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
This statement from STEAM: How art can help math students really struck a chord for me:
“The big point of it is that maths is really not about formulas, it’s about finding patterns in things, and how they influence people. And that’s a big part of art,”Year 12 student Maxim Adams
I am currently teaching geometry (among other maths!) and we are in the sweet zone of shapes: triangles lead to parallelograms which lead from 2D to 3D to the idea of 4D (think twisted paper*). All of this incorporates the effects of angles on shape which affects (and effects) area. Some shapes are more ‘beautiful’ than others. Some shapes and forms lead us to wanting to know more…
I use paper folding to illustrate these shapes and connections. Patty paper is very light and a great beginning medium to illustrate the connections between shape and angle, how changing the angles changes the shape. Origami fascinates me, and I use it to fascinate my geometry students, and to lead them into exploring the relationships found there. The zig zag origami is complex and beautiful, and it takes a bit of time to teach, but as my students learn to fold, we are talking about math. Even their mistakes, when they have to go back and refold, teach them about the shapes and connections. And as STEAM director Melissa Silk says,
“I think it needs to be embedded from a very early age…When I’m teaching design, I’ll try to bring in a mathematical element. It’s worth students trying to make those connections without having to find it arbitrarily later in life.”
These ‘connections’ she speaks of are not innate. Students see without understanding. Teaching reveals and makes obvious; children begin to see math in everything in the world around them. It will allow them to make much deeper connections, more fluid connections, throughout their education and beyond.
Art is not math, it’s not boring. Folding paper, for many math students, isn’t ‘doing math’. While they may struggle to fold straight lines, to make inverse folds, to make it look ‘right’, they are creating something beautiful, and if I can teach them nothing else, I can show them math can be beautiful!
The idea, after introducing them to this complex looking, yet simple, folding technique, is to have them think about what it will take to replicate angles, curves and other shapes with their paper. Then they have to apply some design sense: how many folds? How long does the fold need to be? What direction do they need to fold? What angles will produce a curve? The folds are repetitions of a fold- what is the proportional relationship? In this image, there is a curve. Each fold is a point on that curve. It can be mapped on grid paper, it can be written as a formula. There is a connection!
Don’t be afraid to start with complex looking folds. It’s so powerful for students to struggle with something that seems beyond their abilities. Success is completely intrinsic and will lead to a student who is more confident. This feeling of success is habit forming. It is a primary building block in creating a student who perseveres!
(Check out 7 more art and math connections here.)
We can calculate the area of this beautiful 3D paper construction, we can create more of them by following the ‘formula’ or we can dramatically change or even destroy an image by changing the formula! This creative process stretches the imaginations of my students. The ‘what if’s are endless.
Many of my students will continue with this exercise beyond class, through drawings, foldings, and cut paper. There is a great simple foldable here that actually incorporates cuts to create a 3D effect. It’s a great way to jumpstart paper folding and cutting shapes to students that are not accustomed to paper-folding activities! I want my students to make graphic organizers that are beautiful to look at. Notes as art… What a concept!
Notes as Art… What a concept!
I am an #mtbos -er because I finally found a place where I b
‘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!
This year: new ways to deliver PD; a different way of communicating with my students; improve my Spanish; a second semester in my ELL training (certification in my future!); but one thing at a time!
My #MTBoS mentees: hang on for the ride!
Great thoughts on PD: (credit goes to a brief tweet from @pamjwilson):
“stop [Tchrs] doing good things to give them time to do even better things”
Dan Meyer’s recent post on Swan’s idea that we ‘get worse’in the process of getting better.
Professional Development: Getting Worse Before We Get Better
Think about what happens when we make announcements to the whole class. Does ‘broadcasting’ really work? (Crediting another tweet: @justinaion):
#noTalkWC, Alice Keeler
I started with an expensive program, but ended up using a free app:
I love the way it has improved my accent! (Southerners speaking Spanish can be embarrassing!)
My ELL training is a grad class at UGA… Three semesters (I started this last fall, 2015) later, (and a certification test!) I will add a new certification to my license! #lovethedirectionmycareeristaking!
More than anything, this online PLC is about sharing and supporting great ideas, one teacher to another. I look forward to learning even more!
My newest inspirations are the three lovely people I get to mentor this year: Melanie, Doug(@freeejazzz) and Sandy, Welcome to the group! And thanks to everyone(!) for helping me become better. Happy New Year!
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!😩.
When I was young, I would ask “why?” Not once, but Continue reading “Engaging students: to ask better questions, we must become better listeners”