Six weeks into second semester, I was asked to become the teacher for a group of Algebra II students.
How many more of our students feel this way, but instead of telling us with words, they distract, joke, sleep, or skip class:
…Math is such an interesting subject that can be “explored” in so many different ways, however, in school here I don’t really get to learn it to a point where I say yeah this is what I know, I fully understand it. We move on from topic to topic so quickly that the process of me creating links is interrupted and I practice only for the test in order to get high grades.
Taking Time Learning Math:A Student’s Perspective by Evan Weinberg
Would I want to come to my class?
This question haunts me. What are my kids seeing, feeling, thinking? Why does this kid come, but stay totally uninvolved? Why does this child talk, constantly, but about anything but math? Where did curiosity go? Is my class a class I would look forward to?
My personal enjoyment of math comes from the struggle with ideas and the satisfaction I get from my connection of and understanding of the relationships among those ideas. It’s like a huge puzzle that will take the rest of my lifetime to fully understand. The student’s comments in Evan Weinberg’s post resonated with what I see happening with my students. They are not learning math so much as preparing for a test about math.
They are not learning math so much as preparing for a test about math.
The current situation of ‘learn how to do this; learn how to do that’ mentality is slowwwwly changing over to ‘understand why this is so; why does this relationship work’ exploration. It will need a shift in how we teach, letting kids struggle and connect ideas (we must facilitate this exploration, but not down some tightly designed path), and changing our view of grades and mastery. I can’t say I don’t have the answer- I am working on an answer that works for me and for my students. And I’m sure I am not the only one teacher who has found the path that is taking them closer to the ideal.
This post grew out of my response to Evan’s column. His response,
“I completely agree that this is a shift, and it is ongoing. Clearly, despite the changes I’ve made to the way I teach, students still get the sense that the test is the important part, which means there is still a great deal of improvement yet to be made!”
*Taking Time Learning Math:A Student’s Perspective by Evan Weinberg
I am an #mtbos -er because I finally found a place where I b
As teachers, we have the authority to hold our students accountable for their learning. We do these children no favors by feeding them the answers or rewarding them for less than stellar work.
As someone with authority, we are charged also to show our students how to take on that authority in their own life and toward their own learning. Giving students choice and voice is part of that transference of authority. But:
Only those with authority can convey authority to others.
We must first model that which we wish to convey.
How do you handle your authority? Do you give it away by allowing students to misbehave? Do you hold students accountable for missed work? For making up the work missed when they are absent? Do you hold them responsible for deadlines?
Children know instinctively who is in charge. They see our weaknesses and exploit them. They respect authority– well used, not that which is authoritarian!
Over time, after days and days of class and expectation, they will begin to reflect that which you give them. You may not see it, but others will.
My son was always rebelling against my authority, at home. But when speaking with others, I was told of his joy, his politeness, his willingness to help other adults and friends with the chores he did not willingly do at home.
Model authority. Pass this torch onto your students by expecting them to respect your authority, and you will help the next generation to understand what it means, and how to take up its mantle responsibly.
I know that giving students choices in learning works precisely because we are passing on authority!
Could this be why formal “college seat time” is going away, and has already begun being replaced by things like certificate training, MOOCs, the rise of information available on the Internet, and why a resurgence in apprenticeships is already on the horizon (it just looks different, because entrepreneurs are the new apprentices)!
The title is a quote from a treatise on Tim Urban’s blog about Elon Musk. Urban is trying to understand Musk’s success by examining how he thinks, and how Musk’s thinking affects his choices. Read the full 4-part series here.
As a teacher, and as someone who reads widely on any subject I feel I need to know more about, the title quote makes powerful sense. Author Urban calls this ‘first principles’:
‘A scientist gathers together only what he or she knows to be true—the first principles—and uses those as the puzzle pieces with which to construct a conclusion.’ *
For me, first principles is something I’ve always practiced. I just didn’t call it that. I love to learn, to read, to gather information, and test it out against what I know. I learned early not to take what others say without checking and confirming- maybe because as a kid I was rather rebellious (my dad says I have to learn everything the hard way!), maybe because my naïveté allowed me to be made to feel stupid, and I do not like to feel stupid! As I read further into the analysis of Musk, I found a companion to my own thought processes, albeit worded differently than I would have ever thought:
‘ Musk sees… his brain software as the most important product he owns—and since there aren’t companies out there designing brain software, he designed his own, beta tests it every day, and makes constant updates. That’s why he’s so outrageously effective, why he can disrupt multiple huge industries at once, why he can learn so quickly, strategize so cleverly, and visualize the future so clearly.’ *
Urban’s statement about this awakens my “spidey sense”** as a teacher. All of us have hardware (our physical sense) and software (our brains). As a teacher, I feel strongly that this is the goal of my interaction with my kids: to teach them how to learn quickly, to strategize cleverly, and to visualize the future clearly. This is where success, innovation, and fulfillment (yes- doing that which we are so passionate about) must spring from. Our world’s future is at stake if we don’t bring out in our children these abilities.
‘When your childhood attempts at understanding are met with “Because I said so,” and you absorb the implicit message “Your own reasoning capability is sh*t, don’t even try, just follow these rules… ,” you grow up with little confidence in your own reasoning process. When you’re never forced to build your own reasoning pathways, you’re able to skip the hard process of digging deep to discover your own values and the sometimes painful experience of testing those values in the real world and learning you want to adjust them—and so you grow up a total reasoning amateur.’*
Teaching our children the value of ‘first principles’is critical in education. It’s what our children lack: the need to know for themselves, and not rely on what they are told by others: what ‘conventional wisdom’ says must be true.
‘A command or a lesson or a word of wisdom that comes without any insight into the steps of logic it was built upon is feeding a kid a fish instead of teaching them to reason. And when that’s the way we’re brought up, we end up with a bucket of fish and no rod—a piece of installed software that we’ve learned how to use, but no ability to code anything ourselves.’*
My need for ‘first principles’ has been leading me to those ideas that will train my students for their future- one that will require them to code for themselves, going beyond ‘established’ wisdom’, carving out innovative solutions, and finding a future that enthralled and fascinates them. Without this, it becomes easier for a population to become manipulated by a leadership that does not have their best interests at heart. For more on this idea, check out How the Oil Industry Conquered Finance, Medicine, and Agriculture, by James Corbett. In it, Corbett references an essay by Frederick T. Gates, the man intimately connected to the origins of public schooling as we know it. The essay, The Country School of Tomorrow, Gates lays out his plan for education,
‘… we have limitless resources, and the people yield themselves with perfect docility to our molding hand… We shall not try to make these people or any of their children into philosophers or men of learning or of science. We are not to raise up from among them authors, orators, poets, or men of letters. We shall not search for embryo great artists, painters, musicians. Nor will we cherish even the humbler ambition to raise up from among them lawyers,
doctors, preachers, politicians, statesmen, of whom we now have ample supply.’*
From my view, I think we have chillingly succeeded (thanks to Rockefeller and his billions), in the first quoted section of this essay. I look with hope on what is happening as teachers everywhere have begun to break with tradition and seek to teach for understanding, and to teach students to think independently. Read a little further down Mr. Gates’- Frederick’s, not Bill’s- essay, to find the sweet hope and goal of education that somehow became lost in the ‘monetization’ of public schooling,
‘…all that we shall try to do is just to create presently about these country homes an atmosphere and conditions such, that, if by chance a child of genius should spring up from the soil, that genius will surely bud and not be blighted.’
All of our children are budding geniuses- in their own time and in their own way. Finding and nurturing that bud is my ‘action plan’. I think I’ll continue upgrading my ‘software’ in pursuit of being the most effective teacher I can be, sharing what I find, in order to improve our education system in whatever way I am able. Excuse me while I go fill my ‘goal pool’ so that I can plan my ‘strategies’ to maximize my ‘experience’ and ‘feedback loop’ so that I can move my ‘goal attainment mechanism’ forward.
Let’s keep the conversation going!
*The Cook and The Chef: Musk’s Secret Sauce, T. Urban, Wait, Why Not
**No, I was never bitten by a radioactive spider. I am an intuitive woman (which some of you will no doubt see as a redundant phrase…)
‘One colleague suggested turning to the calculator and using the answers as an investigation. Why does the calculator give this answer? What rules is it following? Can you write a set of rules? What would the calculator say for this sum?’by mrbartonmaths
Student misconceptions are critical to planning successful lessons! In a recent series of posts (I think he is up to 11 posts now) mrbartonmaths explains how he uses diagnostic questions to delve into the misconceptions his students have about basic arithmetic.
The quote above came from #10 in the Insight of the Week series: order of operations. As mrbartonmaths explains,
‘…the misconceptions I think students hold are different to the ones they actually make, and I want to put this to the test on a larger scale.’
While I found all of the responses interesting, I was surprised by the number of responses that indicated students were trying to place parentheses (brackets*) into the problem where none existed! These students were trying to make sense of the problem using familiar notation. The only problem I saw is that students didn’t know the ‘rules’ of brackets!
My comment to mrbartonmaths:
Teach them the rules for brackets!
To return to the thought at the beginning of this little essay, ‘have them use the calculator to evaluate the rules the calculator is using’ would require students to identify and investigate their own misconceptions- an idea I find ultimately rewarding!
For a great interactive lesson- which includes some much needed awareness of when parentheses are needed- try this Make This Number game.
As a further step in ‘teachers as lifelong learners’, I love that mrbartonmaths has embarked on something he calls Guess the Misconception, an email poll he sends out weekly to those who are signed up. What misconceptions are you holding about your student’s misconceptions!?!
I am reminded of the time I asked my Algebra I student, who was having a lot of trouble solving basic algebra problems in one variable (3x + 7 = 13 for example), why he kept wanting to start with 3x first. I had spent time working with him on the ‘unwrapping’ idea, without success. He pointed out that he was dividing by 3 because it was first. Headsmack! (Me, not him!)
Don’t assume! (You remember what that does, right? Makes an a– out of u and me!)
Thank you, mrbartonmaths, for giving us a little bit more insight (and some great ideas) into best teaching practices!
Dan Meyer has struck again:
“I spent a year working on Dandy Candies with around 1,000 educators… In my year with Dandy Candies, there was one question that none of us solved, even in a crowd that included mathematics professors and Presidential teaching awardees. So now I’ll put that question to you.” Dan Meyer’s full post
What is it about a challenge like that?!? Of course I had to follow the thread!
I read the over 100 comments as writers posed solutions, wrecked solutions posed by others, and even wrecked their own solutions! I watched as they systematically used the faults in their solutions as a springboard to better – but apparently still breakable – solutions.
I also heard a ghost of an admission that there may not be a single solution,
as timteachesmath writes, “Which broken algorithm is best so far? An algorithm that fails for ‘720’ but works for 95% of really composite numbers less than 720 might be better than one that works for ‘720’ but only works for 80% of really composite numbers less than 720.”
There is a Lesson here!!!!
I teach algenra 1 and geometry; that means 9th and 10th graders. I want to challenge them with Dan’s problem.
It’s simple, right? We are just talking about a box of CANDY!
I can see you now, shaking your head in disbelief: 9th and 10th graders able to frame an answer to a problem that even 1000 math teachers couldn’t solve.
Not only that, I can give this lesson to both algebra AND geometry!
Here is my explanation of the sequence of activities that would make the most sense to their budding understandings of math:
Essential Understanding: The best packaging involves the least surface area.
- The least surface area results from the tightest (closest) configuration of a cube’s side lengths.
- The surface area is a result of the combined areas of the six sides of the candy box.
- To find the minimum surface area for any number of candies, check for the following conditions: a) if the number is prime: 1, 1, the prime; b) a perfect cube: root squared times six; c) numbers with three primes: use the three primes; d) numbers with four or more primes: Multiply groups of the prime factors back together to find three products. These three products will be the three factors that will be the measurements of the box.
- Calculate the surface area from the measurements of the box.
- The box with the least surface area will have the factors that are closest to each other. It is possible for two of the factors to be the same number.
720 is a great example for (d):
Prime factors of 720 are 2, 2, 2, 2, 3, 3, 5
While they can be multiplied back together to create numerous factors, not all sets of three factors will give us minimal surface area.
Some of the sets of three that can be created are:
4, 4, and 45;
8, 6, and 15;
10, 6, 12;
And so on, until we get the multiples 8, 9, 10;
Checking for the optimal area involves a handshake (multiplying) among each of the three numbers – 8 times 9, 8 times 10, and 9 times 10, adding the products together and multiplying by 2.
Does anybody else see the individual lessons embedded in this process? This one problem is incredibly rich!
It’s not the solution, it’s the building of understanding!
The interactive process of doing this by hand is a wonderful opportunity to teach finding primes (6n-1), (6n+1). Students might also feel the need to learn how to find prime factors (and learning that all numbers are products of primes!). The question would arise about the geometry of area vs surface area. (Think of the manipulatives! I wonder of my kids would feel silly stacking cubes of jello!!!)
We also wouldn’t be able to ignore the eminently practical side of saving the planet through minimal packaging – not to mention the extension of how many candies we should pre-package for the best shipping (i.e, how many boxes can fit into a bigger box? Can we afford to package odd sizes and still keep our costs low enough to generate profit and sales?) (ooh! I can teach my kids to design boxes – quadratics, anyone?) Here we could also lead the class into the sales curve (parabolas – more quadratics! I’m in Heaven!)
By Jove! I think I figured out why Algebra and Geometry finally got together! They complete each other!!!
And I love the fact that once my students come to this understanding of the problem, they could begin to write a viable solution, either in algorithm or in code. Or maybe their understanding leads them to the conclusion that a single algorithm isn’t possible – did somebody just whisper the word “proof”? (You did just think that – you know you did!)
Just think of the STEM project ideas this activity could generate…
As many of Dan’s commenters pointed out, this is tedious by hand. But the truth of the matter is – they knew how to begin solving the problem by seeking to understand the problem to be solved! These are the skills our children need to learn. These are the lessons we need to teach. Let’s quit calling them 21st Century Skills; these skills really are useful for any age, anywhere. I’m living proof! (I’ve made it this far on those skills, haven’t I? LOL!)
Let’s bake a cake. You start.
What kind of a cake are we making today? A chocolate cake? Good! I love chocolate!
We need a recipe.
How can we possibly choose! Which recipe will make the cake that we need? Which ones have the ingredients and flavorings we want to use? Which recipe is simple enough for our skill level?) I don’t know about you, but I am not ready for chocolate angel food!) Have any of us ever baked a cake before – any cake?
Is there a recipe that will push us and our students a little beyond what we already know; what we are comfortable with? And let them build on previous knowledge?
Do we set all of the ingredients out for them, or do we let them go to the cabinet and explore what is there to use? Do they have to follow the recipe exactly? Do we hover as they measure, or do we let them experience the trial and error that can be found in cake baking (or math sense making, or working as a group to design, build, or solve a problem together)? Will we need to premeasure for some students, but not for others? Do we know who will need more guidance than others? Or who will need the ingredients for a flourless cake because of food allergies? To pull this off, we will need to settle on a recipe, or two, or three (or maybe let our students find the recipes that they are most interested in).
Decide together how you will each handle supporting each step in the process.
The key word is together. I am a co-teacher with three wonderful, brilliant teachers. Who have already planned their lessons for the current unit. Without me.
I have to go find them to ask about the lessons, and ask about my role in the class, usually the day before, or morning of, the class. I don’t see the materials without asking for them. By not including me in the materials selection and planning process, they have created double work for themselves. This is not how co-teaching becomes its effective best. It also shortchanges the very students it has been created to help.
A co-taught class is a regular education class with students who need extra support in class.
A co-taught class is a regular education class with special education students – who need extra support in class. In this environment, students with special needs have the opportunity to interact with other students in what is called a “least restrictive environment” or LRE. The key to making the LRE work is having two fully certified teachers, one of whom knows what needs to be done to help the students who need their lessons delivered with a little more support.
When one person handles ALL the planning, no provision will be made for the alternative forms of presentation/activities that may be needed to support ALL students in the class with opportunities to learn.
I don’t think this getting left out is being done on purpose.
Each of us has our own teaching style. We may plan our lessons informally, away from school, over the weekend, perhaps. Some teachers plan obsessively (in a good way-getting copies made ahead of time, or planning from the test backward – all excellent strategies). Letting someone else in on the process may feel like leaving oneself open to criticism.
Sticking with the baking analogy, some of us always make a chocolate cake. It’s the cake that everybody eats. It always has chocolate frosting. Everybody gets it, even if not everybody can eat chocolate. And there’s the rub. Regular Ed teachers are not aware of different needs/ limitations/ legal issues of special education. There is a misconception that the way they teach their lessons will work just fine (perhaps they have been successful in this way, and have test scores to prove it) for any student. And if the student isn’t “getting it” the fault is the student’s behavior, or that the student needs to listen more closely or take better notes.
Along comes the special education student.
This is a child who is perfectly capable, mentally, of keeping up with, or even surpassing the other students in the class, except for one thing – they need more processing time, or they don’t compute numbers the way others do, or they have problems focusing, or they learn by doing, instead of hearing, or they need larger print, or they don’t write fast (for notes – so they need pre-printed materials). Because of these reasons, or others like them, this child has been placed in a room with two teachers: the general Ed teacher and Me.
Here’s where I come in. I know the best ways to help these students learn, to get past whatever wall is causing them to need these extra services. I can build these supports into a lesson. I can make sure that there are activities to support these needs, but they have to be designed into the lessons during planning – with both of us buying in to what is going to be done in the classroom.
It does no good for me to walk into the room on the day the lesson is delivered and find out Marie (not her real name) can’t read the lesson worksheet because the teacher has printed multiple copies on a page to save paper and the print is so tiny we can’t tell whether that’s a division sign or a plus sign without checking what the answer is supposed to be! I will have no file on hand with which to print her a larger copy, because I was not involved or copied in on the materials for the class. The larger copy, by the way, is required for her by law, because it has been written into her individual education program document, or IEP. The reg Ed teacher should know this because she has a copy of the accommodations. I would have said something while the copies were being planned- and we would have been prepared.
The regular Ed teacher needs my knowledge for her classroom to be successful. She (or he) needs me to ensure each child is getting information about the lesson in the most optimal way. When a co-teacher is left out of the planning, the unfunny thing is that all children in the classroom suffer, because multiple learning styles are not being represented. All children benefit when lessons are learned in multiple formats. Not everybody likes chocolate cake every day, all the time.
Another benefit of the co-teacher model is that the reg Ed teacher isn’t carrying all the load, teaching the class, grading papers, and struggling with disruptive behaviors.
Is it hard to let another teacher in your space? A recent Education Week article on good co-teacher practices compared it to a marriage. I think it is more like two horses in harness. One cannot lead without tipping the wagon. We must pull together, in step, to get where we are going without upsetting the cargo. There is no room for personalities, and yet, we need to play to each other’s strengths.
Co-teaching takes communication.
Co-teaching requires a learning curve, and it requires both teachers putting all of the students in the room first. It requires letting go of pre-conceived notions about what special Ed students can and cannot do. These students are ours, not those are yours and these are mine. That diminishes our expectation that all students can learn the material. It causes us to view students who require support as being troublemakers. Did you know the statistics? The bottom line is that more special education students get written up for behavior problems in regular classrooms than other students! A well-run and executed team teaching plan can help, both in educating regular Ed teachers on what’s really going on when a student appears to be misbehaving AND reducing the amount of learning being interrupted for all the students in the class.
Co-taught classes often look and feel different than regular classes, for both teachers!
The reg Ed teacher has no experience of students that have a learning disability. For example, they have no knowledge of what teaching strategies are directly helpful for a student who is dyslexic (colored backgrounds for worksheets or PowerPoint displays reduce cognitive workload for these students, and do not negatively affect the regular students either – the different look is often welcomed by all). I have to make my co-teachers aware of the difficulties a child will face before we present the lesson. My responsibility is first the child, but also being a partner to my co-teacher: not withholding valuable information, or not impeding classroom discipline by making unnecessary exceptions, or undermining my partner’s authority.
Planning together allows the process of differentiation to happen without bringing undue attention to the disability.
This is a critical point: we cannot single out the special education student in these mixed classes. While privacy is the main issue, I believe we have a moral obligation to our children, as well. We are charged with creating a safe learning environment. Children do not learn when they are stressing over being or feeling “different.”
Planning together can go a long way towards eliminating stress – for students AND teachers. Like any good teaching strategy, the process takes work and practice! Cut your co-teacher some slack and treat him/her the way you would want them to treat you if you came into their classroom. Hey! That’s not a bad idea. Can we meet in my classroom tomorrow? I’ll need to get in some extra desks….
I get to be the sponsor for my school’s chapter of Mu Alpha Theta. This is an awesome group! Unfortunately, all but three students graduated last year, and one of those transferred to the new school!😩.
First week of school is down in the history books!
Block schedule classes… Continue reading “And How Was Your Day, um… Week, Honey?”