those epiphanies to spare others the same pain.
But it works both ways — I want you to share insights with me, too. There’s more understanding, less pain, and everyone wins.
Math Evolves Over Time
I consider math as a way of thinking, and it’s important to see how that thinking developed rather than only showing the result. Let’s try an example.
Imagine you’re a caveman doing math. One of the first problems will be how to count things. Several systems have developed over time:
No system is right, and each has advantages:
Unary system: Draw lines in the sand — as simple as it gets. Great for keeping score in games; you can add to a number without erasing and rewriting.
Roman Numerals: More advanced unary, with shortcuts for large numbers.
Decimals: Huge realization that numbers can use a “positional” system with place and zero.
Binary: Simplest positional system (two digits, on vs off) so it’s great for mechanical devices.
Scientific Notation: Extremely compact, can easily gauge a number’s size and precision (1E3 vs 1.000E3).
Think we’re done? No way. In 1000 years we’ll have a system that makes decimal numbers look as quaint as Roman Numerals (“By George, how did they manage with such clumsy tools?”).
Negative Numbers Aren’t That Real
Let’s think about numbers a bit more. The example above shows our number system is one of many ways to solve the “counting” problem.
The Romans would consider zero and fractions strange, but it doesn’t mean “nothingness” and “part to whole” aren’t useful concepts. But see how each system incorporated new ideas.
Fractions (1/3), decimals (.234), and complex numbers (3 + 4i) are ways to express new relationships. They may not make sense right now, just like zero didn’t “make sense” to the Romans. We need new real-world relationships (like debt) for them to click.
Even then, negative numbers may not exist in the way we think, as you convince me here:
You: Negative numbers are a great idea, but don’t inherently exist. It’s a label we apply to a concept.
Me: Sure they do.
You: Ok, show me -3 cows.
Me: Well, um… assume you’re a farmer, and you lost 3 cows.
You: Ok, you have zero cows.
Me: No, I mean, you gave 3 cows to a friend.
You: Ok, he has 3 cows and you have zero.
Me: No, I mean, he’s going to give them back someday. He owes you.
You: Ah. So the actual number I have (-3 or 0) depends on whether I think he’ll pay me back. I didn’t realize my opinion changed how counting worked. In my world, I had zero the whole time.
Me: Sigh. It’s not like that. When he gives you the cows back, you go from -3 to 3.
You: Ok, so he returns 3 cows and we jump 6, from -3 to 3? Any other new arithmetic I should be aware of? What does sqrt(-17) cows look like?
Me: Get out.
Negative numbers can express a relationship:
Positive numbers represent a surplus of cows
Zero represents no cows
Negative numbers represent a deficit of cows that are assumed to be paid back
But the negative number “isn’t really there” — there’s only the relationship they represent (a surplus/deficit of cows). We’ve created a “negative number” model to help with bookkeeping, even though you can’t hold -3 cows in your hand. (I purposefully used a different interpretation of what “negative” means: it’s a different counting system, just like Roman numerals and decimals are different counting systems.)
By the way, negative numbers weren’t accepted by many people, including Western mathematicians, until the 1700s. The idea of a negative was considered “absurd”. Negative numbers do seem strange unless you can see how they represent complex real-world relationships, like debt.
Why All The Philosophy?
I realized that my **mindset is key to learning. **It helped me arrive at deep insights, specifically:
Factual knowledge is not understanding. Knowing “hammers drive nails” is not the same as the insight that any hard object (a rock, a wrench) can drive a nail.
Keep an open mind. Develop your intuition by allowing yourself to be a beginner again.
A university professor went to visit a famous Zen master. While the master quietly served tea, the professor talked about Zen. The master poured the visitor’s cup to the brim, and then kept pouring. The professor watched the overflowing cup until he could no longer restrain himself. “It’s overfull! No more will go in!” the professor blurted. “You are like this cup,” the master replied, “How can I show you Zen unless you first empty your cup.”
Be creative. Look for strange relationships. Use diagrams. Use humor. Use analogies. Use mnemonics. Use anything that makes the ideas more vivid. Analogies aren’t perfect but help when struggling with the general idea.
Realize you can learn. We expect kids to learn algebra, trigonometry and calculus that would astound the ancient Greeks. And we should: we’re capable of learning so much, if explained correctly. Don’t stop until it makes sense, or that mathematical gap will haunt you. Mental toughness is critical — we often give up too easily.
So What’s The Point?
I want to share what I’ve discovered, hoping it helps you learn math:
Math creates models that have certain relationships
We try to find real-world phenomena that have the same relationship
Our models are always improving. A new model may come along that better explains that relationship (roman numerals to decimal system).
Sure, some models appear to have no use: “What good are imaginary numbers?”, many students ask. It’s a valid question, with an intuitive answer.
The use of imaginary numbers is limited by our imagination and understanding — just like negative numbers are “useless” unless you have the idea of debt, imaginary numbers can be confusing because we don’t truly understand the relationship they represent.
Math provides models; understand their relationships and apply them to real-world objects.
Developing intuition makes learning fun — even accounting isn’t bad when you understand the problems it solves. I want to cover complex numbers, calculus and other elusive topics by focusing on relationships, not proofs and mechanics.
But this is my experience — how do you learn best?
This section by Kalid Azad was made under a Creative Commons Attribution-NonCommercial-ShareAlike license.
Using virtual reality in the classroom
We learn through lectures and reading. We especially learn through illustrations, photographs, diagrams, and animations. But a limitation is that so many of these images are flat, two-dimensional. Not surprisingly, many folks have trouble visualizing what a system is really like, if they only have two dimensional pictures.
An obvious practical solution is to make a lesson hands-on: Students can take a field trip to see gears and machines in a power plant; see ancient ruins on site; travel to a valley and fly over a vast ecosystem to see different parts of the environment. But there’s only so much that a school can do in practice: we can’t purchase every manipulative and lab, or travel to see every place that we talk about.
Yet with today’s technology we can actually model machines, cells, valleys and volcanoes, ecosystems, distance cities, and archaeological sites, in three dimensions – and then bring all of this into the classroom. We bring these models in to a virtual space that students can explore.
And that’s what we are already doing in our classrooms! First, let’s learn a few terms: XR, AR, and VR.
XR- Extended Reality
the emerging umbrella term for all immersive computer virtual experience technologies. These technologies AR, VR, and MR.
Augmented Reality (AR)
When virtual information and objects are overlaid on the real world. This experience enhances the real world with digital details such as images, text, and animation. This means users are not isolated from the real world and can still interact and see what’s going on in front of them.
CRISPR enzyme floating in three dimensions.
Virtual Reality (VR)
Users are fully immersed in a simulated digital environment. Individuals must put on a VR headset or head-mounted display to get a 360 -degree view of an artificial world. This fools their brain into believing they are walking on the moon, swimming under the ocean or stepped into whatever new world the VR developers created.
Mixed reality (MR), aka Hybrid Reality
Digital and real-world objects co-exist and can interact with one another in real-time. This experience requires an MR headset… Microsoft’s HoloLens is a great example that, e.g., allows you to place digital objects into the room you are standing in and give you the ability to spin it around or interact with the digital object in any way possible.
Excerpts of these definitions from Bernard Marr, What Is Extended Reality Technology? A Simple Explanation For Anyone, Forbes, 8/12/2019
Augmented reality in Ecology & Environmental Science
When students actively participate in augmented reality learning, the class is effectively a lab, as opposed to being a lecture. Here we are studying ecosystems with an app from the World Wildlife Foundation, WWF Rivers.
This student has their head in the clouds 😉
Here we are using the Google Expeditions app, on a Pixel 3A smartphone. The plug-in is “Earth Geology” by Vida systems. For more details see Google Expeditions – Education in VR.
AR in Earth Science
As we walk around the room, we see the Earth and all of it’s layers in a realistic 3D view. Here we stood above the arctic circle, and took screenshots as we moved down latitude, until we were above the antarctic.
AR in Physics & Engineering
A simple machine is a mechanical device that changes the direction or magnitude of a force. They are the simplest mechanisms that use mechanical advantage to multiply force.
Here we are examining gears, including bicycle gears.
Related Special Education topics
If someone can’t visually imagine things, how can you learn? We know some people can’t conjure up mental images. But we’re only beginning to understand the impact this “aphantasia” might have on their education.
A discussion of an inability to form mental images , congenital aphantasia. This is believed to affect 2% of the population.
by Mo Costandi, Jun 2016, The Guardian, UK
What kind of learning standards will students address when using augmented reality science lessons?
NGSS Cross-Cutting Concepts
6. Structure and Function – The way an object is shaped or structured determines many of its properties and functions: Complex and microscopic structures and systems can be visualized, modeled, and used to describe how their function depends on the shapes, composition, and relationships among its parts; therefore, complex natural and designed structures/systems can be analyzed to determine how they function
Massachusetts Digital Literacy and Computer Science (DLCS) Curriculum Framework
Modeling and Simulation [6-8.CT.e] – 3. Select and use computer simulations, individually and collaboratively, to gather, view, analyze, and report results for content-related problems (e.g., migration, trade, cellular function).
Digital Tools [9-12.DTC.a] – 2. Select digital tools or resources based on their efficiency and effectiveness to use for a project or assignment and justify the selection.
American Association of School Librarians: Standards Framework for Learners
1. Inquire: Build new knowledge by inquiring, thinking critically, identifying problems, and developing strategies for solving problems
Advanced Placement Computer Science Principles
AP-CSP Curriculum Guides
LO 3.1.3 Explain the insight and knowledge gained from digitally processed data by using appropriate visualizations, notations, and precise language.
EK 3.1.3A Visualization tools and software can communicate information about data.
EK 3.1.3E Interactivity with data is an aspect of communicating.
As we all know the NGSS are more about skills than content. Confusingly, though, they ended up also listing core content topics as well – yet they left out kinematics and vectors, the basic tools needed for physics in the first place.
The NGSS also dropped the ball by often ignoring the relationship of math to physics. They should have noted which math skills are needed to master each particular area.
Hypothetically, they could have had offered options: For each subject, note the math skills that would be needed to do problem solving in this area, for
* a standard (“college prep”) level high school class
* a lower level high school class, perhaps along the lines of what we call “Conceptual Physics” (still has math, but less.)
* the highest level of high school class, the AP Physics level. And the AP study guides already offer what kinds of math one needs to do problem solving in each area.
Yes, the NGSS does have a wonderful introduction to this idea, (quoted below) – but when we look at the actual NGSS standards they don’t mention these skills.
In some school districts this has caused confusion, and even led to some administrators demanding that physics be taught without these essential techniques (i.e. kinematic equations, conceptual understanding of 2D motion, kinematic analysis of 2D motion, vectors, etc.)
To help back up teachers in the field I put together these standards for vectors, from both science and mathematics standards.
– Robert Kaiser
Massachusetts Science Curriculum Framework (pre 2016 standards)
1. Motion and Forces: Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work).
Mathematical and computational thinking in 9–12 builds on K–8 experiences and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions.
- Apply techniques of algebra and functions to represent and solve scientific and engineering problems.
Although there are differences in how mathematics and computational thinking are applied in science and in engineering, mathematics often brings these two fields together by enabling engineers to apply the mathematical form of scientific theories and by enabling scientists to use powerful information technologies designed by engineers. Both kinds of professionals can thereby accomplish investigations and analyses and build complex models, which might otherwise be out of the question. (NRC Framework, 2012, p. 65)
Students are expected to use mathematics to represent physical variables and their relationships, and to make quantitative predictions. Other applications of mathematics in science and engineering include logic, geometry, and at the highest levels, calculus…. Mathematics is a tool that is key to understanding science. As such, classroom instruction must include critical skills of mathematics. The NGSS displays many of those skills through the performance expectations, but classroom instruction should enhance all of science through the use of quality mathematical and computational thinking.
Common Core Standards for Mathematics (CCSM)
High School: Number and Quantity » Vector & Matrix Quantities. Represent and model with vector quantities.
Represent and model with vector quantities.
(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
(+) Solve problems involving velocity and other quantities that can be represented by vectors.
Perform operations on vectors.
(+) Add and subtract vectors.
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
Understand vector subtraction v – w as v + (-w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
(+) Multiply a vector by a scalar.
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
Operating with Symbols and Equations
- Become fluent in generating equivalent expressions for simple algebraic expressions and in solving linear equations and inequalities.
- Develop fluency operating on polynomials, vectors, and matrices using by-hand operations for the simple cases and using technology for more complex cases.
9B9-12#5: When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes more than one, and sometimes maybe not at all.
12B9-12#2: Find answers to problems by substituting numerical values in simple algebraic formulas and judge whether the answer is reasonable by reviewing the process and checking against typical values.
12B9-12#3: Make up and write out simple algorithms for solving problems that take several steps.
#vectors #teaching #standards #kinematics #physics #kaiserscience #pedagogy #education #NGSS #Benchmarks #scalors #highschool
NGSS has three distinct components: 1. Disciplinary Core Ideas, 2. Cross Cutting Concepts, and 3. Science & Engineering Practices.
A Way to Think About Three-Dimensional Learning and NGSS
From Carolina Biologica Supply Company,, by Dee Dee Whitaker
The National Research Council (NRC) went to science and engineering practitioners and gathered information on how they “do” science and engineering. That information was organized and the resulting framework is the Next Generation Science Standards.
- What scientists do is Dimension 1: Practices
- Concepts applied to all domains of science is Dimension 2: Crosscutting Concepts
- Big, important concepts for students to master is Dimension 3: Disciplinary Core Ideas
Each dimension is further refined into specific behaviors, concepts, and ideas. Below is a list of the three dimensions with an accompanying explanation and a brief rationale for each.
|Scientific and Engineering Practices
||Disciplinary Core Ideas
The broad, key ideas within a scientific discipline make up the core ideas. The core ideas are distributed among 4 domains:
Applicable to all science disciplines, crosscutting concepts link the disciplines together.
Tangible evidence of demonstrated student learning. Artifacts need to be durable. A report, poster, project, and an audio recording of a presentation can all serve as artifacts.
Resources from New York City