Pre-preplanning

School officially starts back for my district on August 6 this year. Before that happens, teachers gather. Today and tomorrow some teachers from my math department are gathering even earlier. We will take two days to fine tune our plans, to collaborate, to build a bit more strength into our teams. Next week begins the official preplanning. The agenda is set. About 6 hours of the five days has been designated for curriculum teamwork. We’ll take it now, early. Next week will be quite busy enough.

I am ending my summer a little bit early. Why? Because I want my students to have their best year ever, and I’m going to plan for it!

Our curriculum team makes this possible through pre-planning grants that are available through Title I funding. To find out if there are funds available at your school, check with your county’s Title I coordinator!

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Asking the right questions of ourselves…

We question our students to elicit and engage, to push their sensemaking, to activate prior knowledge, and to get them thinking about their thinking. But do we question ourselves and our pedagogy with the same focus?

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. Writing Team: Steve Leinwand, Daniel J. Brahier, DeAnn Huinker, Robert Q. Berry III, Frederick L. Dillon, Matthew R. Larson, Miriam A. Leiva, W. Gary Martin, and Margaret S. Smith. http://www.nctm.org/principlestoactions

Table taken from National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all.

In the rush to use new ideas, incorporate technology, or just ensure our students are doing math things from the minute they walk through the door, are we giving enough thought to why we are choosing an activity? What is that activity suppose to achieve? Has it earned the time it will take, not only for the students to complete, but for the grading and feedback? In other words, will it move these students further towards the goal?

Learning Targets are Not Just For Kids

My colleagues and I have been asked to share learning targets with our students; to write the daily lesson or goal on the board and go over it, so students know what successful learning looks like.

As you think about what your students will be doing, do you ask yourself just how the activity will provide the experience you want for the student? What will it tell you or your student about their learning? How will it move them closer to understanding? How will it engage the learner to produce the desired result? What will the failure of this activity tell you? Tell your student?

For the math classroom, skill builds upon skill. Knowledge is formed by understanding how the old can be reshaped or used to fit into the new. It’s like reading, but with numbers. In learning to read, we begin to decode shapes that are letters, that make sounds, that can be arranged into words (patterns of letters), that are then arranged into larger groups of sentences to convey information.

In mathematics, we learn our numbers, which instead of sounds convey amounts. At first, these amounts are concrete, as we count fingers or toes or toys or blocks or cheerios. At some point the numbers begin to represent the amount. We put these numbers together into groups that define patterns, and we put these patterns into sentences that convey information. Every time we learn a new concept, we can place that building block in context and do more than we could. A concept in math means we’ve identified a relationship, a cause and effect, a reasoning about how one action effects an outcome. When we understand the connection, we can extrapolate, interpret, compare, contrast, synthesize, and create. All of those things that we say we want our students to be able to do, no matter the subject.

Is this what your materials and lessons are teaching?

Every teacher reading this has bemoaned the lack of time we have in the classroom. How we spend that time often forces us to cut our activities, explorations, and conversations about the material. We have to hit the ‘most important’ standards, or the ‘big ideas’. Yet we know that knowledge is built by exploration, by focusing on a problem or situation, by playing with ideas. But how often have you asked ‘how will this activity increase understanding of the concept?’ ‘What is it teaching?’ ‘How does it allow showing the learning, or mastery?’ ‘Will it allow for connections to what has already been done, or to what is to come?

Is the idea or activity sticky?

By sticky I mean will this be something the student will think about longer than the activity itself. Will a student come in a day or a week or a month later and say, ‘I’ve been puzzling about this idea and I think I finally get it.’ And yes… I believe all of our ideas can be sticky – not necessarily for everybody at the same time. If we are truly giving each part and moment of our lesson thoughtful care, there will be more sticky moments than not. Those moments are what build interest, knowledge, and understanding. One of the best examples of this is the Four Fours activity. My students worked on that for over a week!

How do we get there?

1. Put yourself in your students’ shoes.

Think about what is happening to them daily. When they walk into your class, where are they coming from? Have they had time to process their last class? (Probably not.) Are they looking forward to math or dreading it? Did they do their homework – or even understand it? Is your class a relief, a chore, or an interesting, thought provoking space in their day? As I write this, I see the faces of my students, and can easily see the few that truly look forward to this class – but there are occasions where my lessons have let them down, too!

What do they need from you in that moment, to get their mind off of what has happened up to the moment they walk through your door?

For a start, pretend you are a student and walk into your classroom. Pick up that starter. Take it to a desk and try your own lesson. Where is your student brain? How does it make you feel? Does it do you want it to do? Loosen them up? Assess yesterday’s lesson? Review a skill they need in the main lesson? How will you check the outcome? This shouldn’t be a ‘take up and grade’ – it’s very name implies short, sweet, and to the point. What you want to accomplish must guide what you do. What you do sets the tone for the rest of the class. If the starter isn’t working for you, it isn’t working for them. Stop. Just stop doing what doesn’t work.

One teacher I know

has a great routine. She has trained the kids to pick up a starter (half page, every day) on their way in. Some fill it out. Some don’t. She goes over each problem, quickly working it on the board. She asks a few questions about the numbers or the process. Usually she has to get the class’ attention, many are off task. The kids who knew how to do this are already zoned out. The ones who don’t know how either copy her work down exactly, or don’t write anything. She does this quickly, and in her mind this is circling back around as a review for weak skills or concepts. She is practicing good classroom management by getting kids in their seats and working. She has trained them that math is boring.

During one week, she gave the same material as a starter and as a quiz to check for Learning. To her frustration, it did not result in increased knowledge for those who hadn’t already learned it (and I suspect it was extremely boring for the handful who did!) This is a 15-20 minute activity every day. How could she change this activity to get the desired outcome, i.e. strengthening this skill?

2. Ask yourself what your lesson is teaching: Process or Concept?

A process is a pattern of activity. A concept is the explanation or reasoning for why we do the process. Teaching a concept should lead to the process. In the interest of time, students often learn the process. Then it’s practice, homework, and a test. Are you teaching process or concept? Are you reviewing process or concept? Are you practicing process or concept? Concept is harder, takes more time and doesn’t work well on a worksheet. It is much more interesting, however. Concept is sticky.

You do not have to reinvent the wheel!

No time to write those magnificent lessons? Have I got a tip for you! You do not have to do this alone! Lessons and resources are out there and so many are FREE. Check out these links (courtesy of Matt Vaudrey and the #MTBoS:

Not only will you find good lessons, you will find teachers who are constantly looking for better ways to share this wonderful world of mathematics!

Here are a few more questions for you to consider, (and which I will be grappling with while planning my next classes):

3. What is my lesson intended to do? How do my materials (problem set, delivery, class activity and structure, timeframe, sensemaking, etc) support this goal?

4. Where are my kids likely to fail? What can I do beforehand to support the weak spots (Starter idea!)?

5. What does the learning of the concept look like? What do I do for those that ‘get it?’ What do I do for those that need more? How will I know (formative assessment). If they don’t start, WHY not? If they don’t finish, WHY not? Do they really know how to do it? Is homework appropriate- i.e. will this truly extend the learning?

6. Does my lesson connect this idea to what they already know? Does it give them a peek into a future idea?

7. When/how will I give them time to process what they’ve done?

8. When will I revisit? How will I revisit? (Yes! Plan for this!)

I leave you with this:

The moment you can really know a student has internalized a concept/learning target is the moment you hear/see them sharing what they’ve learned with another student. Plan for that, too.

L-Q-E vocabulary (or how to drive your OCD students crazy!)

Algebra I, Unit 5, Compare and Contrast Linear, Quadratic, and Exponential functions…

This simple vocabulary lesson generated a surprising result: my kids thought the end result was too messy, so many of them didn’t want to follow the final step!

The activity uses a simple alphabet mind map. It is a 5×5 grid with a letter of the alphabet in each square. The last square contains X and Y. You can print a copy here, or have the students draw their own.

Here is a simple alphabet mind map.

The standards for this unit were that students should be able to identify linear, exponential, and quadratic functions from equations, graphs, tables, and contextual situations, and be able to compare each function with regard to rates of growth.

To achieve these skills, students needed to be able to identify the key characteristics and key vocabulary associated with each type of function. They also needed to be able to discern small differences in equations, the shape each functions takes when graphed, and the changes in a table that would indicate what type of graph the table would produce. Given contextual scenarios, they needed to identify which type of situation would produce a linear change, a parabolic track, or a classic J-curve from exponential growth or decay.

When we started, they could barely list the key characteristics, much less identify which function was associated with each characteristic, what those characteristics looked like, or how to tell them apart. They needed stronger, more fluent use of the vocabulary!

Step One, The challenge: using each letter of the alphabet, fill in the grid with the names of as many key characteristics of each type of graph as each student could think of. (We’d made lists over the previous several days, along with examples, so I knew they would be able to come up with several familiar words.) I encouraged them to start with any word they could think of that they associated with graphs or equations. As they wrote, I then passed out colored pencils for the second part of the task.

Step Two: After about 10 minutes of individual work, we came together in a large group, and I asked each student to share one item from his/her list. I encouraged the students to add new words they heard to their papers, and to use a different color  pencil than they used to write their initial lists. After going around the room about two times, we asked kids to popcorn choices that they had on their papers that hadn’t been covered. We had a few letters that remained without words, so we again asked for ideas from the whole group that would fit for those letters, reminding the students to stay within the linear, quadratic, exponential, and graphing parameters.

We found that we had to ask a few thought provoking questions to make sure some important terms weren’t left out.

(At this point, because we were talking about why these words were acceptable, what they meant, and how they were related to LQE, I had a pretty strong idea of where my kids needed additional help and lessons!)

As my students shared their words, I wrote them on a poster sized alphabet chart that I had prepared beforehand. I gave the students a few moments to make sure that they copied all the words from the collaborative chart onto their personal charts.

Step Three: each child labeled the outside of their chart with the words LINEAR, QUADRATIC, and EXPONENTIAL.

I explained that we were now going to match each term with the function to which it belonged by drawing a line from the word to the function. I warned them that some words might belong to more than one function!

They each picked a color to use for the line that would connect the appropriate words to LINEAR.  The first word under A, asymptote, was determined to be related to exponential, not linear. Not only did they have to decide which function, they had to say WHY and in what way the word connected with FUNCTION. The word Axis was next. Everyone could get behind that as a graph term that could belong to any of the function types, but we only connected words to one type of function at a time. We would come back to ‘axis’ two more times as we matched words with the other two functions! Colored lines were drawn from Axis to LINEAR. This happened with several more words, before the students began to realize this was going to get messy. I was drawing the same lines on my big poster, but I was totally surprised as students began color coding each word with dots, or making these neat lists on separate pieces of paper, sorting out each of the characteristics, because they didn’t like the tangled mess that was happening on my poster. (I explained that they were actually drawing the map for their brain, not their eyes. They were somewhat skeptical…). ‘I can’t read it,’ was the standard response!

Here was our resulting map!

 

I  encouraged the students to use a different color for each category, and we progressed in order through each function, so no one would end up confused. Throughout the matching, as students popcorned answers regarding which words to connect,  I continued to ask for agreement, disagreement, (thumbs up, thumbs down) and ‘why, how do you know,’ from the whole group. This was a very intense, fast paced portion of the activity, with even some of my most blasé students getting involved!

We followed this activity with a neat card sort, that was another intensive activity in and of itself, and was spread over two days. By the end of these activities, I could tell that more of my students were fine-tuning their selection processes, looking more closely at the details of each equation, graph, or table, and applying the key characteristics lists they’d made to their compare and contrast process!

Here were some lists they made of the key characteristics:

Close Reading in math; and the “after” math…

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.)

  • “I liked it because it was different from the way we usually do math.”
  • “Why can’t we do math like this all the time?”
  • “I felt like I really understood the story.”
  • “I remember more about how many fruits Jack planted.”
  • “The table helped me understand.”
  • “I think it will help me in my reading in other classes.”

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.

To read Jack’s full story, visit Anno’s Magic Seeds, by Mitsumasa Anno. For more of her series of math books click here.

Are you a 1, 2, 3, or a 4? What’s numbers got to do with it?!?

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!

 

 

Gwinnett County Public Schools is hiring…

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!

http://publish.gwinnett.k12.ga.us/gcps/home/public/employment

Good Hunting! Your next job is out there.

Knowledge Machines are here; How will you use them?

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.

Continue reading “Knowledge Machines are here; How will you use them?”

What do you mean, it “…can’t be solved?!?”

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.

Murder Mystery Solved with Trig!

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:


Each student had to use the prop to arrange themselves into a triangle. The other students watching were, um, helping. (that’s what they called it!)

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!

We’d like you to take over this class, six weeks into second semester….


Six weeks into second semester, I was asked to become the teacher for a group of Algebra II students.

I met the Continue reading “We’d like you to take over this class, six weeks into second semester….”