Two roads converged in a yellow wood, and I- I took the road less traveled by, and that has made all the difference.
From “The Road Less Traveled”, a poem by Robert Frost
Two roads converged in a yellow wood, and I- I took the road less traveled by, and that has made all the difference.
From “The Road Less Traveled”, a poem by Robert Frost
Literacy is everywhere, including math class. And I don’t mean those clever (and not so clever) word problems.
I used a simple story book, predictions, and paired discussions for my 9th grade Algebra I -ers. I passed out colored pencils for annotation, and paper for predictions.
We started with the brief prelude, a paragraph, about a ‘happy-go-lucky young man’ who meets an old man who tells the young man,’I have a gift for you.’
There were a few other clues in the paragraph, so after asking the kids to read and annotate the paragraph individually, a volunteer read the paragraph aloud. With no further discussion, I asked each student to write down their prediction of what the story was about. Then I asked them to share their predictions – and their reasons why they thought that – with a partner. Then they would listen to their partner’s prediction and decide if they agreed or disagreed.
I modeled having a conversation…
To remind them of how to have a meaningful conversation, I modeled listening, responding with questions about what was said, and commenting on the information. We talked about how conversations were about listening and responding, not a contest of spouting information. Everyone got to practice, with a little (okay, a LOT of,) prompting!
We turned to page 2. My students were now eager to read further (we had predictions to fulfill!), and since the book’s format was laid out in sections of two to four sentences, with lots of images, I was able to let all have a turn. We took the four sections on pages two and three rather slowly, and we pulled apart the actions of the old man (he gave the boy two seeds), and the young man (he cooked one seed and planted the other seed). We talked about the passage of time – winter, when they had met, and the boy planted the seed; spring as the ‘sturdy’ plant appeared from the ground; summer, when the plant produced two flowers; and fall, when two fruits became evident. Each one of my questions caused them to return to the text for details.
Before we turned to page four, I asked my students about their predictions, and whether they had changed their predictions based on this new information. The conversation among the students turned to the details they just reviewed: how long it took to grow the seeds, how one seed could feed someone for a whole year, and why couldn’t he get food where he lived. New predictions came from the ashes of the old, with several students predicting that the plant would grow huge, up to the sky, and the boy would climb it. This was an obvious scaffold onto a familiar story. They told me the key word ‘sturdy’ meant that it would be strong and big. We turned the page.
New predictions came from the ashes of the old…
As each student read a section, we learned that the plant the young man had grown produced two more seeds from the fruit. He cooked and ate one and planted the other. We noticed from the picture that he put a hammock in a tree. Apparently he was planning on staying put. I pointed out that this was something that usually happened when people planted, instead of hunting for food. Another scaffold, as they agreed with me that he would need a place to sleep while the plants grew.
Again, we turned the page. This time, the readers learned, Jack (that was the boy’s name) had managed to grow another plant from the seed, which yielded two fruits, which yielded two seeds, one which was dutifully cooked and eaten and one planted in the ground. No changes here, but we noticed in the drawings that Jack seemed to be getting fat. Also, the plant wasn’t getting any bigger, which some of the boys seemed disappointed with. I think they were still waiting for the plant to grow up to the sky. One boy persisted in asking why Jack just didn’t move to a place where he could buy some food. We made more predictions, adjusting our expectations based on what we’d read. I asked them if their mental picture of the story, and of Jack, was changing. One of the girls agreed, and then we turned the page.
Their mental picture of the story, and of Jack, was changing…
The next two pages supported the story line. Jack continued for two years to cook and eat a seed and plant the other. The plant never got any bigger and never produced any more than two flowers, two fruit, and two seeds. It looked as though Jack would be living out his life, year by year, cooking, eating and planting seeds. I had the students make predictions anyway, and talk about what they thought Jack should do, given the circumstances. Some of the children thought Jack was stuck in a rut. On to the next page…
My readers were still eager to read. They continued, aloud, to devour the slim text (it was great to get a window into their abilities to read and decode the text – you know, for those word problems we will get to someday!) It was momentarily exciting to hear that Jack was as bored with his existence as some of my students! He said, and I am paraphrasing here, ‘if I always do what I have always done, I will never get anything different than what I’ve gotten.’ (Here I looked at my math students to see if they had taken anything of a personal nature from this comment. It appeared they hadn’t felt a connection.)
‘What,’ I asked, ‘Do you think he is going to do?’
‘He’s going to plant both seeds,’ spouted one girl. ‘But what will he eat while he waits for the plant to grow?’ I asked. They hadn’t thought about that, and in thinking about it, several students were seriously confused about how many seeds he could plant if he ate one, and he only had two to start with. (This may point to a reason that so many students struggle with adding negatives and positives.)
A serious discussion ensued as to what he was giving up if he didn’t eat the seed. Would it be worth it? We talked about how sacrifice is sometimes needed to affect change. One of the kids said he would be cranky because he was hungry – the voice of experience talking? The kids had gotten into the spirit of the lesson and were ready with their predictions. The Jack and the Beanstalk contingent were ever hopeful. There was still one boy who wanted to know why he didn’t just move to a place with food and forget about the seeds. We turned the page.
The story continued with Jack explaining that he decided to go hungry so he could plant two seeds. To assist the verbal process, and give structure to the next round of predictions, I drew a table on the board and labeled the two columns ‘seeds planted’ and ‘seeds produced’. I then filled in the first line. Under ‘seeds planted’ I wrote 1, and under seeds produced, I put 2. I asked the kids to review the current situation: is this correct so far? After some discussion of my column titles, the students agreed. I asked them to fill in the next line. Now that Jack has decided to go hungry for a year, how many seeds will he plant? The students agreed he would plant both seeds.
I wrote a 2 on the second line under ‘seeds planted’.
‘So what goes under seeds produced?’ was my next lead in, ‘and why?’
Another look at the text produced the facts: two seeds produced two plants, each with two seeds. How many different ways could we count this, and still get answers varying from 2 to 6?!?
More fingers, more math. One girl supported her position loudly by mentioning multiplication; two plants times two seeds would be four. Several students had made the same conclusion by different routes, but getting the kids to share their explanations that they had discussed with each other was the hard part. The confidence that they can be right is so difficult to encourage! The student who mentioned multiplication was encouraged as another student agreed with her. A 4 was written in; satisfaction all around.
So, how many seeds will he plant next? And ‘why?’
‘Four’ was the immediate answer, until a single student voice reminded us that he had to eat. So began another discussion about how to calculate the number of seeds, and how many will he eat, and how many will he have to plant? Some students felt he was going to eat two of them. (We had doubled the amount of seeds grown – we must double the amount he could eat!)
We returned to the initial instructions on page 1. A student read the evidence and told everyone definitively that only one seed would feed Jack for a whole year. By this point, I was listening to these children teaching each other how to support their facts, correct their misconceptions, and expose and correct mathematical errors. Every child was involved in this discussion – every child!
One seed eaten, three seeds planted was the ultimate conclusion. (If I rendered their discussion here, you would be as bored as we were with three years of Jack eating a seed and planting a seed!) The chart was filled in, and we were ready for the next question: if he plants three seeds, how many seeds will he have to plant the next year?
While the whole process was a conversational struggle, before I left them to this final prediction, I pointed out the table and had them write it in with their predictions. We had just begun defining a function the day before, so I asked them if they thought this was a function? I was rewarded with a student identifying the input and output terms of seeds planted and seeds grown, as x and y. Several students then made the next connection to the fact that none of the inputs we had so far would repeat, so it must be a function.
Without giving away any more of the story, I will tell you that they were able to successfully calculate the answer. We stopped here as I had them make a final prediction about the rest of the story. We will revisit Jack and his adventures in the coming unit.
Here is the ‘after’ math:
Our school lit coach came by two days later and interviewed some of the students that had participated in the close reading. Here are some of their comments (I stayed across the room, out of their line of sight. I was curious to hear their honest responses.)
Read that last bullet again.
‘I think it will help me in my reading in other classes.’
Wow. Now, that’s some kind of ‘after’ math!
Editor’s note: the standards for this lesson had to do with identifying functions; recognizing a function in multiple forms (such as tables), identifying functions from contextualized settings; the literary standards were identifying supporting information and using contextual clues to support mathematical arguments (a Mathematical Practice, as well). Additionally, the conversation supported mathematical reasoning, practicing vocabulary, and reasoning aloud. The constant predictions were embraced by the students, who stayed involved for what amounted to about 60 minutes of close reading. They had to know each outcome! No one read ahead, which surprised me a little. They really wanted to maintain the suspense! I can’t wait to revisit Jack and the rest of his story!
For those who notice the reading level, I wanted to use a text that would provide a low floor, and that would allow me a high ceiling – the actual math is writing and calculating an exponential equation. This was a great text for my ninth graders! The text didn’t cognitively get in the way of the activity.
I, along with a couple of other teachers, are piloting a grading strategy that is generating some interesting conversations on a DAILY basis with our students!
We’ve all read that grades do not improve or motivate learning. In fact, once a grade is given, the student assumes that idea is ‘done’ and drops it, moving on to acquire the next grade.
What I am about to share with you has MY KIDS talking about how THEY can improve their learning…
First, I have to give credit for the base of this idea to an amazing educator that I work with every day: Rebecca K. She, of course, credits it to an idea she learned in a workshop some years back. Anyway, she started the year off with a cool bulletin board, that looks something like the image above, which I used to create a powerful way to motivate my babies to take more responsibility for their own learning!
The students I am talking about are your average 9th grade (yes, FRESHMAN!!) students, that run the gamut of every freshman stereotype you’ve ever met. Really. (This includes students with personal learning plans and students whose first language is not English!) AND we’ve got them talking about growth – THEIR growth – as learners. When we hand back a paper, instead of the ‘crumple it up and put it in the bookbag or the trash’ mentality, the comments are varying forms of, “..tell me what these results mean!”
Here’s how it works:
Four numbers, four learner identities: 1. Novice, 2. Apprentice, 3. Practitioner, 4. Expert
Novice: I’m just starting to learn this and I don’t really understand it yet.
I explain to the students that this is where everybody in class starts out. Algebra I will have lots of things that are new to them, and we expect that they won’t be familiar with the material! We don’t expect them to know it all before we teach it. Sounds obvious, right? Sometimes you have to be explicit with Freshmen. I think that’s where the name originates!
Apprentice: I’m starting to get it, but I still need someone to coach me through it.
The apprentice is the beginning of the learning phase. When a student gets a 2 on a problem or a whole assignment, they are in the initial learning stages. As a teacher, I’ve just told them (by marking it a 2) that I know they still need help with the concept, and that I will be supporting their learning. This also tells them that they are not there YET – and that they have room to continue learning. Sometimes we have to give kids permission to not know things YET!
Practitioner: I can mostly do it myself, but I sometimes mess up or get stuck.
This is a proud moment for most of my students. That little 3 next to a problem or on a paper, tells them so much more than a traditional grade. This sends them the message that I get it that they’ve got it! This affirms their learning. This affirms their work. This is personal. Better than that, this motivates them to keep going, to keep learning. They ALL want to be….
Expert: I understand it well, and I could thoroughly teach it to someone else.
Isn’t this where we want our babies to be? You know that if they know it well enough to teach it – THEY KNOW IT!! That peer tutoring thing is for real! Please notice that there are TWO parts of this level: knowing and teaching.
How does this work?
My (totally awesome) co-teacher, Stephanie W., and I, use the following grading process. Feel free to modify it to fit your students, and what is happening in your classroom. We know that what we are doing is working for our kids – you may want to start with this, and then modify as you see what is working for you.
We give an assignment or quiz. We grade each problem with a 1, 2, 3, or 4. We add up all the grades and divide by the number of items. That gives us a number between 1 and 4. Many times that will generate a decimal, say 1.8 or 2.5, or even 3.8. Here is an important point: we DON’T ROUND UP! We DO EXPLAIN the process to our students. It is important for them to understand that this is not arbitrary. They must own the process for this to work. These conversations happen EVERY time we return an assignment. That’s a GOOD thing!
Our goal for our students is mastery, so unless the resulting average is an actual 2 for example, the child is still a NOVICE (1, 1.2, 1.8, 1.9 – doesn’t matter. They are still a 1). Same with 2 point anything – they are still a 2, same with 3 point whatever – still a 3. The ONLY exception is 3.8 and above. If the student has one or more 4+ answers, with clear justification statements, then, and only then, will we round up to a 4. See below for the PLUS explanation!
Our evaluation goes something like this:
a) Answer that is incorrect, No work shown, or No answer at all: give it a 1.
b) Answer with some work shown (they attempted a solution) but it is incorrect in major ways and answer is incorrect or incomplete; give it a 2 (remember they are still learning and need more help!)
c) Answer given is incorrect, but work is also shown. (OR answer is correct, but NO work shown to support the answer). Student did pretty good, but minor errors and/or mistakes caused the incorrect answer; give it a 3. This student is obviously getting it, but he/she is letting errors get in the way. Maybe they are lazy, maybe in a hurry. The 3 tells them that they are getting it – but they NEED TO BE MORE CAREFUL! (The 3 for NO work shown is to allow us to ensure students are not ‘borrowing’ answers from another student! We are giving them the benefit of the doubt until further notice.)
d) Answer and work is shown and is completely correct. This baby gets the 4! The student can feel the glow of being an expert. But wait, there’s more! This only satisfies HALF of the description. What about the ‘teaching’ part?
Four “+”? What is Four Plus???
‘Four +’ is that special designation for the child who not only knows the material, but can prove to us that they are able to teach the material to another student. Time dictates that we don’t have the opportunity for EVERY student to demonstrate teaching ability (although we do try to build in those opportunities!). We have explained to our students that the way to demonstrate this ability is to justify the work they’ve shown, with brief written explanations.
Written Justification sets the student up for PROOFS in Geometry
Algebra I is a class of foundations. It is important to teach with an eye to the future courses our kids will encounter, and proofs are some of the most difficult lessons for students. One of the Algebra I standards is to be able to justify the steps taken to solve simple one step equations. This is an important step to understanding that there is a mathematical reason for being ABLE to take that step – and not just because the teacher said so! By building this into the idea of EXPERT, we are modeling the concept that understanding – that is, the realization that there are solid REASONS for why math ‘works’ – is a valuable part of the learning process.
What WORDS do you use to tell a parent how their child is doing in your class?
I know this is just a brief overview of this process, but I wanted to share because I feel it is the first solid step in moving towards talking about GROWTH and LEARNING, instead of grades. I believe it is important that we take the focus off of grades, for students and parents. To do that, we, as teachers, have to stop using GRADES as the unit of measure in communicating with our students and parents. Unfortunately, our grading systems, and I’m talking the actual computer systems we have to use, are not set up to show mastery – they are set up to show GRADES!
I already changed my conversations, my wording, my language, with my students. It will happen with my conversations with parents in my next phone call/email home, as well. Will YOU?
What’s the downside?
My school still uses a grading system built on averaging traditional grading numbers. That means I can’t just put in 1, 2, 3, 4, or 4+. I have to turn these numbers into a grade between 0 and 100 that will accurately translate and describe my students’ mastery of the curriculum.
My solution is two-fold. The grades in my gradebook are tied to one of the required standards, and each of the above levels is tied to a number that has already been given meaning by how it is used as a grade. While the first is fairly easy to accomplish, the second is based on how parents and students interpret grades. A 100, for example is the ideal. That sends the message that the student has mastery of the assignment, or the course. In fact, anything above 93, in my County school system, is an A, and as such, denotes pretty much the same thing as a 100. Same for a B, or a grade in the 80 range. Those two grades are obviously acceptable to most parents and students. The grade of C is a little more ambiguous. The C denotes that the student is somehow less than perfect, but still passing. While a student may be GLAD to have a C – it does denote that the student is doing the work and IS mastering the concept – it doesn’t have the same cache’ as the A and B grades.
So how do I reconcile the grades with the numbers?
A novice receives a grade of 65. The Apprentice receives a 70. The Practitioner has earned an 80, and the Expert, a 90. The 4+ student will earn a 100, as long as all problems on the assignment or quiz show justification, evidence that they have not only mastered the concepts, but have gone above and beyond to be able to communicate their knowledge with others.
The final issue I will address here: What happens when a student makes no effort at all. Our students never do that, do they??? In that instance, the student has given us no information on which to base a grade. Effectively, they have NOT TURNED ANYTHING IN. The grade in the book becomes an NTI, and we are made aware that we need to step up our efforts with that student. An NTI is a zero, until the student completes an assignment on that material, and we can assess mastery. From there, the averaging work of the gradebook takes over, and the grade reflects the whole course mastery. Grades in this context are fluid, and can be changed by future mastery as evidenced by quizzes or testing situations.
The system is not perfect, but the teachers with whom this is working believe that we have created a system that truly tells us where our kids are with the curriculum, and allows us to modify our teaching darn near immediately, so that we can address the areas in which they need further help – which is the actual point of all this grading, isn’t it?
Here is the poster we use in our classroom to explain the levels. Our students get their own mini copy for their notebooks. We utilize a small chart of “I can” statements for each unit – no more than 3 – 5 statements – that allow the students to chart their progress. Here is the chart for our Unit 1 standards. The kids get this, too. You can use any “I can” statements you need for your particular units.
At the beginning of each unit, the STUDENTS determine their pre-assess level, the quizzes give them the mid-assess levels, and then the unit tests are the post-assess level. The students keep track of these themselves. We incorporate a running conversation DAILY of what their goals are, where they think they are with these goals, and how they are going to get to the 3 and 4 levels. I have personally found this is a great way to have the students tell me where they are at the end of instructional and practice periods throughout class. I simply ask them where they think they are – 1, 2, 3 or 4. The majority of students are incredibly honest, because we are all speaking the same language. The ability to quickly assess and modify my teaching is been made incredibly easy! Grading has become a process of assessing growth, not despairing over what they don’t know. I LOOK FORWARD to grading the work, knowing most of my students WANT to have a conversation about where they are, and what they need to do to get to the next level. Let me know if you would like the rest of the “I can” levels we are using with this course. I’ll be glad to share!
They are looking for math, science, and special education teachers.
I work with absolutely excellent educators! We need teachers who teach students the subject matter, not the subject matter to students. There is a difference, and it matters.
Our county has been the recipient of the Broad Prize for excellence, not once, but twice. We excel. We are Gwinnett. It’s what we do.
Please put my name in the comments of your resume to let them know you heard it from me!
Good Hunting! Your next job is out there.
There was a time when school was about learning the three R’s: reading, writing and ‘rithmatic. Sounds like the beginning of a long ago time story, doesn’t it?
After reading this 1993 article from Wired, I realized that Papert’s ‘Knowledge Machines’ are, in fact, here.
If you are a teacher, that is.
On August 8, hallways and rooms will fill with the wriggling eager bodies of their parents’ best! I can hardly wait! During the summer, I’ve been loading up on great ideas, reading about effective teachers, discovering new tech and new resources, and creating learning plans that will put them into practice!
I am teaching my students Algebra I this year; 9th graders, some returning 10th, and I want them to feel the excitement, the sizzle that I feel with math. This is a new year, a new crop of children, a new chance for me to share what I love- math – with children who never fail to delight me (and challenge me, worry me, turn my hair gray, and, well, you get the idea- but that’s another post!)
The year I have planned, this year, will be different. This will be the year that every student tests proficient on the EOC, aka Georgia Milestones. My lessons will start with Wonder/notice, there will be lots of student conversation, with roles for small group work, and conversation starter posters on the wall! My class will be fully engaged, will actually complete their assignments, will receive thoughtful feedback, and grades that really show how well they’ve mastered standards. I’ll make all the calls, on time, to the parents.
My IEPs will have clear goals, my re-evals will be works of art! I’ll handle my discipline issues with skill and compassion. This year, I’ll have strong closure routines, include literacy in every lesson, hold awesome number talks, and have nimble responses to my formative instruction.
This year, my room will be organized. I’ll have study centers, whiteboard walls, standing desks, and engaged, curious students! This year – well, this year will be everything I was hoping last year would be…
So, you see, teachers really do celebrate New Year’s in August!
As I rest and recharge myself this summer, I am like one of those little chipmunks- stuffing my cheeks with ideas, thoughts, and nubbins of lessons.
I start a list of those items that MUST go into my lessons next year: good starters, exit strategies, literacy activities… You, too?!?
At some point, I need to actually plan! And I will. But not today. Today I am going to read a book. Not a ‘teacher’ book! This one is purely for pleasure: a new mystery novel. But first, let me check out this new email from Edutopia…
…make PBL [problem based learning] disappear. In an inquiry-based school, it should be nearly indistinguishable from general instruction.
Problem based learning, inquiry based learning, question based learning… There are lots of different names for a specific type of classroom activity that places one item, image, expression, or situation in front of a group of kids and asks them to engage with it. The teacher steps back. Instruction is minimal: come up with a question, any question. That’s it. Not what type of question, not what the question needs to pertain to, not how many questions.
As a teacher, that is really hard. And it takes time to let this process happen, and, oh yeah, you need classroom rules like no judging questions, just write them down without changing them, don’t try to answer the questions, just write them down, and every question matters, so write it down even if it sounds silly or stupid.
Because when students start asking questions, curiosity is piqued.
It is okay to promise that as many questions as possible will be answered. It’s okay to go off in a direction that wasn’t in the original lesson plan. Because when students start asking questions, curiosity is piqued. And when that happens, well, all I’m sayin’ is, never underestimate curiosity!
Once students are curious, lessons can turn one of two ways; you control this knob. You can kill curiosity just as quickly as you can pique it. However, it takes courage to let this curiosity take hold, to guide it gently, and to allow the students to run with their questions.
The students have asked. The cards (one question per) are up on the board, sprinkled around the room, arranged by subject matter… However you feel they should be arranged. What next?
I like this moment. While the questions are being shared, I am gaining new insight into the minds of my charges. What do they know? What connections are being made to the image or situation or equation (not necessarily a math connection, by the way)? How do my students think? Where are the questions going to take us? And my teacherly question… Where are we in relation to the knowledge I want my students to gain from this? Will we get there, or (and this is the scary, let go and teach part) will we get somewhere else just as valuable?!
Making PBL disappear means building this type of student led inquiry into every part of the course and creating a daily attitude of curiosity.
At this point, I would take a moment to address the title of this discourse. Making PBL disappear means building this type of student led inquiry into every part of the course and creating a daily attitude of curiosity. What are we going to explore today? What are we going to learn how to do today? Students come into class prepared to ask questions, questions that they know will be answered.
This is not a pipe dream. This does take a belief on the part of the teacher that the students’ questions are worth exploring, discussing and answering. It also takes a little bit of classroom setup- students need instruction in the rules of the game and, most importantly, the focus for the lesson – that picture, equation, situation, idea – needs to be well-chosen, the possible questions and directions prepared for, and the possible math directions imagined and worked out ahead of time. This kind of lesson takes time to prepare, a commodity in short supply for most of us.
How do we get from using PBLs as sometime specials to a technically invisible way of doing business in the classroom?
The biggest thing I can say is that no teacher has to do this alone. The body of resources is growing by leaps and bounds. There are informational blogs with ideas, examples of lessons, stories of actual class experiences (some with scans of student work), videos, websites, and instructional books. There are multiple teacher groups on Twitter that share ideas and discuss lesson results. These teachers are more than willing to give feedback and help.
Having said that, integrating PBLs into the classroom requires:
1) a knowledge of the standards you want to teach, translated into “I can” statement goals,
2). A willingness to encourage student led questions and discussion,
3) A portfolio of PBL activities that cover one or more standards,
4) the ability to facilitate “what happens next” and use the questions generated by students to unwrap the ideas that will lead to understanding of the standards, AND
5) the courage to let students choose how they will investigate/learn what you are setting before them.
Why do our children have to complete four PBLs in a two month period, separate from their ‘regular’ schoolwork? Why can’t their ‘regular’ schoolwork be taught in such a way that they learn and can draw parallels to their world outside of school?
Not that the content should match their lives, but the way they learn that content; the way they organize and make it a part of who they are in school should have some relevance to how they organize and deal with the stuff outside of school.
These two parts of their lives should mesh, not be two such disparate worlds that they cannot be reconciled.
Here is one solution:
I came across an Algebra I review problem the other day on Classworks. The challenge was to solve a quadratic using the Quadratic Formula. One of the answer choices was “can’t be solved.” Which I did not notice.
I was working with three students who did not understand what to do. Once I wrote out the quadratic formula, (actually, all I had to write was the negative b plus or minus!) they began to remember. One boy immediately told the other two how to find a, b, and c. That required a discussion about standard form, so we had to do a little rearranging of the problem given on the screen. Once we got the formula equal to zero, the second student plugged the numbers into the correct places! The third began offering solutions to various parts. I thought we were doing pretty good! Until we came up with a negative under the radical.
Like the music in Jaws… Dum, de dum, dum… They looked at me, dumbstruck.
“What do we do, Ms. Maxcy?”
I asked them if they had learned about imaginary numbers. (Of course they hadn’t – yet. This was only Algebra I! But sometimes I forget which level I am teaching… Which is another story altogether!!!)
Still not checking the given answer choices, I blithely proceeded to give them a brief ‘reminder’ lesson on real and imaginary numbers. They continued to look at me blankly.
As I magically (to them) unraveled the answer as 2 plus/minus 2i sqrt 11 divided by 3, they stared at me. Then they stared at their answer choices. They looked back at me.
“It’s not there, Ms. Maxcy.”
At this point, admit it, we teachers think, “it’s got to be there, that’s the right answer; why is it not there? Gosh, did I do it wrong?” And then we doublecheck our answer. And then it hit me. This was Algebra I. We don’t teach imaginary numbers. Yet. It was then that I finally looked at the answer choices…
The correct answer was there, but it wasn’t the correct answer at all!
Right there in front of me, there was the answer that the students were supposed to choose: choice “D) Can’t be solved.”
Right there in front of me, there was the answer that the students were supposed to choose: choice “D) Can’t be solved.” This is a terrible choice! It’s not the right answer! It’s not a good answer! Okay, so we don’t teach them imaginary numbers in Algebra I, why don’t we just list the result with a negative under the radical as the answer?!?
The kids get used to seeing the beast (negative radical) and we teach them how to simplify in Algebra II or geometry, depending on your school system. But, please, NOT “can’t be solved”!
That is just setting them up for trouble ahead! Lay the foundations, don’t build a wall that will have to be torn down later. Please!
Rant finished. Thank you for listening.
Dateline: April 14, 2016
The murder of Maria, whose body was conveniently found at right angles to Leg Streets A and B, has been solved! Investigators found the weapon across the river, apparently thrown there by her assailant while he was running down Leg Street A in an attempt to escape. A quick thinking officer (who had majored in math at the police academy) was able to calculate an angle measure for the angle made by the throw from the perp and the street leading to the victim. Another savvy investigator was able to determine the distance from the suspect to the location of the attack.
With the mathematical evidence in hand, investigators were able to triangulate a conviction. Math teachers everywhere weighed in, saying it has the proportionate ability to change the way investigators do business!
Dimensions of the prisoner’s defense will be released at a later date. Film at eleven.
Okay, so I don’t really have film (we forgot to assign the job of reporter!) What I do have are a room full of kids who can now set up the proper proportions for trig problems!
Here’s how the crime went down:
Scene 1: Before the murder, I handed six students a few props:
A short q&a followed:
Me: Okay leg A, are you opposite or adjacent to angle b?
Hapless Student holding Leg A sign: “I’m opposite, um, no, I’m right next to him (indicating student holding the angle b sign)! What does adjacent mean, again?”
We were able to sort out the definitions, and the students holding the leg signs got pretty good at determining whether they were “opposite” legs or “adjacent” legs. A big moment came as students noticed that they could be opposite OR adjacent. More importantly, they were able to articulate WHY the status would change.
More importantly, they were able to articulate WHY the status would change.
Scene 2: The next six students were given the cards. This time, I stood back and let the first group help position the players. A little skirmish ensued as Leg A and Leg B were being positioned. After a brief discussion about whether or not leg locations could be interchangeable (did Leg A have to go in the same place as the first triangle?), it was decided that as long as a leg were placed on each side of the 90 degree angle, it didn’t matter what we called them.
The opposite and adjacent discussion began again. It was fun watching students correct these new players, or making them guess by giving them tantalizing clues!
(If you ever want to know what you look like teaching, give your students the reins. Mimicry is not dead!)
Scene 3: With everyone up to speed on definitions, the murder could now commence! Maria was positioned. Ryan was immediately suspect, as we put the crime scene tape in his hand and instructed him to escape a bit down the hall. The “weapon” was given another piece of crime scene tape and told to take off in the opposite direction. The “hypotenuse” was asked how far the “perp” had thrown the weapon. We stretched the crime scene tape from the suspect to the weapon location. It was at this moment that I heard several students say “Hey, we made a triangle.”
It was at this moment that I heard several students say “Hey, we made a triangle.”
(Scary, I know, right?)
After a bit more discussion, the students determined that we needed an angle and we needed the distance from the body to the suspect to set up a proportion to solve for the distance. Two students were dispatched with the piece of crime scene tape that had been held between the victim and the suspect (Leg A, for those of you following along). Twelve inch square floor tiles assisted in the crime scene measurement. I used my oversized protractor to come up with the angle measure, and we were ready to set up some proportions!
Back inside the room, our eager detectives checked their trig proportion info sheet (yes! They used their NOTES!) and settled on cosine, adjacent and hypotenuse. I stood back and watched them argue over who was going to set up the problem, exactly how to set it up, and how to enter the information into the calculator. Then I watched them convince one another which answer was correct.
Concrete to representational to modeling AND peer tutoring…I love it! I would say that a murder wasn’t the only thing that got solved today!
Through my voice, I encourage you to find your voice.
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Through my voice, I encourage you to find your voice.
Through my voice, I encourage you to find your voice.
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Through my voice, I encourage you to find your voice.