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]]>Have you ever heard someone say that teachers need to lecture more to students in middle or high school so they are prepared for being lectured to in college? I have and I’m guessing that you have too. While I think that people who say this mean well, I think that this belief is misguided at best.
How many people look back at their college lectures and remember them as being the time they best understood what they learned? If you also felt that lectures were a challenging way to learn, why would we want to do more of it?
I asked people on Twitter what they thought about this:
Random vent: I am not a fan of the argument that b/c students will be lectured to in college, we should prepare them by lecturing in HS & MS
— Robert Kaplinsky (@robertkaplinsky) October 31, 2017
You can click on the tweet above to read some of their responses and I’ve shared a few of my favorite below:
A better strategy would be to teach college and university 'lecturers' better pedagogical practices
— Kelly Bairos (@kellybairos) October 31, 2017
One of the most difficult things for me is to go from my math methods class taught which is discussion based to my stat lecture.
— Ms. Cromer (@carmencromer12) October 31, 2017
“Let’s ignore best practices and make sub-par instructional decisions to match subpar instruction they might get in college.”
— Jack Leonard (@JackLeonardEdu) October 31, 2017
Even as an adult, sitting in a full day session of sit & get doesn’t better prepare me for the next time I’m in this same scenario.
— Matt Sanders (@mr_sanders78) October 31, 2017
I’m not saying that there is never a place for lecture. Rather I believe we should be thinking of ways to do less lecture at all levels (including college) rather than acclimate students to it. While universities are held in high esteem, many of the professors who work there are primarily interested in research and have little training in education.
If educators hate to be lectured to in professional development, why should we subject students to it? Might students be better off if we took a more balanced approach so that they had more opportunities to experience and interact with mathematics rather than sitting and listening to it? Is it possible that the best thing we could do to prepare them is actually lecture as little as possible and instead spend more time on discovery, discussions, and active learning?
What do you think? Please let me know in the comments below.
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]]>The post You’ll Never Guess My Most Popular Lesson… And Why appeared first on Robert Kaplinsky.
]]>Recently I was looking at data that Google shares about who visits your website including what pages they go to and how long they stay on your site. I had some hunches about which lessons would be the most popular and I was curious to see if I was right.
SPOILER: I was very wrong.
As it turns out, my most popular lesson by FAR is this one that uses a stick of butter as a context for identifying fractions on a number line. I’ve always liked the lesson and thought it was strong, but this was still surprising. In fact, this butter lesson is so popular that it had more visitors than my second, third, fourth, fifth, and sixth most popular lessons combined.
I wanted to figure out why it was, because I don’t hear about a lot of people using it. I was hoping that looking at more data might provide a clearer picture as to why it was so popular. It did.
What popped out at me was that people who came to this lesson did not stick around. In fact, of the 87,000+ of people who came directly to this lesson in the last year, more than 97% of them didn’t look at any other pages. That was strange. In contrast, most of my other lessons had percentages closer to 60%, meaning that people were more likely to stick around after seeing the other lessons.
So, next I decided to look at what people were searching for in Google that took them to my site. This is when I could not believe my eyes. Here are the top ten Google searches that took people to my website over the last year:
Of the top ten searches that took people to my site, SIX of them were people searching for information about how much a third of a cup of butter is! The picture this was painting was too funny to believe, so I had to do one more check. I went to Google. I searched for “1/3 cup butter?” Guess who you’ll find?
Yep. It’s my site. My best guess is that people are not sure how to determine how much one third of a cup of butter is. They go search on Google, find my site, and then either figure it out or have a “What the heck is this? Is this a math lesson!?” type reaction and immediately leave.
I had to laugh. Maybe I should also cry.
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]]>The post What I Wish Teachers Knew About Living In A Group Home appeared first on Robert Kaplinsky.
]]>I’m writing this blog post to get people thinking about how they support their students living in a group home or foster care. Few educators have personally experienced this or deeply understand the experiences of someone who has. So, here’s what I wish educators knew about teaching youth who live in foster care or a group home.
I write this as a person who lived in a group home for 3.5 years, from the middle of 9th grade until my first day at UCLA. If you’re interested in learning more about that part of my life, I shared it here in my short ShadowCon talk.
While I hope to be helpful and provide context and strategies that are applicable to all students, I can’t emphasize enough that my thoughts are extremely biased by my own experiences. I’ve broken this post into three parts: relationships, behavior, and academics.
I don’t know how to say this subtly, but in the time that I lived in the group home, two residents committed suicide. I wasn’t close to either of them, but when that is part of the environment you live in…
Building and maintaining relationships with the adults was also hard, but for a different reason. While most were very kind, ultimately it was a job for them and there were at least 20 people who were my guardian in 3.5 years. In that kind of situation, it feels like no one understands you and it makes you hesitant to open up because that person will be gone soon enough. Can you imagine how much I valued my relationships with the teachers who made an effort to connect with me? They were the most stable people in my life.
Making friends at school was also challenging. Maybe I shouldn’t have been, but I was embarrassed to live in a group home and I tried to hide it. Living in a group home also limited my ability to get together with these friends outside of school. I wouldn’t let them come to where I lived and there were also significant restrictions like non-negotiable bed times and limited free time to leave the facility. This made it hard to spend time together and made you feel like an outsider.
I remember also being embarrassed about my clothing. When I came to the group home, the entirety of my possessions were the clothes I was wearing and what fit in my backpack. Initially, I didn’t want to go to school because I would have to wear the same clothes multiple times a week and didn’t want to be teased. I didn’t tell anyone that. I just seemed like a kid who refused to go to school. Obviously this issue did get resolved, but I think it speaks towards where students are coming from. Maybe they have bruises they want to fade away. Maybe they haven’t gotten a haircut in a long time. Maybe all their clothes are hand-me-downs. The reality is that these students may appear to be defiant or non-cooperative, but this could be a symptom of a problem you wouldn’t suspect when you haven’t walked in their shoes.
Holidays were rough while living in the group home and even afterwards. It really made you self-conscious as it seemed like everyone else was excited to spend time with family, go on vacations, or receive gifts. Many kids living in group homes become sad because they feel left out or recall easier times from their childhood. For me, this actually continued into college. People would go home for the holidays like Thanksgiving and I wouldn’t have anywhere to go. For spring break I could at least stay in my dorm, even if there wasn’t food service. However for winter break I had to leave the dorms and sleep on my friends’ couches for two weeks. As a result, when I rented my first apartment during my junior year, I appreciated it far more than most. So, when you’re doing a holiday celebration and you’re thinking everyone is feeling happy, they may not be. Remember, if school is where it feels safe, then the last thing they want is to be in the group home for the next two weeks. Check in with them to see where they’re at.
Other things I wished I had in retrospect were more mentors. Remember that for a lot of students living in group homes or foster care, there aren’t role models to guide them. Going to college may not even be on their radar because it’s hard to think about the future when you’re concerned with just getting through the day. I really needed someone to build a relationship with me at school that lasted through my time there. Maybe someone who would have advised me on which courses to take to be competitive or even just basic life advice that guardians tend to pass on to their kids. While I love my life now, I occasionally wonder about how much farther I could have gotten with the right environment.
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]]>The post Empowered Problem Solving Workshop appeared first on Robert Kaplinsky.
]]>I’ve got some exciting news to share! I’ve created an online workshop to help mathematics educators implement problem-based learning and higher Depth of Knowledge problems and I want to tell you more about it. I first ran the workshop in September 2017 and had over 200 educators from all over the world participate including the United States, Mexico, China, Vietnam, Spain, Qatar, and Canada. You can also earn up to four units of graduate level professional development credits through a partnership with Brandman University, so it will help you move over on the pay scale.
Before I get into the details, if you’d prefer to listen to me talk about my workshop, click on the image below:
At first, I thought that maybe I was just lazy for not incorporating the new ideas. But I see it differently now. The real reason was that I didn’t believe what I was learning was worth the effort to incorporate it. These ideas didn’t seem that much better than what I was already doing and I didn’t understand why I really needed them.
And that is the problem I see with other trainings. I believe that much of the professional development that educators receive is not effective because it primarily focuses on WHAT you need to do and HOW to do it, but little on WHY you would ever want to.
As a result, what you learn winds up on a list of things you hope to get to one day… a day that may never come. What I believe is missing from many workshops is the why. Why should you teach this new way? Why did the old way not work?
Believing in the why takes a training from “This is nice, I’ll get around to implementing it when I have time.” to “This is absolutely critical to implement right away. I cannot keep doing things the way I’ve always done them.” In this workshop, I definitely cover what you need to do and how to do it, but I also spend time focusing on why we need to make changes. When you finish this workshop, you’ll better understand why these strategies are worth investing time on.
My online workshop is six weeks long and has six sections, called modules. I’ll explain each of them now by telling you what it’s called, how it will help you, and what you’ll learn.
Module 1 – Introduction to Problem-Based Lessons
How will this help me?
How do problem based lessons help students develop deeper understandings of mathematics?
What will I learn?
Module 2 – More on Problem-Based Lessons
How will this help me?
How do I ensure that my students become true problem solvers and not just math robots?
What will I learn?
Module 3 – Preparing for Problem-Based Lessons
How will this help me?
How do you go from seeing a problem-based lesson you like to one that you are prepared to use?
What will I learn?
Module 4 – Dealing With Worst Case Scenarios
How will this help me?
Will you know what to do if your lesson doesn’t go the way you plan and students struggle?
What will I learn?
Module 5 – Depth of Knowledge in Mathematics
How will this help me?
Would you like to challenge all your students and spot hidden misconceptions?
What will I learn?
Module 6 – Advanced Problem-Based Lesson Tips
How will this help me?
How will you integrate these strategies into your curriculum?
What will I learn?
Each Monday morning I release a new module so that every workshop participant is focusing on the same content at the same time. I’ll also send you a weekly reminder email to let you know which workshop resources you’ll want to print out ahead of time, and point you towards a short overview video where I cover what you’ll learn in the module, how you’ll learn it, why it’s important, and where you might get stuck.
The workshop costs $297. This is less than the registration cost for many large conferences or university teacher education courses. This includes all the electronic resources and handouts you’ll need for the workshop.
Also, if you work at a school where you are only allowed to teach what is in the textbook, then then this workshop may not be a good fit for you. I will be sharing strategies that will supplement or replace parts of your textbook, and without that flexibility, you may not be able to implement these strategies.
Also, people who have already attended one of my full day workshops will find that this workshop covers much of the same content you’ve already seen. So, unless you’re thinking of taking it as a refresher, I wouldn’t take this workshop. If you’re not sure if this applies to you (for example, maybe you attended one of my 3-hour workshops), feel free to email me at robert@robertkaplinsky.com and we can discuss potential overlap.
This workshop will help you implement these strategies with your own students, get them to enjoy problem solving, and provide you with techniques that will push kids to keep trying when they would normally give up. It will help you both prepare for facilitating classroom conversations and also help you handle uncomfortable situations, like when a student solves a problem in a way you don’t understand.
Maybe you’ve wanted to incorporate some new ideas and techniques, but you already feel like you’re out of your classroom way too much as it is. Maybe you’ve found some workshops you’d love to attend, but the costs are too high when you include travel expenses. Maybe you’re looking to earn professional development credits to help you move over on the pay scale, but many of the university courses don’t seem useful.
If any of these situations sound familiar, then my online workshop could be what you’re looking for. With my workshop, you don’t have to be out of your classroom another day, there are no travel expenses, and you can earn up to four units of graduate level professional development credits from Brandman University, though that has an additional cost and requires additional coursework.
First, I’ve made a problem-based lesson planning checklist. To give you a sense of why this would be valuable, if you had to use a problem-based lesson in your classroom tomorrow, would you know what’s essential to get done so you’re ready to go, or what would be very helpful to do if you have more time? I’ll break down everything you’ll need to do to in a PDF you can have handy when you need it.
Second, I’ve made a problem-based lesson troubleshooting guide for the eleven most common worst-case scenarios when using a problem-based lesson including what to do when:
I realize that in those situations, you’re not going to want to look back at your workshop notes. So, I’ve written up the strategies I use in these situations into a PDF you can keep close by. One part breaks down the issues that happen before students start working on their own and the other part covers the ones that happen after they start working on their own.
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]]>The post Never Ask A Question A Horse Could Answer appeared first on Robert Kaplinsky.
]]>If you’ve ever asked students a question only to have tentative responses come back like “Yes… No… Uh… Yes…”, then you’ll enjoy this story.
In the early 1900s, there was a famous horse named Clever Hans that amazed people because of his intelligence. At first Hans was able to count objects and then tap his hoof to state the number of objects. So, if he saw 4 objects, he would tap his hoof 4 times. Over time his owner Wilhelm von Osten was able to teach him how to add and subtract. Then he was taught to “multiply, divide, work with fractions, tell time, keep track of the calendar, differentiate musical tones, and read, spell, and understand German.”
This amazed audiences of people who came to observe this feat. Obviously, people suspected fraud, because how could a horse do something like this? Here’s what happened:
As a result of the large amount of public interest in Clever Hans, the German board of education appointed a commission to investigate von Osten’s scientific claims. Philosopher and psychologist Carl Stumpf formed a panel of 13 people, known as the Hans Commission. This commission consisted of a veterinarian, a circus manager, a Cavalry officer, a number of school teachers, and the director of the Berlin zoological gardens. This commission concluded in September 1904 that no tricks were involved in Hans’s performance.
They even ruled out fraud by writing the questions on paper, having someone else ask the questions, and removing von Osten from the room. For a while, no one could figure out how the horse was able to answer these questions.
Eventually the breakthrough came when they realized that Clever Hans “got the right answer only when the questioner knew what the answer was, and the horse could see the questioner.” If the person asking the question knew the correct answer, Hans correctly answered the question 89% of the time, but when the person asking the question didn’t know the answer, Hans got it right only 6% of the time!
[Oskar] Pfungst then proceeded to examine the behaviour of the questioner in detail, and showed that as the horse’s taps approached the right answer, the questioner’s posture and facial expression changed in ways that were consistent with an increase in tension, which was released when the horse made the final, correct tap. This provided a cue that the horse could use to tell it to stop tapping.
Furthermore…
After Pfungst had become adept at giving Hans performances himself, and was fully aware of the subtle cues which made them possible, he discovered that he would produce these cues involuntarily regardless of whether he wished to exhibit or suppress them.
I love this story for so many reasons. First, when I heard it, I was consistently wrong in guessing what the trick was. I figured that the owner was intentionally communicating the answers to Hans. Then when I heard what was really happening, I started thinking about where else the Clever Hans Effect might happen. For example, a UC Davis study has shown that the handlers of drug sniffing dogs can influence their ability to spot drugs.
I believe that this is also happening in our classrooms. Going back to the beginning of this blog post, could it be that we are doing the same thing with our students? Sometimes I feel like we ask students questions and then stand there waiting for them to tap out the answers with their hooves. Rather than really thinking about what we’re asking, could it be that they’re just reading our body language and adjusting their answer until it appears that they answered it correctly? It sure seems that way when we get responses like “Yes… No… Uh… Yes…”
Thinking about how to mitigate this possibility, perhaps instead of practicing our poker face to prevent this from happening, we should instead be thinking about how to ask questions that Clever Hans could not answer. When we primarily ask yes/no questions or ones with numeric values, we set ourselves up for these situations.
I’ve found that most questions which begin with “How…?” or “Why.. ?” get kids talking about what they know and would be unanswerable by Clever Hans. It takes practice to incorporate them, and if you’re interested in some professional development tools to do so, check out these free questioning scenarios which allow you to practice questioning in trios using role-playing.
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]]>The post How Do We Know When Kids Understand? appeared first on Robert Kaplinsky.
]]>You’ve probably heard that there are tournaments for board games such as Scrabble. These winners are often held in high esteem for their intelligence, knowledge, and strategic thinking which they use to propel themselves to victory. They are thought to be experts of the languages they use… but I wonder if that is always the case.
Consider the story of Nigel Richards. He won the 2015 French Scrabble championship… which doesn’t sound that remarkable, except for the fact that Nigel does not speak French.
So what do you think about this? Does Nigel understand French? Does it matter?
You can make the case that, “Who cares if he understands French? He won.” So, yes, if your goal is, “Win the French scrabble tournament.” Then you are right, it doesn’t matter.
However, if you’re goal is to speak and understand French, the scary possibility is that he may appear to be a French speaker by various assessments including a French spelling test.
For me, this makes me reflect on what it means to understand mathematics. We can’t assume that assessing students on procedural knowledge is enough to demonstrate understanding. We also need to assess deeper understanding by using problems like the ones on Open Middle or my real world problem-based lessons.
What do you think? Let me know in the comments.
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]]>For as long as I can remember, people have been fascinated by robots that can think for themselves and be “intelligent.” This has amazing potential but also comes with some concern that robots will think for themselves and no longer be controllable. In extreme cases, we imagine possibilities like you find in Terminator 2.
Part of this is a curiosity about when the line will be crossed from unintelligent to intelligent. People have debated various tests to measure when this line has been crossed. I want to share two of the most famous examples and focus on one of them, as I think it has application in mathematics education as well.
Perhaps the most famous test is the Turing Test, created by Alan Turing in 1950. You may remember Alan Turing as the main character from the movie The Imitation Game. He proposed a test where a human would write a question which would then be given to two recipients. One recipient would be another human and the other a computer. Those recipients would send back a response, and if the person who asked the question could not tell which was from a human and which was from a computer, then the line of intelligence would be crossed.
Another test, from John Searle, is the Chinese room. In this thought experiment, imagine a man who does not speak Chinese sitting in a room. He has boxes of Chinese characters and a book which gives him a list of the characters he might receive and what character he should send back in return. So, a native Chinese speaking woman comes up to the room, writes a character on a piece of paper and slips it under the door. The man inside picks it up, looks up what he should do in the book, and slides a character back to the woman outside.
From the perspective of the woman outside, she asked “Do you speak Chinese?” and received the response, “Yes, fluently.” So, from her perspective, the man inside the room understands Chinese. However, from the perspective of the man inside the room, he has no idea what he was given or what he sent back. He would likely say that he does not understand Chinese.
This makes me think about how the Chinese room thought experiment might apply to math education. Consider my own experience. I have a Bachelors of Science in Mathematics from UCLA. So, that proves that I understand college level mathematics. Or does it?
Unfortunately for me, the majority of my middle, high school, and college mathematics experience felt like I was in the Chinese room. My professors handed me numbers to use. I plugged them into a formula. I gave back the corresponding value. From their perspective, I understood mathematics (or at least understood enough to pass the class). From my perspective, I often had no idea what I was doing. I was essentially taking in a character, looking it up in a book to see what character I should send back, and sending it back.
I am concerned that we may be creating our own Chinese rooms in education when we mistake students who give us correct answers to our problems with students who are “intelligent” and have deep understandings of mathematics. I have more questions than answers at this point:
I’d love to hear what you think about my questions or whatever else you’re wondering. Please let me know in the comments.
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]]>The post I Hope You’re Embarrassed appeared first on Robert Kaplinsky.
]]>When I look back at my first years of teaching, I’m shocked that I wasn’t fired. I don’t mean that figuratively. I mean it literally. Times were a bit different in 2003. When I was hired, I was the third teacher of the year for those students. The first teacher quit and the second was fired.
I had no business being a teacher. I was an unemployed programmer after the dot com crash trying to find a job. I was hired with an emergency credential and had never taken a single education class nor did any student teaching. I did have a math degree, loved my students, and put a lot of heart into what I did, but basically the school needed a body and I needed a job.
I had very little idea about what I was doing. I lectured to middle school students, pretty much only taught procedures, and used far more worksheets than I’d like to remember. While I was improving and working hard (I was at school from 6 am to 6 pm), it didn’t seem like I was growing fast enough.
Reflecting now, I’m embarrassed. I wish I could have done better for those students. I wish I knew then what I know now. It’s a hard feeling to deal with.
Strangely enough, I’m happy that I’m embarrassed. The reality is that if I’m embarrassed about how I used to teach, it probably means that I’ve improved. If I was still teaching the same way as I had been back then, then I likely wouldn’t have the perspective to realize that my methods were not ideal. Weird, right?
This isn’t just for the first years of your career either. I am embarrassed by things I’ve done throughout my years in education. Sometimes it was as a teacher of students and sometimes as a coach of teachers. But again, these are good things because they have to mean that I have grown so much that I can look back with the perspective of someone who has improved.
So, if you look back at how you used to teach and feel embarrassed, congratulations. It means you made progress.
Of course, this also leads to some other thoughts:
So while much of this post is somewhat tongue-in-cheek, I think it is important for educators to reframe something that might appear negative and turn it into a positive.
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]]>The post Here’s How You Can Have An Extra Month To Teach Students appeared first on Robert Kaplinsky.
]]>Most teachers believe that there is not enough time to teach all their grade level standards in a single year. I’m guessing that this doesn’t come as a surprise to you. So, the question then is what do we do about it?
I think that the most common reaction is to want to skip certain topics and standards. For example, a teacher might skip sections from geometry and/or statistics. To be honest, I’ve definitely done that. It isn’t a good feeling, but it seems like you often run out of time to teach it all. So, you might proactively skip certain parts.
However, what if I told you that there might be another way to do it where you wouldn’t have to skip topics because you’d have the equivalent of a whole extra month of time to teach students?
I remember how I used to spend the first five to ten minutes of class going over the previous night’s homework. Now, let’s put aside the reality that many kids didn’t even do their homework and got nothing out of the experience. Let’s also put aside the possibilities that students who did do the homework may not have gotten anything out of the experience.
Now, let’s think about the total time spent on homework. 5 minutes per day x 140 days per year (assuming that there isn’t homework every day) is 700 minutes a year on reviewing homework. At 10 minutes per day, we’re at 1400 minutes per year. Assuming a 50-minute period to teach math, we’re talking 14 to 28 days per school year were being spent reviewing homework! Imagine reclaiming that time for other things! It wouldn’t be in those large chunks, but you could re-purpose it for something like problem solving or number talks.
It doesn’t have to end there. Matt Vaudrey has shares some wonderful ways he uses music to make his classroom more efficient. If you’re wondering about when you could use music, he explains simply:
Think about the stuff in your class that takes longer than you think it should. A music cue could smooth out that transition. My students also appreciated a “talk to your neighbor” song for several reasons:
- It mandates “wait time” for the teacher; I can’t call on anybody until it’s over.
- It allows students with language needs or disabilities time to process the prompt and think out a response.
- It provides squirrelly students a chance to get out of their seat and chit-chat. Even if they burn through my prompt and talk about something else, they’re more likely to focus after the song ends.
He goes on to share a list of all the transitions for which he used music, which includes tasks like “clean off your desk” and “take out notebook and turn to page ___.” How much time (and stress!) would you save each day if students completed these tasks quicker? Could you save 1 minute each day? That’s over three extra days of time per year.
Matt’s point (and mine) is that small increases in efficiency have huge impacts when scaled out over an entire year. So, perhaps it’s time to take another look at what you’re doing in class and think about areas that can be tightened up.
If you’ve tried any of these strategies or have some of your own to share, I want to read about it. Please let me know in the comments.
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]]>The post Read This If Teachers At Your School Are Burnt Out appeared first on Robert Kaplinsky.
]]>I want to share another lesson (here was the first) I learned from Stephen R. Covey’s book, The 7 Habits of Highly Effective People. Dr. Covey defines effectiveness as a balance of production and production capacity (“P/PC balance”). To explain what he means by that phrase, he shares Aesop’s fable of the goose that laid the golden eggs.
The fable is the story of a poor farmer who one day discovers in the nest of his pet goose a glittering golden egg. At first, he thinks it must be some kind of trick. But as he starts to throw the egg aside, he has second thoughts and takes it in to be appraised instead.
The egg is pure gold! The farmer can’t believe his good fortune. He becomes even more incredulous the following day when the experience is repeated. Day after day, he awakens to rush to the nest and find another golden egg. He becomes fabulously wealthy; it all seems too good to be true.
But with his increasing wealth comes greed and impatience. Unable to wait day after day for the golden eggs, the farmer decides he will kill the goose and get them all at once. But when he opens the goose, he finds it empty. There are no golden eggs — and now there is no way to get any more. The farmer has destroyed the goose that produced them.
So, in this case the golden eggs are the production (P) and the goose represents the production capacity (PC). You need a balance of both of these to have long term success. Focusing just on the golden eggs at the cost of the goose will leave you without the ability to receive more.
Well not so much for the factory and whoever replaces her. During this time period, the focus was so heavily on production that production capacity was hurt. Maybe the other factory employees are burnt out, machines have not been maintained, and in general the factory has lost some of its long term ability to produce. That’s not good.
Sometimes, it feels like districts are so focused on production, that they neglect production capacity. I have seen many districts where so many new initiatives are adopted that is seems like teachers are implementing every program under the sun. Sure, initial results may seem great. But it often results in a staff that is burnt out from initiative overload.
I’ve literally seen districts where multiple administrators quit mid-year or get life threateningly sick from the stress. At some point you have to take a step back and think about whether you no longer have a balance of production and production capacity.
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]]>A homeowner tiled her floor with pennies to make the beautiful design seen below.
These questions may be useful in helping students down the problem solving path:
This beautiful penny-tiled floor was created using approximately 13,000 pennies. That information is provided to us by the homeowner and the actual dimensions needed to calculate the total number of pennies are not provided. So, while at first glance this may appear to be a problem about area, the real focus is on determining how much the value of 13,000 pennies is.
There are multiple ways to approach this problem including multiplying 13,000 pennies x $0.01 per penny for a total of $130. Another way would be to set up a proportion where ($x / 13,000 pennies) = ($0.01 / 1 penny).
Where this problem gets interesting though is what happened when people posted about it on Twitter. Someone initially shared the story and people loved it…
This person used 13,000 pennies to create this peng floor pic.twitter.com/t1yNGQObLd
— Fatss (@Fatimaa__z) December 2, 2016
However some people reached a different conclusion about whether it was worthwhile because of how much they believed it cost (“smh” means “shake my head”)…
Jokes on him because now he just wasted $13,000 smh. https://t.co/RX8Es3t0Uk
— dean. (@vxrnvn) December 2, 2016
This person (and several others) calculated the pennies’ value to be $13,000 instead of $130, resulting in comical blog posts being written up. This tweet provides us with a great way to challenge students’ ability to articulate their reasoning and intellectual autonomy. I suggest that when you first show students the “correct answer”, you show them the wrong answer where it was calculated to be $13,000.
Try your best to keep a straight face and see what happens. Are they dying to explain why they are right and he is wrong? Or, even worse, do they stop believing in themselves and think he is actually correct? Either way, it should give students an opportunity to implement Math Practice 3 where they need to construct a viable argument and critique the reasoning of others. It also gives you the opportunity to have a conversation about the importance in believing in yourself and being able to articulate your reasoning.
Ultimately, you can end with the tweet below where someone asks Twitter to “never change” because it provides her with such good laughs.
I'm taking a mental break from data analysis class and see someone on here stating that 13,000 pennies = $13,000. Never change, twitter.
— Cher Underwood (@TrixieLadelle) December 3, 2016
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]]>Rapper Nelly owes money to the IRS for failing to pay his taxes. Someone came up with the clever idea that they could help him pay his debt by playing his songs over and over which would earn him money.
These questions may be useful in helping students down the problem solving path:
This lesson uses Nelly’s debt as a context for dividing decimals. It begins with students having to determine what information they will need. This should include finding out how much money Nelly owes the IRS and how much money he makes every time a song is streamed on Spotify. We’re also ignoring that he may earn money from other sources. Once students determine that this information is needed, show them this screenshot from a CNN article.
The actual amount Nelly owes is a bit higher than the rounded amount shown in the article ($2,412,283) (see TMZ). Now with this information in hand, students can divide the $2,412,283 debt by $0.006 (the low end per stream) to find the highest number of times the song would need to be streamed and by $0.0084 (the high end per stream) to find the fewest number of times the song would need to be streamed. Conceptually, we are trying to see how many $0.006 (or $0.0084) we would need to have $2,412,283.
The leads to the results of about 402,047,166 times at the most and about 287,176,547 at the least. Now students are ready to compare their answer to the reported answer.
If students wonder why the exact answer was not reported, it is a good opportunity to talk about rounding when communicating information.
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However, that would be wrong. This is the actual quantity of each type of chips.
My hope is that after students have a conversation about each strategies’ pros and cons, they will conclude that surveying their classmates will enable the fairest results (and conveniently incorporates CCSS 7.SP.1 and CCSS 7.SP.2). Here is an example of how this might go in class once they know they need to take a survey. Start by showing them a picture of the chip package with the quantities blanked out:
Next, have students vote for their favorite type of chips and collect the results. Let’s say that in a class of 36 students you get:
Accordingly, the favorites breakdown as:
With 54 bags in a package, a distribution of chips that matches the class’ preferences (based on the voting percentages) would be:
It is important to note that the breakdown above includes a half bag of Doritos (Cool Ranch) and a half bag of Lay’s. It is worth having a conversation about there being an extra bag and how to decide what happens to it. The reality is that there would not likely be half bags and a decision would have to be made about which flavor would get an extra bag.
Once students have reached their conclusion for how many of each type of bag there should be, show them the actual number of bags in the package:
In this example, all of the flavors had a different quantity of bags. It is important that kids have time to reflect on why the answers are different. Some may think that they have miscalculated or had a mathematical error. Make sure to facilitate the conversation towards realizing that their classroom may not be a representative sample of the preferences of people buying the chips.
Here are two extensions to consider:
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Where the math gets interesting is figuring out what password requirements make them more “complex,” or more specifically, are most effective in increasing the total number of possible passwords a hacker has to try. For example, if you took the previous scenario and now allowed lowercase and uppercase English letters, there are now 52 possible passwords (“a” through “z” and “A” through “Z”). However if you instead required passwords to be two lowercase letters long, there would now be 676 (26^2) passwords “aa”, “ab”, “ac” through “zy” “zz”). That is because there are 26 passwords that begin with “a”, 26 more passwords that begin with “b”, etc. all the way through “z”.
Potential password requirements include:
You may want to begin with students working as a whole class to agree on a certain set of password requirements and then calculate the total number of potential passwords based on those requirements. There are a number of methods available to figure out the total number of potential passwords. A tree map is very effective at developing conceptual understanding when there are very, very few possible characters. Permutation formulas will work as well. Eventually they will lead to a generalized formula to figure out the number of potential passwords similar to:
(number of possible characters) ^ (number of characters in the password)
For example, if a password has 8 characters and lowercase, uppercase, and numbers are the only characters allowed, the total number of potential passwords would be:
(26 lowercase characters + 26 uppercase characters + 10 numbers) ^ 8 characters in the password
(26+26+10)^8 = 62^8 = 218,340,105,584,896 ≈ 218 trillion potential passwords
Perhaps it goes without saying, but this is not the time to have students calculate the totals by hand. The time spent calculating totals by hand will reduce the time students have to consider how the requirements affect the total number of potential passwords. Once there is understanding and agreement as to how to calculate the total number of potential passwords, let students play around with the different password requirements to see how they affect the total number of potential passwords. For example:
At this point, students may have a false belief that the seemingly large total number of potential passwords implies that the passwords are secure. So, have them guess how long they think it would take for a computer to guess their own password. You may need to ask them for low and high guesses if they have trouble coming up with an answer, allowing them to say, “My low guess is at least a year and my high guess is a century.”
Where the lesson starts to get very real is when you show them the article (click on the “Download files” button towards the bottom) about how quickly computers can check passwords. The image below shows the first page of the article with a computer that can “cycle through as many as 350 billion guesses per second.” That adds a serious dose of reality because while 218 trillion potential passwords seemed like a lot, at 350 billions guesses per second it can be hacked in less than 11 minutes.
Have them now calculate the amount of time it would take to guess a password with the requirements they came up with earlier. Depending on what they chose, they may be surprised that many of them can be guessed in less than an hour. Give them additional time to adjust their password requirements. Eventually they should come to the conclusion that the single most important factor in making a password more complex is the password’s length. For example, the password “ThisIsMyLongPassword” has 20 characters and uses only lowercase and uppercase letters. Accordingly, to calculate the total number of potential passwords that have the same requirements, you get 52^20. There are approximately 2 x 10^34 passwords with those requirements and it would take 1.8 quadrillion years to crack it via brute force on that computer.
This Desmos graph has adjustable sliders to shows a visual representation of how the total number of potential passwords change based on four requirements:
It is worthwhile to note why this type of hacking is not more common in real life:
Finally, one sobering fact is that the article on the password guessing computer was written in 2012. You can be certain that it won’t be long before 350 billion passwords per second is considered slow.
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In the movie clip there are 8 hot dogs and 12 hot dog buns in a package. If he doesn’t want to have any extra hot dogs or hot dog buns, he will need to buy 3 packages of hot dogs and 2 packages of hot dog buns for a total of 24 hot dogs with buns. Something to keep in mind is that students may give the answer of 24, which is the least common multiple of 8 and 12. However the question is specifically asking about the number of packages, so anticipate needing to have a conversation about what units we are measuring with.
Below I have provided images with examples of hot dogs and hot dog buns in various packaged quantities to use as an extension.
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You will notice that some foods do not neatly tally up to 2000 calories so you can assign different foods accordingly.
One last thing to consider when doing this problem was brought up by Lucam Chups and is shown below:
Check out Merryl Polak’s implementation of this lesson with her seventh grade students and her reflections on how it went.
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]]>These questions may be useful in helping students down the problem solving path:
This lesson uses the coinstar change counting machine as a context for decimal operations. The coins in the video of the boy using the machine are not the coins we are actually counting and is only provided for students who are unfamiliar with the machine.
After showing students the video and/or talking about the machine, provide students with the problem solving framework and establish that the question we are trying to figure out is, “How much money are the coins worth?” We want students to realize that they need to know how much of each coin there are before providing them with that information. You can accomplish this by having them take a guess for the total, which should bring out the necessity of this information. The picture below has the total number of each coin.
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]]>These questions may be useful in helping students down the problem solving path:
The image below shows a woman using a Coca-Cola Freestyle machine. The drink dispenser allows you to take Coca-Cola soft drinks and add flavors to them including:
Not all soft drinks have all of those flavors available as options as most let you only pick a few of them. The image below shows all of the available options for people to choose from on most Coca-Cola Freestyles (a few locations have an additional customized drink). Note that the large circle above each of the sets of flavor choice is also counted as that is the regular flavor.
How you modify the task at this point will determine whether what grade level standards apply. By asking students “How many soda combinations are there on a Coca-Cola Freestyle?” it gives students a context for composing and decomposing numbers. Strategies students may use include (in order of complexity):
I would definitely look for these three strategies as well as other creative strategies and think about the order you have students present so that the answers build upon each other. Ultimately the total number of flavors comes to 125 flavors. The image below, from Boston.com validates their counting.
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]]>The post What Should The Freeway Sign Show? appeared first on Robert Kaplinsky.
]]>A freeway sign is missing the distances to the upcoming exits.
These questions may be useful in helping students down the problem solving path:
This lesson is designed to get students arguing about mathematics by using a rich context. The basic premise is that students will compare the freeway sign to the corresponding map to determine the distances that should be listed. Many assumptions will need to be discussed as these non-driving students may not have enough experience to know that:
Other issues that are guaranteed to come up during the lesson include:
Also, students may find it useful to make a fraction number line that lists all the possible fractions (and their decimal equivalents) on a freeway sign including 0, 1/4, 1/3, 1/2, 2/3, 3/4, 1, 1 1/4, 1 1/3, 1 1/2, etc. so that students can have a tool to help determine which fraction is closest to the value they came up with. Look out for students who extend their number line towards the negatives. Since distance cannot be negative, the values should all be positive. Perhaps we should call it a number ray.
This map involves a reasonably horizontal freeway with perpendicular streets running across it from north to south. Note that Campus Ave has a very peculiar path and appears to go around something as it approaches the freeway. The other two streets, Carnelian St and Archibald Ave are straighter on this map. The scale to show the length of 1 mile is on the bottom right of the map picture.
Using the two images above, I chose to use centimeters and measured one mile to be 3 cm. I measured the distances from the sign as follows:
Since every 3 cm is one mile, then:
Next we need to convert those distances to mixed numbers with denominators of 2, 3, or 4:
The image below shows the original freeway sign. Interestingly, none of my answers match the sign. I would expect similar results from students. It is now a great opportunity to use Math Practice 3 and “construct [a] viable argument and critique the reasoning of others.” A lively conversation should ensue. Honestly, I don’t know why my distances are different. I reason that it could be a combination of:
At the very least, have students defend their answers and state why the sign should be changed. Students could also debate each other if they have different distances.
This map has a freeway running diagonally from the northwest to southeast. The other freeway (Riverside Fwy 91) that runs from east to west is not the freeway we are using. Note that there are four streets (Magnolia St, Brookhurst St, La Palma Ave, and Euclid St) and only three distances on the freeway sign. That is because Brookhurst St and La Palma Ave have the same distance listed. Every street except La Palma Ave runs from north to south. La Palma Ave is not labeled on the map but it runs from east to west and intersects Brookhusrt St perpendicularly where the “5” is located. The scale to show the length of 1 mile is on the bottom right of the map picture.
Using the two images above, I chose to use centimeters and measured one mile to be 3.6 cm. I measured the distances from the sign as follows:
Since every 3.6 cm is one mile, then:
Next we need to convert those distances to mixed numbers with denominators of 2, 3, or 4:
The image below shows the original freeway sign. Again, none of my answers match the sign and I would expect similar results from students. It is still a great opportunity to use Math Practice 3 and discuss whether the answers are off because of a combination of:
At the very least, have students defend their answers and state why the sign should be changed. Students could also debate each other if they have different distances.
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