# Do you think teaching math causes you to over think??? Almost like “going insane “…NaNa Dunn from a post on Visual Math’s FB page

Oh yeah! I used to dream about my lessons… over and over during the night. I’d wake exhausted. Here’s an example of what I do now…

When you realize the students can’t ‘see’ the thing you are teaching them

My kiddos are learning to write quadratic equations. This requires them seeing patterns of growth. The first lesson in this series was to help them see the growth in a pattern of cubes. I used a YouCubed lesson that involved coloring each growth step. Just that. This exposed several issues. The lesson with numbers was the next day, and I was pleased by the intensity of their interest, but here’s why I think that happened: I had prepared them for what they were looking for.

Quotes taken from the original FB post…

Beth Hanna McManus writes “My 8th graders… totally missed the concept that the altitude to the base of the triangle goes through the vertex and hits the base at a 90 degree angle. I explained this in mathematical terms, in layman’s terms, using words like straight to the base, “shortest distance”, had them draw them on the white boards (checking and helping each drawing) multiple times, had them label triangles. When it came time for homework, no one knew what to do. I flipped out and spent the rest of the day in a dither wondering what is wrong with me that I can’t communicate with these kids…”

When I plan a learning experience, I look for the base skill/image/idea a child needs to be able to have to participate in a lesson. It may be a simple warmup – for the triangle problem above, after realizing they didn’t get it, I might have them draw the altitude, pointing out the way the angle looks, and how it starts in the vertex, in several different triangles. Just that. No math, just letting them learn to see what they are looking for.

Starla Adams writes, “Yes, yes, and yes….says the teacher who is out of strategies to teach long division after Friday. I actually thought I was going a little bit insane for an entire hour.”

Teaching long division is challenging. I’ve worked with 9th graders who do not understand what dividing does. If they don’t have an understanding of partitioning and regrouping, long division is just a nonsensical set of steps that they must follow – and memorize. A warmup using manipulatives (coins buttons beads) to set the stage for dividing into groups would help them understand what they are looking for. Then long division can be taught as a routine to get there.

Another common student skill gap with division is poor factoring skills. A warmup (or preview lesson the day before) with a factoring math problem string like this video from Pam Harris, might help strengthen their math fact fluency.

While working on any new skill that requires factoring, try giving students a factor chart. They won’t be using all their working memory on remembering number facts, but on learning the process of the task at hand.

Learning plans should empower students…

I realized that my 9th graders didn’t yet know that they had the power to create their own understanding. They were waiting for me to tell them what to do. Danielle Love and Kay Butler point out ways to shift the heavy lifting (and learning!) to the students!

As teachers, we spend lots of time creating learning plans. Many of us already know what misconceptions kids have, and what errors are going to get made. So lets plan ahead to expose -and remediate and preview- so that these issues don’t cause student failure during the learning of new material. These little discovery sessions and warmups are critical to building understanding and are often worth every minute of time we spend on them!

By all means, overthink and go crazy, in a most productive way! Math teachers, you rock!! I would love to hear how other teachers prepare for these misconceptions and gaps!

(And no, I don’t have those recurring math dreams nearly as often anymore!😂)

Thanks to Shana McKay of Scaffolded Math and Science, and and this really interesting thread on her fantastic FB Visual Math!

# 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?

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?

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.

# 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!

# New Year’s will be in August, this year.

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!

# I let the kids teach today. They (well, most of them, anyway) rocked it out of the park!

#MTBoS30 A simple review activity created positive energy in the classroom today.

This is a difficult week: End of Course tests are done, finals are a week away. Students tend to relax and give into the end of the year lassitude. There are still a unit test and a performance (writing) assessment to go before the finals, but the students are completely burnt out on anything with the word review in it.

To combat the blahs, I designed a student teaching assignment. Students were paired up and given three brief instructions: create a lesson, teach it to the class, and ask the class to perform a brief activity to show understanding. We assigned each pair a problem from the sheet of review problems for the unit (on probabilities) and gave them 15 minutes to create their lessons.

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Some students required help with their understandings, and as I walked around listening to the students planning their strategies (20 points for sharing the teaching responsibilities), I was able to discern which students needed help and which ones didn’t. It was an enjoyable day of formative assessment for me, and having them teaching freed me up to spend more time facilitating the learning.

It was fun to watch the students mimicking me (and boy did they!), but I also saw a different side of many of the students as they led their peers, answered questions, and walked around the room helping other students understand the problem. They were reviewing without a single complaint! (Well, maybe just fewer complaints!)

This lesson took a block period, which for us means two 50 minute back to back sessions (a regular geometry and a strategies class). This gave us time for a brief introduction to the types of problems that we were going to review, a setup of the task and partnering, and 15 minutes for the pairs to plan. The students really needed about 2O minutes to plan, and we were able to give each pair about 5-7 minutes for lesson presentation and follow up.

Assigning the problems was made easier by taping the problem number to every pair of desks. As students came in, we sent the students where we wanted them to sit, which streamlined both the pairing and assigning problems portion of class.

The photos show the wide range of visuals the students employed to teach their lesson. They also used the smart board, however, that was more an electronic chalkboard for them!