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!

 

 

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There is this “buffet of choice” for teaching and learning of mathematics: How can I choose?!?

First: just breathe.

Second: start at the end, and work backwards. Reflect on these questions (posed by Brian Bushart During a #mathrocks event:

1. What are your goals for yourself for your math teaching Continue reading “There is this “buffet of choice” for teaching and learning of mathematics: How can I choose?!?”

Sixth Grade, Fractions, and Fair-shares

I have been assigned a room full of bright six-graders. They are successful; achieving high proficient scores – they are students who are “on the bubble” of scoring at the advanced level. A review of their work reveals the problem: Operations with fractions.

My goal is to help them create a stronger conceptual understanding of fractions, in order to allow them to be more successful in their fraction interactions!

The initial lesson involves some baby steps, in order to formatively assess the understandings about equal sharing, naming shares and rebuilding wholes. Materials will include the students themselves and a fair share box (see attached image- thanks to Jere Confrey).

I tell them they are going to gather in teams and ask them to come up with ways that we can divide the class so that each team has an equal number of students, WITHOUT COUNTING FIRST. At this point I am not looking for anything but suggestions on grouping.

I will use thinking questioning to get them started, clarifying q’s to help them express ideas and repetition: students repeating what others have said. The students will then be asked to critique, analyse and decide how best to group.

They will use a method they decided upon to create the teams. We may need time for trial and error here, as well as how to divide ‘leftovers’ when the class does not divide evenly!

Once they have successfully created even teams, I will pose questions (I will use the family feud style of questioning, so every student gets a chance to answer, can ask for help, can challenge another group’s answer, etc)

First question: Have we divided the teams evenly? Are they ‘fair’? (There may be discussion about this, especially since students sometimes want ‘smarter’ kids in their group!)

We will have to talk about ‘leftovers’ and how to divide them. (Options: let the students adjust groups until we have even amounts, or give the extra students jobs before the exercise begins so we don’t have to divide any bodies!)

Q: (work on in groups) How would you describe your team in relation to the rest of the student teams in the room? ( this is where I will go around and listen to what the students think about their relationship to the other groups. I want to find out about their level of expertise with the language of groups- I anticipate that I will get anything from ‘one group of five students out of 20’ to ‘1/4 of the whole class’.

I will have the students share their comments and ideas. I will then let the students give each other feedback on the reasonableness, accuracy, and will take the opportunity to guide the vocabulary for the discussion, by restating, and asking students to restate.

At this point, I expect to have identified any conceptions about fair sharing and language of sharing the students have.

Now we will fill in the chart:
How many students in each team?
How many teams?
What was the total number of students in the class
So what can we say about each team?

I will have the students tie their comments to the chart, walking through the naming conventions they use. If there is not a connection to fractions in the discussion, I will shape my questions: ex. Do they see another way we could name each group in relation to the whole?

Once we have a fraction on the table, we can start rebuilding. (At this point, I would have them go back to their seats.)

I would flip the table at this point, to show the connection to the fractions they usually see. I want them to make the connection from part to whole.

The next step will be to reconstruct our groups: if one group is 1/nth of the whole class, how will we talk about two groups in relation to one? What about the whole class in relation to one group? (n here represents how many groups a class actually comes up with, but this could also be the goal, to start with two teams, three teams, four teams, and have the students introduced to the mathematical extension of ‘n’!)

After this discussion, I would reinforce the problem solving with the penguin problem, (how do we share 30 snowballs among three penguins?) using small groups. At this point, I will be checking in with each group to hear what they are saying and to reinforce the ideas of sharing, naming and reconstructing. I will be listening for understanding.

All of this should only take a fraction of the lesson time. I am asking the students to think about fractions in a primary way, to connect the more elaborate work of operating with fractions. The students will be encouraged to use the language of fractions, wholes and reconstructing as they work with grade level problems involving fractions. Students will be encouraged to break down fractions or reconstruct them to assist in solving problems, with drawings, with manipulatives, or using the fair share table. Then they could begin to operate with the fractions, understanding how to combine groups, and in further lessons, what happens when groups are multiplied and divided.

I see my understanding of how children instinctively do this work informing my ability to identify students who have “lost” or buried the skills and to assist in reconstruction of this knowledge to enhance and strengthen their future conceptual understanding.

 

So let’s say a television is falling on your head. Will a bigger TV kill you faster?

You are standing on the sidewalk. Somebody yells, “watch out!” And you look up and realize a television is hurtling towards your head. You have 2 seconds to move out of the way. What floor was the television dropped from?

This is a reverse of the classic egg drop problem, which asks the student to figure out how long it will take an object to fall from a specified height. The formula for gravity and time is usually provided. We are going to turn this lesson on it’s head, literally!

The lesson is designed for two 45 min lesson periods. I’ll give the standards and practices at the end (or maybe I’ll let you tell me which ones this lesson hits!)

The lesson starts with an exploration of gravity and a ‘where does the formula come from’, and moves to the exploration of the above scenario – quickly, before somebody gets smacked on the head by a TV! (The title of the blog is the question that kicks off the gravity part!)

Start with the question: So let’s say a television is falling on your head. Will a bigger TV kill you faster?

Have the students make conjectures in their own minds. Have the silent thumbs up when they are ready. Pair students up and have them share their conjectures, giving reasons that the other person can articulate. Go around the groups and ask what they decided: what the question was asking, whether a bigger tv would fall faster than a smaller one, give the reason(s) they felt that was true or false.

If your facility offers a one or two story drop (football stadium announcer stairs?) conduct this next experiment empirically.
Or, use this great video called Misconceptions about Falling Objects.”

First, watch only the first part of the video, where the interviewer is asking random people whether a heavier ball will fall faster or the same as a lighter weight ball.

Stop the video before they get to the proof part. Survey your students. See how many agree with the heavier is faster theory. Ask them why. Have them explain their thought process.

At this point, you can take them outside, let them try it themselves with two different weight items, or you can play the rest of the video, or both.
If you get to do the outside drop, bring a stopwatch. Set up some students to drop and some students to time the drop and determine how fast the ball drops. Have the students estimate the height (if you can, measure it ahead of time, so you can work the math first) and come back to the class.

The task ahead of the students is two- fold:
1. To understand that items fall at the same rate, no matter the weight, and to perhaps extend to the connected idea that gravity is a pull that has nothing to do with weight – weight has to do with density of matter. (If you have a wonderful science teacher, maybe she will work on a connected gravity lesson!)
2. Develop an understanding of the parts of the formula used in so many of their quadratic one-variable math problems, as they develop their speed of gravity based on the tests, or in discussion from the information from the video.

I will tell you that the video gives the formula in meters per second. The formula in the Alg II Word problems are generally given in feet per second so take some time here to work the students through a conversion process, so they will see the variations as different ways to write the same formula.

Rest or break here. Debrief the students, let them write out their understanding of the question asked in the title- will a bigger TV kill you faster? See how many students changed or enlarged their views. Have them articulate what changed, or if they thought objects would fall at the same rate, how did what they did support their initial thought.

Pick up the next day with the formulas. Let the students (whole group) come up with thoughts as to how they are different and why one formula might be more appropriate than another.

Now, pose the second scenario:
you are standing on the sidewalk beside a multi-story building. Above your head, someone has knocked a window air-conditioner out of the window. If you don’t move it will land on your head. You have 2 seconds to move. How high up does that window have to be to give you time to get out of the way? If you double the height, does that double the amount of time you have to move?

Have the students each think about the problem individually and observe their ideas. You may need to ask some questions of those students who can’t get started, to help them find an entry point. Ask them if they could draw the problem, or have them list things they will need to know (rate of gravity) to solve the problem. (Working the problem looking for height requires re-arranging the formula, which is why I wanted the students to really understand the parts and where the numbers come from.)

The next step is group discussion. In groups of 3 or 4, have them share their ideas and work together to discuss how they feel they can solve the problem, discuss which version of the formula will be appropriate to use, and details, like the average height of floors in a multi story building.

Encourage through questions some hypotheses, maybe a drop from the second floor. At this point there will be some discussion about how to mathematize the answer. One of the recurring problems for students deals with the square root solutions required to solve quadratics. Students should have already worked through the various ways to solve quadratics, so you will want to listen to the conversations surrounding these issues.

Some students will be able to solve for the square and some will not. Have the student who does solve the square explain his/her understanding of the process. If students in each group come up with differing ideas, let them question, critique, convince, until they can agree on some solution. After all, their life depends on knowing how fast they must move!

As you listen, decide which groups to call on for sequencing purposes. Bring the group together and let each group talk about what information they used to solve the height issue. As students speak, if someone realizes their mistake, let them correct it, then iterate the thought process – why did they think the mistake was reasonable?

Continue to ask questions to guide the students through the ideas, asking students to reiterate what they heard another student saying; asking a student to clarify a point, to dig deeper into the thought process.

They should, of course, come up with two answers, positive and negative. So the next question is which answer makes sense and why? What does the negative answer represent? What would have to happen for the negative answer to make sense?

The students can practice their skills by asking them the same problem, only transferring the location to Mars. If the television dropped from the same height on Mars, would they have more time or less?

Don’t forget to ask about doubling the distance. Let the students confirm whether or not that is true and why.

This is a good time to pull the students together and have students restate the ideas they’ve heard. Watch and listen for evidence of a change in student understanding. Are they making the same errors or different ones?

To cement the knowledge, have this similar problem ready: you are in a boat at the foot of a cliff. You look up and see a man tumble straight down. Develop some different scenarios for the height of the cliff and the time it takes him to fall.

Okay, I won’t make you guess the standards and practices- but if you see any I missed, let me know!
The CCSS task that is directly addressed here is Algebra II: Quadratic Equations in one variable (standard A.REI.4)
The activity supports all eight mathematical practice standards. The various individual, small group and whole group tasks give plenty of opportunity for formative assessment of math skill levels and understandings. The Mars thing – have either an internet access to look up the gravity on Mars, or have it ready and let the students who get done early work on the Mars formula for the rest of the class to use. They can do a mini present on how they got it.

There are sure to be multiple paths students took to get the answers. Let whole group discussion offer opportunities for students to share how they thought about solving. Put the paths on the board and let students compare the paths – they will find ways to add some of these new paths to their own toolbox if skills!

As always – feel free to use, modify, comment, and question. Thanks!

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Do You Speak Words of Life or Death?

“Thoughts become words, and words have the power of life and death. Think to speak life giving words to yourself and others.” Joseph Prince

These words put me in mind of how we as teachers have the power to create hope or plant failure in the minds of our students. Our students believe us. For good or ill, we hold all the answers (even when we don’t).

We can use that belief to inspire our students, to enable them to reach beyond anything they might be willing to do on their own. Here is one way:

Fixed vs Growth Mindset

Do your students have a Fixed Mindset or a Growth Mindset? Do they eagerly tackle problems, curiously start investigating the challenge placed before them, generate ideas for possible solutions?
If you give students a choice of assignments, do they pick the easy ones or the hard ones?
Do your best students want class discussion, or do they just want you to show them how to work the problem, so they can do their work and keep their A?

It breaks my heart when a student walks into class and says ‘I’m not good at math. Don’t expect much from me.’
Or the kid who told me, ‘whenever I’ve done enough to pass, I’ve been told that I usually quit trying.’ Or the A student who is afraid to try the problem until you have shown them how they are supposed to think about it. These students are exhibiting Fixed Mindsets. They have been given the idea, by teachers or parents or grouping or grades that they are dumb or smart or not good at math, or science, or too smart to fail at math, or science, or any other subject. Girls, especially, tend to get the message: math is not your ‘thing’; children are grouped by what teachers believe they can do: advanced, remedial or in-between – all a form of ‘silent’ stereotyping, as deadly as anything we say out loud. What message am I sending when I too quickly provide an answer, instead of asking good thought-provoking questions? How can I encourage a growth mindset?

What is a growth mindset, and how does it benefit our kids?

A growth mindset is the ability to see possibilities. It is a confidence in one’s ability. It is the MacGyver in all of us, to use a modern example. And it can be taught!!!

Teachers can encourage a growth mindset by changing the messages they give to their students:

“I believe in you”

Don’t let grades define your student. Instead of a grade, give specific feedback on problem areas. Tell them – put it in writing – that you are giving them the feedback because you believe in their ability to fix their mistakes.

Celebrate mistakes!

Your students are scared to death of making a mistake, getting it wrong, so they sit on their hands in class. Am I right? How frustrating is it to ask a question about the material and be greeted with blank stares?

It is time to celebrate (maybe even reward) mistakes. Sounds crazy, I know, but hear me out…

Tell the kids that every time they make a mistake and struggle with fixing it, their brain gets smarter. (Don’t worry, it’s true, check out this link: Brainology)

Reinforce the message:
Give them a picture of their head in profile with a brain drawn in it. Make sure the brain has lots of little empty spaces in it, like the one I’ve posted below (or break out the projector and let the kids make their own personal profiles on blank paper).

Tell them that each time they find a mistake, they get to color in a section of their brain. Celebrate the fact that their brain has grown! (Younger students like stickers; older students may want stickers, too! It doesn’t have to be anything huge.) I would even keep a big poster of a brain on the wall- on a really tough day, where the kids have made it through with lots of effort, color in a space on the class brain- they have become collectively smarter! This is also a great community builder. It’s like those scaling a wall, walking on ropes exercises where you can’t make it without everybody pulling together – without the parental permission forms!

And for those of you who aren’t sure whether you have a fixed mindset or a growth mindset, here’s a link to a great article, with a short mindset assessment at the end. Growth Mindset

And don’t worry if it tells you that you have a Fixed Mindset- it is only temporary!!

What did my results say? What do you think? Am I a Fixed or Growth mindset kind of person? I can’t wait to hear what you think!

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