Remember Uncle Drew

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The second day of school we jumped head first into an investigation. Students were presented with this context. Within groups of four they worked on VNPS investigating each scenario. This was my first time using vertical non permanent surfaces, although I had been encouraged to use them a couple of years ago. I loved the easy view it allowed me to have of student thinking. Walking around to converse with each group seemed easier. 

This investigation provided insight to several things: 

  • How students work in group settings. 
  • How students can use a context to make sense of a new concept. 
  • Students’ background knowledge of solving equations. 
  • Students’ ability to persevere when things are difficult. 
  • Can students write equations from a context. 

*Our first unit dealt with solving multi-step equations with special cases.  

Students were given access to Algebra Tiles and Algeblocks during the investigation. We discussed the conventional meaning of the blocks, which helped groups like this make sense of the quantities within the first scenario. 


While students worked, I observed their strategies and listened to their explanations. I used this monitoring sheet to identify which groups were thinking what way all the while considering in which order I would have them present during math congress.  

During math congress, 3 people representing 3 different groups shared their findings. This process hit some many realms of instruction, formative assessment (like an informal pretest), use of manipulatives, SMPs (especially 1, 2 and 3), peer corrections, writing equations from real-world situations and solving equations with special cases. 

The lesson took 2 days, one full day of investigation and preparing for math congress and the second for math congress and focused instruction. On that second day, I used the students’ findings to lead focused instruction on equations with special cases. The use of the context helped it make sense to students and gave the procedure for solving equations a purpose. I was able to provide formal vocabulary such as one solution, no solution and infinite solutions based on students findings. Throughout the rest of the unit, we were able to always connect our thinking back to Uncle Drew’s points and “our” points. 

The two posters above were not shared during math congress. 

Creating a Remediation They Want to Come to!

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At the end of every 1st semester, students who failed a course are invited to participate in Comet Academy, my schools grade recovery/remediation sessions.  This year, those attending Comet Academy come for 2 1/2 hours for 6 consecutive Saturday mornings.  During our first session on January 30th, students were given a 30 questions, multiple choice pretest covering 5 key standards from 1st semester.  I was assigned the 8th graders who failed to which I was very excited.  8th grade is the one grade level I haven’t been able to infiltrate with the use of manipulatives and building conceptual understanding.

Students completed the pretest in about an hours time, followed by checking the answers of a partner.  Stunned by the results, I knew immediately, 6 sessions, even at 150 minutes a pop, wouldn’t be enough to get the students to pass the post test.  The highest score was 11 out of 30 correct, ugh!

After proposing to the math AP about the need for a math boot camp for students who failed to stabilize their foundation for high school, I got busy planning.  8th graders who failed 1st semester, whether they attend Comet Academy or not, would be pulled twice a week for 30 minutes each time for 5 weeks.  This would occur during their Connections classes (electives, specials, not the core classes).  This boot camp would only offer additional support and would not be rewarded with a grade, extra credit or anything tangible outside of a better understanding of the content.  For students who are often motivated by outside factors, I needed to ensure the activities were enticing enough to get them to come week after week.

Meaningful Practice

The standards are not new for students, therefore a mix of meaningful practice and concept development is necessary.  In their regular classroom, students are subjected to worksheet after worksheet or textbook page after textbook page for practice.  In boot camp we used a Solving Equations Bingo game to practice solving equations.  Students filled in the answers: x=3, x=11.5, x= 1 1/2, x=5, x= 17, x=4, x=17 and -72 =x into the Bingo board.  Then I read off equations in which students needed to solve in order to cover the correct answer.

We also played a game of Knockout!  This idea was taken from the basketball game Knockout!  In the basketball version, participates line up behind the free throw line, the first two people in line have a basketball.  The goal is for the second person in line to make a shot before the person in front of them in order to knock the player out of the game.  In boot camp, students sat in a straight line and the second person tried to correctly answer the math problem before the person in front of them did in order to knock them out of the game.  Here’s the PowerPoint with the game: Boot Camp 2-9 and 2-11.

To encourage collaboration, students were grouped (they chose girls against boys) and given whiteboards to record the answers to problems.  The group representatives would hold up their whiteboard showing their answers.  In order to receive a point, teams had to get the correct answer, but also shot a tiny basketball into a toy hoop.

Concept Development

To help to continue to build student understanding of solving equations and integer rules, I used lessons from Hands-On Standards and incorporated color tiles and algeblocks with lessons like this Exponent Activity.

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Formative Assessment

I have to know where they are from day to day because I don’t have a lot of time to cover the material.  Therefore, each session has a connected formative assessment.  This helps me to plan differentiated lessons even within the small group of students I see on the different days.  I’ve used a two question quiz, a portion of a FAL and a Ticket out the Door pictured below.  Students were able to choose what type of question they wanted to answer which is a formative assessment within a formative assessment :-).

 

I’m Not a Special Education Teacher

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I’m not a special education teacher but I’ve been around a few in my career. I’ve been around those who push their students to meet goals and go beyond them. I’ve been around those who believe their students have great potential and we just have to provide support to get them there. And I’ve been around those who limit their students capabilities by believing they “can’t do it”. 

I’m not a special education teacher, I’m a teacher who believes all kids can make sense of the mathematics when provided the opportunity and support. I remember working with a group of students at an elementary school. This was a targeted group, meant to increase their conceptual understanding and pull up their math scores. Within the group were regular education and special education students. The stand out kid was Ezra, a special education student.  Ezra had strategies that would impress any math teacher. The strategies taught through the lessons for the Numeracy Project books he would develop an understanding of them quicker than the other students. And just like most kids, when allowed autonomy, he would choose the strategy that made the most sense to him. 

Fast forward to present day.  From my observations, it seems some special education students aren’t making sense of the mathematics. This could be caused by numerous factors such as, teachers no longer allowing opportunities for students to make sense of the mathematics, teachers exposing students to only one way of thinking, not enough wait time resulting in the answers being given by the teacher and teachers literally taking the pencils out of the hands of students to do the work for them. 
When I model instructional strategies for teachers, I go in believing the students will be able to make sense of the mathematics when there is effective scaffolding and questioning (which takes a lot of pressure off of the teacher). *Sidebar: it takes less work to scaffold understand through a mini lesson and ask guiding questions while students work independently of you than it does stand at the front or back of the class walking students step by step through the process. Less work, more effort. It’s the difference between pulling a bus along a path to a destination and pulling Radio Flyer wagons along the path. It’s more effort to pull multiple wagons at different times, lighter work though. *

This brings me to the CGI lessons I conducted in a SLD (Specific Learning Disabilities) class and a MID (Mild Intecllectual Disabilities) class. Each class contained 8th graders. In both classes I observed some practices mentioned above which gave the appearance that students couldn’t make sense of the mathematics. Within the SLD class, students were given an expectation (draw a picture to make sense of the problem), provided scaffolding through guided practice, shared their thinking with those within their groups and derived the steps for solving one and two step equations. 

   
The students transitioned to contextualizing a given equation and solving. 

   

  

  

  

  

 For students who once struggled with solving equations and math in general, it began to make sense. 

In the MID class, students were provided an organizer to help with processing the activity. Teachers were encouraged to allow students to think through how they wanted to represent each problem and to not take the pencil out of the kids’ hands. 

  
Together we read the first problem and students were asked, “what pictures popped in your head first?” Most said the 7 cars. Students recorded that image on their paper as I recorded it on the board. When ask what did they picture next, most agreed it was the 12 cars. They drew 12 total images to represent the 12 total cars. Lastly, we labeled what each portion of the drawing represented. Because one of the goals the teacher set for this model lesson was verbal to written explanation, I had a student explain his thought process and recorded word for word what he said. Pictured above are examples of how students made sense of those problems. Students were successful when provided support through guided questions. 

During the post conference with the MID teacher, she expressed how she was surprised by the students abilities to make sense of the mathematics and by how much the students were engaged. She has begun to incorporate components CGI into her everyday mathematics instruction. Her next step was to target division problems highlighting start unknown, change unknown or result unknown using manipulatives.

CGI in Middle School 

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If I could do last year’s lessons on solving equations all over again I would begin with CGI. As a whole, 7th graders at my school struggled with solving equations last year. And as I’ve been in 8th grade classes and had conversations with 8th grade teachers I see the struggle continues. I’ve concluded that many students do not understand the real meaning of the equal sign.

When you ask students what the equal sign means the overwhelming answer is “equals”. When you ask if they can give another meaning the look of confusion covers their faces as they respond, “equal to?”

Starting with CGI, Cognitively Guided Instruction, helps students to make sense of a concept through making sense of a context. Through this process of thinking, students are able to conceptually understand a mathematical idea such as dividing fractions, addition and subtraction and even solving equations.

This past week I worked on a few CGI problems for middle school concepts. My plan is to do a couple of professional learning sessions where we can focus on this instructional strategy. 8th grade is my first group. An 8th grade teacher asked if Number Talks could be used to introduce systems of equations. What a perfect fit for CGI.

Here’s a problem I wrote. Students were instructed to only draw diagrams with labels to solve the problem.


Pictured above are two students beginning thoughts.


How two students served in a special education resource class thought about it. 
 More processing going on.



Students realized the answers needed to be 18, not more and not less. They had to use reasoning to determine how much of each fish was needed in order for it to be equivalent to 18. Within their explanations, students expressed they had to determine one unknown in order to determine the other unknown.  Imagine students being able to develop their understanding of systems of equations in this manner. Making the bridge to the abstract would make more sense.

Here are other CGI problems I have written and examples of more. If you have a source for more please let me know.

This Game Has Broken Up Friendships!

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Each time I’ve participated in a @Globalmathdept webinar I’ve learned something I could take back to my classroom and use almost immediately. This past Tuesday, I participated in the “Helping Struggling Students” webinar in which the game “Grudge” was introduced. As soon as I heard it, I knew I had to try it. So the very next day I did.

Feeling the name “Grudge” had a negative connotation, I changed it to “Last Person Standing” and explained the rules to the students.
Rules:
1. Each person starts out with 3 x’s.
2. Because we were working with solving equations, I would read the equation in word form. Students had to interpret the equations and solve for the unknown. This was done in their math notebooks to capture for note taking purposes.
3. Students were to record only the answer on the white board and hold it up when prompted by me.
4. Each student who solved the equation correctly would have a chance to erase an x on the board.
5. If you lost all 3 x’s you could no longer win. However, you could still play as a “ghost” or “zombie” and erase x’s.
The goal was to be the last person with an x by your name.

Excitement filled the air and the plotting began.

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When watching this video, listen carefully to the buzz happening after students have determined the correct answer.

Last Person Standing.

I don’t want it to seem like getting the correct answer was important to me. What was important was what the students were doing in order to position themselves to erase an x. SMP 1: students had to interpret the written equation in order to solve for the unknown (multiply 1/2 by a number then add 6 which is 10). SMP 2: students had to reason about the quantities. I did not teach them procedures for solving equations. Prior to this math review activity, we modeled equations using balance scales. Students determined two ways in which to determine the unknown, guess and check and use known information. The latter prompts the reasoning. For example, when solving 1/2x +6=10 a student explained, “I have to get to 10, so I did 10-6 and got 4. So I know 1/2x has to equal 4. Then I used guess and check to get 8.” We also discussed what we could do to get the unknown by itself. We call it keeping both sides equal. If we remove or add something on one side it has to be done to the other in order to keep both sides balanced. SMP5: students choose tools to use when they needed them. When guess and check didn’t work, they used a different method. For certain questions they recorded the information, others they could do mentally. And for some they whipped out a calculator. SMPs 6 and 7 were in there as well.

Our principal joined in the action during my 3rd academic class (I’m upset I didn’t get a picture). Upon leaving class, students would tell those waiting to come in what a fun game was awaiting them. There were many, “I had so much fun!” But my favorite one was, “Ms. Sexton, this game has broken up friendships!”

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