Are You Ready For More?

Are you ready for more? That is the question the authors of Illustrative Mathematics and Open Up Resources ask students following engagement in problem solving routines and instructional activities.  This week my students responded with a resounding, “BRING IT ON!” Not only that, they attacked the activities and problems with such zeal this educator was left with goose bumps and happy tears!

I teach 6^th – 8^th grade middle school students with disabilities and have the honor of looping with them throughout their junior high years.  My current eighth grade students were my first sixth grade group. From day one I have incorporated Dr. Jo Boaler’s Growth Mindset research, teaching students the tenants as we have explored the Weeks of Inspirational Math. We have embraced the belief that with struggle, mistakes, problem solving and a growth mindset everyone can learn maths to high levels.  In addition, my pedagogy has included constructivist ideals with heavy doses of productive discourse, collaboration, and joint construction of knowledge. I have consistently utilized the routines of Notice/Wonder, Estimation 180, Which One Doesn’t Belong, Visual Patterns, Empty Number Lines and Grayson Wheatley’s Quick Draw as tools to teach students the problem-solving process, and  have been utilizing curriculum that is inquiry based and student centered.  To this end my students have become familiar with self-talk and working through the problem-solving process in a way that makes sense to them. We have developed the following anchor chart as a guide if we get stuck.

We have also created an anchor chart to remind us what we expect from each other when working in groups and collaborating.

When I discovered Open Up Math Resources, a beautiful curriculum grounded in routines for reasoning, research-based practices, student centered, world connected, inquiry based instructional practices that resonate with so many of the constructivist philosophers I have come to passionately embrace, I became an instant zealot for the curriculum!

Following is my attempt to capture a moment in time that happened this week in one of my classes. For this educator this is evidence that the paradigm shift educators are being asked to make concerning their pedagogy is vital and life altering!

Are You Ready For More?

When students completed the exploration of the instructional activities in lesson 6.1 Tiling the Plane they were given the following prompt:

On graph paper, create a tiling pattern so that:

• The pattern has at least two different shapes.
• The same amount of the plane is covered by each type of shape.

My students have disabilities and they have learned that there are many tools available in our classroom to aid them in the removal of barriers to their access to the mathematics we are exploring. Many of my students have dysgraphia, dyslexia, fine and gross motor challenges as well as a plethora of other disabilities that require supports. For my students with gross and fine motor barriers, drawing with conventional paper and pencil as well as on a computer is too restrictive.

With that in mind one of my students went straight to the pattern blocks to tackle this problem. The student expressed the desire to create a tessellation that would satisfy the prompt.

The tiles chosen by the student were hexagons and trapezoids.  After a little while working with the chosen tiles the student created the following pattern.

While I was circulating among the students to monitor understanding, strategies and misconceptions I found this student working diligently. They asked me what I thought of their creation.  I find I always channel my mom in these situations and turn the tables by saying, “It doesn’t matter what I think of it, what do you think of it?”  Who knew mom was a constructivist?  The student said they really liked their design, but was not sure if it was correct.  I reread the prompt and asked, so what do you think?  Are your shapes covering the same amount of the plane as the question asks?  The student counted the trapezoids and said they know it takes two trapezoids to make a hexagon so there were too many trapezoids.  They then decided to try an easier problem to help them break down the design. To do so they pulled out a portion of the pattern to critique (solve an easier problem).

The student then said, “I notice that for every one hexagon there are two trapezoids. So, in this pattern there is a two to six ratio.” I asked, what relationships do you notice or wonder about that information?” The student said, “Well, I will need to think about common multiples and maybe factors.”  They thought for a while and then said, “I know I can multiply 2×3 to get six, so I wonder if I start with six hexagons and 12 trapezoids, will I be able to create a hexagon pattern with them where the yellow and red cover the same amount of the plane?”  With that I left the student to explore on their own for a while.  When I returned, the student was experimenting with several patterns, and was starting to create a straight-line pattern like the following using six hexagons and twelve trapezoids.

At this point I was happy to see that the student was showing understanding of decomposing a shape into different shapes, and that the new decomposed shapes still cover the same area.  My student on the other hand was not happy.  They did not like the design and expressed the desire to create a hexagon, and a more elaborate pattern.  They rotated, translated and wondered aloud about orientation and were quickly on to something! After a short period of time the student created the following beautiful piece of mathematical artistry that met the requirements of covering the plane!

The student was disappointed that they could not physically draw their design, but was thrilled when I suggested they take a picture of their work and upload it to our Google Classroom with their assignment.  They were also not happy with the gap in the middle and said they were going to work on this more at home!

This one moment in time is what every educator lives for.  It is a moment when all that is learned before, and what is being learned come together cohesively and flawlessly. This moment would not have been possible if this student did not feel safe to take risks, make mistakes, make conjectures, develop strategies, reason abstractly, and problem solve.  This child that has been identified as having significant learning disabilities has a beautiful mathematical mindset and disposition as well as a growth mindset! This child is a critical thinker, a problem solver, a risk taker, and a world change maker! This child proves that everyone can learn maths to their highest level and if educators will make the shift to teaching students to think critically, problem solve, collaborate, communicate, take risks and have a growth mindset, the possibilities for learning are limitless!

So again, I ask, Are you ready for more?  I know I am!  I can’t wait to provide routines for reasoning and instructional activities for my students so they may become amazing mathematicians and thinkers! As a bonus, I will have the opportunity to stand as witness to their mind-blowing awesomeness this school year!

 

First Week in Review

A new school year always makes me a little nervous because there are so many unknowns. Who will my students be? What are my class sizes going to be? Who won’t return this school year, and why? What triumphs and tragedies will my students have experienced over summer break, and what will that mean for their school and life experiences? After so many years teaching these, and so many more worries, run through my mind for weeks before our first day of class.

Now that the first week is in the books I am breathing a little easier while at the same time feeling stress from other worries! This profession that I, and so many others have chosen, is a 24/7 way of life!

The first week also has me celebrating a plethora of successes!  I always spend the first several weeks establishing norms, teaching the problem solving process, and building a safe learning community.  First experiences indicate that this year’s community is going to be amazeballs!

Here are a few of the activities the MNM Math Nerds have engaged in thus far:

Kid President’s Letter To A Person On Their First Day Here

Time to put away FORTNITE// “I Gotta Feeling” The Black Eyed Peas Parody

Name Tents thanks to Sara Van Der Werf!

Me…By the Numbers, Math Activity by Donna Boucher

MINDSET ASSESSMENT PROFILE TOOL

Math is….. snowball fight ala Sara Carter

Go Ahead Break the Ice from Twitter Math Camp 2016

You Are an Important Piece of Our Puzzle

Give Us Your Champion adapted by me from a game I played at a workshop this summer

 

All of these activities were not originally mine, but they are gems in helping establish community! I am so grateful for a wonderful Math Twitter Blogosphere, #MTBoS,  community that so readily shares and collaborates!  In addition to these activities MNM Math Nerds also learned the process of problem solving though www.estimation180 activities as well as developed a class set of expectations for one another while working in groups.

These are the ideas we brainstormed for our group work expectations:

And here is our final product:

The final activity for the week was to jointly create a class creed. From these collaborations  I realized that I am going to have the honor to learn with some dedicated and serious mathematicians this school year!

Our final product:

 

We start every class period with a Take Five because we are a Great Expectations school,  and this  routine sets the stage for our learning. I am excited that my MNM Math Nerds already have Great Expectations for themselves and one another!  It is going to be a great year to be a Zebra! #GoZebras

I Am An Open Book: Take From My Pages What You Need

I recently had a frank conversation with a friend that has caused me to reflect on my teaching practices outside of the classroom and my personality. From those reflections I feel compelled to write a disclaimer blog concerning me.  I am fully aware of my idiosyncrasies and short comings because I am the analytical, problem solving, giving, surviving me that I apparently was created to be.  Readers, the things that annoy you about me, also annoy me about me.  I am a fixer, and believe me, if I could fix me, I would have been “fixed” decades ago!  I am a work in progress, and I am constantly trying to be a better me. Hopefully the following insights will help those around me tolerate, and maybe even embrace me, with open arms.

First of all, I am the oldest child in my family, and I display the typical organized, reliable, achieving, and tightly wound characteristics as such. This was not a choice I made or embraced. It is hard to believe, even for me, that I was once a quiet and shy child that wished to disappear into the paint on the wall. My childhood circumstances caused these characteristics to surface as I came from a dysfunctional family, and in order to survive I had to adopt these traits in order to help my family members.  I will not share gory details, but the surface story goes like this:

My parents got separated/divorced seven times from each other. I have attended sixteen different institutions for learning. My mother had bi polar disease, my brother had schizophrenia, and my father was a recovering alcoholic.  When my parents finally divorced when I was sixteen, my mother chose to live with an alcoholic drunk and my father moved in with a woman who had six children of her own.

One day, when I was in second grade in California, my mother came to school with two suit cases and checked my brother and I out of school. We went to the Greyhound bus station and for the next three days we were on a cross-country journey to Oklahoma. Unbeknownst to my father, my mother had sold everything in the house that she could, and gave the rest to charity. My father came home from work and had no idea where his family had vanished to. As a second grader I thought this was just one big adventure, but as the years went on I learned how crisis and change devastate lives and families. Amongst all of this I became the parent, the caregiver, the rescuer and fixer of my birth family.

Fortunately, among all of this dysfunction and upheaval I had teachers that chose to see past the dirty little transient kid sitting in the corner of their classroom. They looked past my terrified eyes, my dirty second-hand clothes and saw me. They saw my potential, they saw my talent, they saw a future that I dared not dream for! They taught me to see me, to believe in me, and to dream for me. I believed and I became.

The becoming was no joy ride as I still lived in a dysfunctional family embroiled in poverty. I still had emotional scars that needed healing. I still was living in a situation that is typically statistically impossible to overcome. If you have ever read the books “Running With Scissors”   or “ The Glass Castle: A Memoir” you have had a glimpse into my childhood. Despite these barriers, educators stepped up and helped me with the bureaucracy and paperwork that comes with getting an education. Unfortunately, no teacher EVER did this for my brother, and the differences in our adult lives are staggering!

I am the lone survivor from my birth family. My mother passed away fifteen years ago due to a complication with her mental health. My father passed away ten years ago as a complication of smoking, and my brother died from lung cancer in a mental hospital five years ago. In contrast, my husband and I have been wonderfully married for 33 years. We have not been without heartache as our first-born child died from congenital heart defects.  We are blessed that God sustained us and nurtured us through our loss.  We have two hugely successful college educated children that are living their dreams. On the surface it would appear I live a fairy tale life. I feel truly blessed and grateful to be living this life and do believe God has granted all of my dreams come true. The reality is, on the inside, I am still that little dirty transient kid in the classroom corner just trying to make a better world for myself and others.

I recently had the opportunity to participate in a True Colors® personality exploration as part of a professional development conference.  I discovered that I am considered to be a Green/Gold. The following table shows how Green/Golds see themselves as well as how others see them.  This was reaffirming while at the same time disheartening.  While I see myself as someone who is exuberant, passionate and sharing, others see me as a know it all, uptight, snob. Nothing like a slap in the face to wake you up!

See Self	Others See
Tough-minded Efficient, powerful Original and unique Rational Great planner Calm not emotional Precise not repetitive Under control Able to find flaws objectively Holding firm to policy Stable Providing security Dependable Firm Always have a view Efficient Realistic Decisive Executive type Good planner Orderly, neat Punctual, expect same	Intellectual snob Arrogant Afraid to open up Unappreciative, Stingy with praise Doesn’t consider people in plans Critical, fault-finding Cool, aloof, unfeeling Eccentric, weird  Rigid Controlling, bossy Dull, boring Stubborn, pig-headed Opinionated System-bound Unimaginative Limiting flexibility Uptight Sets own agenda Rigid idea of time

My favorite quote sums me up in a nutshell. “I am part of all that I have met” Tennyson.

I am me because of everything and everyone that I have encountered in life.  I would not change anything that has happened to me because I would not be me. So many people have given of themselves to help me through this life, and now it is my turn to give of my time, talents and resources to help others. I work hard not to be an over sharer. I am contentious about how much I talk and share. I try desperately not to dominate a conversation or situation. If you have been a “victim” of my over zealousness, and are turned off by it, I truly apologize! Please know that my enthusiasm comes from a place of authenticity and passion for humanity and the teaching profession. So many opportunities, so much help, so much love and acceptance were withheld from my brother, mother and father that I cannot fathom holding back ANYTHING that I have that may help you be a better teacher or person.  I have no other agenda, no other motivation, no other egotistical reason for sharing, other than I care deeply, love wholly and am passionately on fire to make the world a better place like so many educators did for me.

So, to state again, I am an open book: Take from my pages what you need.  The only thing that I would ask is, please, don’t judge this book by it’s cover. I am so much more than the hard and loud exterior that you see on first glance.

Now that you now know that I am an over sharer, I hope some of my favorite things will be of benefit to you and that you are not offended or turned off that I want to share with you.

MNM Math Resources

#NoticeWonder

Open Up Resources

Mindset

Great Expectations

True Colors

Co-Teaching

DESMOS Activities

My Growth Mindset Channel

Team Building Channel

12 Things You May or May Not Know About Mrs. Naegele

Instructional Routines and Geometry

As the world quickly moves from the industrial age into the age of STEM (science, technology, engineering and mathematics), it is becoming imperative that today’s schools prepare students to become critical thinkers who can reason, problem solve, and collaborate.  Weckbacher and Okamoto assert that spatial skills, the foundation of geometrical thinking, is paramount to success in STEM careers (Weckbacher & Okamoto, 2015). According to Smith and Stein (Smith & Stein, 2011), children in America are not required to think critically, problem solve or wrestle with concepts or ideas. Instead, they are taught rote procedures that require memorization of ideas and facts that do not prepare them for college and careers where problem solving and critical thinking will be expected. The National Council of Teachers of Mathematics (NCTM)  (National Council of Teachers of Mathematics, 2014) puts forth that effective teaching practices must include:

1. The establishment of mathematics goals to focus learning
2. Implementing tasks that promote reasoning and problem solving
3. Using and connecting mathematical representations
4. Facilitating meaningful mathematical discourse
5. Posing purposeful questions
6. Building procedural fluency from conceptual understanding
7. Supporting productive struggle in learning mathematics
8. Eliciting and using evidence of student thinking.

This review will examine instructional routines as a teaching strategy as well as the Quick Draw and Block Building instructional routines/tasks that aid in the development of critical thinking, spatial reasoning and geometric thinking and utilize the effective teaching practices called for by NCTM.

Mathematical Instructional routines are learning activities designed with a predictable flow and structure of the learning experience.  As outlined in Routines for Reasoning (Kelemanik, Luenta, & Creighton, 2016), the predictability of these routines supports students by answering these questions, “What is it that I am supposed to be doing?  What question will I be asked next? or, How will things work today in the lesson?”  When students internalize the components of the routine they are free to attend to problem solving, critical thinking and the strategies of their classmates.  Likewise, educators are able to attend more effectively to the strategizing and sense making by students as the structure of the lesson is predictable.  Furthermore, the authors assert practicing predictable instructional routines allow for the strategies, questions, methods, and skills used to become intrinsic.  Thus, creating a mathematical disposition that students use in other problem-solving situations. Effective instructional routines should be focused on the development of critical thinking and should support all learners by including the following components:

1. Articulation of math practice goal- math action process identified.  This is the why.
2. Think time – provides adequate processing opportunities, especially for those with disabilities and those who are English Language Learners, promotes independence, builds on strengths. Think time should be provided throughout the routine.
3. Partner work – aids in developing collaboration skills, critiquing the thinking of others, constructing viable arguments and allows for more processing time
4. Whole group discussion – is not just explaining the process or strategy, but what students noticed or wondered during strategizing and how those observations helped in the strategizing. Also aids in language development by exposure to vocabulary, restating and comparing and contrasting strategies.
5. Math practice reflection – utilization of sentence starters to support language and focus the reflection.
6. Access through multiple modalities and representations – integrating universal design for learning by utilizing concrete and abstract, graphic organizers, gestures, tables, graphs, and drawing.
7. Liberal use of math practice-focused prompts – emphasis on thinking and not answer getting.

When utilized correctly, Kelemanik, Luenta, and Creighton, indicate the use of quality instructional routines remove barriers for learning mathematics by providing “authentic contexts, multimodal techniques, rich opportunities for language use embedded in mathematical learning experiences and, an instruction that scaffolds students’ development of increasingly abstract thinking,” When instructional routines are facilitated as outlined, students will participate in cognitively demanding tasks that will develop conceptual understanding and math pracitices.

Quick Draw is a geometry instructional routine developed by Grayon Wheately.  In this routine the teacher projects a geometric line drawing for three seconds and then instructs students to draw what they saw.  Student drawings are completed on lineless plain paper.  Once students have been given time to draw, the image is revealed for a second quick look, and students are again instructed to draw what you saw.  When students are comfortable with their drawings the teacher facilitates a whole class discussion about what students saw and how they drew what they saw. According to Richardson and Stein (Richardson & Stein, 2008) the use of the Quick Draw routine promotes the development of spatial sense and communication skills in students.  The predictable questions students can expect in this routine include, “How did you see it? and How did you draw it?” The conversations generated by these questions aid in the development of a common language that becomes an intrinsic part of students. Expanding on the benefits of this routine, Wheatley and Reynolds (Wheatley & Reynolds, 1999) espouse Quick Draw is a low floor, high ceiling routine that promotes growth mindset, flexibility, and each of the mathematical action processes. In addition, they support Weckbacher and Okamoto and believe that the industrial age ideals of rote memorization and mass production no longer serve the best interest of students in our modern STEM driven world.  The use of Quick Draw promotes the development of spatial sense, mathematical language and vocabulary, offers think time/processing time, encourages attention to precision, and inspires the joint construction of knowledge. Therefore, Quick Draw meets the five guiding principles set forth by Kelemanik, Luenta, and Creighton by being a high cognitive demand task that builds on students’ strengths, uses multimodal access in an environment that is rich in language and promotes growth mindset. Utilizing Quick Draw then promotes the critically needed spatial sense needed in our modern STEM world.

Problem Solving with Cubes is another potential geometry instructional routine.  This activity is outlined by Browning and Channell (Browning & Channell, 1992) as follows:  A picture of a 2-D cube model is given.  In addition, a pictoral representation of the front, right, top, and base view is shown on dot grid paper.  Students are provided with fifteen one-inch cubes and a piece of paper.  Students are instructed to work in their groups to construct the cube model.  This activity was presented as a lesson/activity and progresses from the perceptual level through the representational level of geometry by scaffolding the images and eventually removing the blocks and encouraging students to visualize rather than build. Throughout the activity, students are encouraged to communicate, explain, and reflect. The authors also suggest that this activity be followed by abstract activities available on the internet. Browning and Channell indicate this activity supports the development of reasoning skills, the ability to visually manipulate images for understanding, and forming generalizations.  These are vital skills when students are exposed to images and visualizations in subjects such as mathematics, science and history as well as in their perceptions in the world.   The Problem Solving With Cubes activity has the potential to be a great geometry routine with adaptations such as creating a predictable routine and questions to support students and teachers and by delivering the images one at a time over several class periods rather than a multi-page worksheet.

Instructional routines are a powerful tool for promoting the mathematical action processes.  They support students as well as teachers with their predictable nature and provide a framework for students to take into other mathematics explorations.  In my class the math questions that have become intrinsic are, “What do I notice and wonder?  Have I done a problem like this before? Is there an easier problem I can solve that will help?  Will working backwards help? What tools, tables, graphs, pictures or models will help me? and What patterns and relationships do I see, and what do they tell me?”  The use of instructional routines is a non-threatening and often fun way for students to practice problem solving and critical thinking strategies and practices and they lead to the development of growth mindset.  I was disappointed that I could not find more peer reviewed instructional routines that are specific to geometry.  I wonder I there is more research and exploration needed in this area.  I was able to locate several resources that are not peer reviewed, but meet all of the requirements as established by NCTM and Kelemanik, Luenta, and Creighton.  These resources and others may be accessed here:

High Yeild Geometry Routines

Sky Scrapper Spatial Sense

Sky Scrapper Puzzles

Designing Instructional Routines to Support the Math Practices

Routines for Reasoning

Instructional Routines

Quick Draw

References

Browning, C. A., & Channell, D. E. (1992). Problem Solving with Cubes. The Mathematics Teacher, 447-450: 458-460.

Kelemanik, G., Luenta, A., & Creighton, S. J. (2016). Routines for Reasoning; fostering the mathematical practices in all students. Portsmouth: Heinemann.

National Council of Teachers of Mathematics. (2014). Principles to action: ensuring mathematical success for all. Reston: National Council of Teachers of Mathematics.

Richardson, K., & Stein, C. (2008). Developing spatial sense and communication skills. Mathematics teaching in the middle school, 101-107.

Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston: The National Council of Mathematics.

Weckbacher, L. M., & Okamoto, Y. (2015). Discovering space in the elementary school. Journal of education and learning, 35-39.

Wheatley, G., & Reynolds, A. M. (1999). Image maker: developing spatial sense. Teaching Children Mathematics, 374-378.

 

Fraction Strips Before Geometry

On first thought one might be hard pressed to connect the ideas of rational numbers and geometry, and yet a strong case for a connection is made by the authors of Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 6-8 (Sinclair, Pimm, Skelin, & Zhiek, 2012).  The book authors put forth that there is a plethora of connections between geometry and ratio. They state, “much of geometry involves the idea of ratio – how one thing compares to another.” Often learners of geometry are asked to compare the ratio of side lengths to diagonals and conjecture about relationships and congruence based on ratios.  Geometry, at its core, deals with measurement, properties and relationships.  Before students can delve deeply into these concepts it is paramount that they have a conceptual and fluent understanding of rational numbers.  With this in mind, and as part of the rational numbers unit that precedes my geometry unit, I posed the following multi part task/lesson to my students.

• Use strips of paper 8 ½ inches long. Each strip represents 1 whole. Fold the strips to show halves, thirds, fourths, fifths, sixths, eighths, ninths, tenths, and twelfths. Mark the folds so you can see them easily, as shown below.

• As you fold your strips, think about the strategies you use to make them, and about the relationships you notice.

On the surface, this task may not seem to demand higher order thinking or be considered a high cognitive demand task.  According to Stein and Smith (Stein & Smith, 1998),

Tasks that ask students to perform a memorized procedure in a routine manner lead to one type of opportunity for student thinking; tasks that require students to think conceptually and that stimulate students to make connections lead to a different set of opportunities for student thinking.   High cognitive demand tasks involve making connections, analyzing information, and drawing conclusions. High-level tasks require students to think abstractly and make connections to mathematical concepts. These tasks can use procedures, but in a way that builds connections to mathematical meanings and understandings.

Therefore, when this task was posed to middle school students with a wide array of disabilities the rigor begins to skyrocket as it was anticipated that they would make connections to their previous study of common factors and common multiples when creating unit fractions. It is predicted that this foundational task will lead to connections in creating equivalent fractions and ratio relationships.

When launching the task, I only provided the above directions and an endless supply of strips of colored paper.  I told students that they could work with their shoulder partners as well.

I anticipated that students would be able to create a 1/2 strip and would discover pretty quickly if they folded that in half they could create a 1/4  strip, followed by a 1/8  strip.  I also anticipated they would believe they would create a 1/12 strip by folding the 1/8 strip in half.  Not all students made this error, but most did.

When students are failing to move forward, or when an error is quite common, we stop working and have a whole group discussion.  We utilize these questions when we are problem solving and discussing in our class.

We noticed the following and decided to put the information in a table.  When the information was in the table the students noticed that each time we folded a fraction strip in half the denominator doubled, and that the denominator was a multiple of the previous denominators.

Noticed	When Folded in half created
1/2	1/4
1/4	1/8
1/8	1/16

We wondered then how we could create the remaining fraction strips of thirds, fifths, sixths, ninths tenths and twelfths.  The students conjectured that if they folded a  third strip in half that they would make a sixth  strip and if they folded a fifths  strip in half they could make a tenth strip .  They wondered what they would fold in half to make a ninth  strip, but when they thought of half of nine they concluded that there would be no strip that would have unit fractions that were half of nine. They decided that this would be their next big challenge and set to work to discover a solution.

Students struggled for quite some time in their efforts to create the  third  and fifth  strips.  They tried several ways of folding the strips that depicted the desired unit fraction, but wrestled with creating equal units.  It was fun to hear them struggle, encourage one another and also critique one another.  In anticipation of struggles I had these questions on hand to promote the thought processes of the working groups:

• What fraction strips can you make by partitioning halves strips?
• Which strips can you NOT make by partitioning halves?
• What fraction strips, besides halves, could you refold to make twelfths?
• Which strips can you NOT make by partitioning halves?
• Can you see a relationship between repeated partitioning (folding) and equivalent fractions?
• What happens when you fold fifths and then fold the fifths in half? What happens when you fold others in half?

When students are working in groups or independently and ask me for affirmation about their work or a correct answer, I almost always ask them to convince a friend or ask a friend before they ask me for help.  This practice encourages students to collaborate with one another and work to communicate their ideas effectively.  This also aids in the development of self-reliance skills and confidence.  During this task, when I noticed groups becoming too frustrated I asked them the above questions as well as what strategies and methods they used to create their halves and fourths.  Students could explain the process of lining up edges so that the units were equivalent.  I then asked if they could use the same strategy for the other unit fractions.  These questions and support served to move the collective thinking along and allowed reentry to  the task.

This task took two full class periods to complete. Students grumbled and struggled, but encouraged each other to persevere. In one of my seventh-grade classes, a student declared, “Mrs. Naegele, you are evil!  This is making my brain hurt!”  He quickly followed that statement with a “just kidding, I know you are letting us struggle so our brains will grow.”  Eventually the students were able to discover what they called a “Z” fold that allowed them to create equivalent unit pieces for the  1/3  and 1/5  strips .  Once this was shared among the group the fraction strips began to come together quickly.

When students are working I utilize the collection sheet below both as a way to conduct formative assessment, and as a tool for deciding how to scaffold follow-up discussions.

 

The follow-up discussion entailed some of the ideas shared about numerators doubling when unit fractions were folded in half.  In addition to using the Notice and Wonder Routine, I also regularly ask my students what relationships and patterns that they notice.  Students were able to notice that the fraction strips that have equivalent parts are those that have common factors and multiples.

The following day I introduced my students to Annie Forrest’s Fraction Strip Creation Lesson and challenged them to create a stop motion animation movie that would teach someone how to create a  1/12 fraction strip.  I introduced them to the Stop Motion app and showed this tutorial video about creating stop motion movies using the Stop Motion Studio app available free for Apple and Android.  Students were instructed to create a plan and write a script before they began filming.

I chose the 1/12   strip because there are multiple ways students could choose to create the strip, and I chose the stop motion project because my students often have difficulty explaining in writing their ideas and strategies.

I started this task and lesson during a four-day week.  I felt that two days for folding the strips and two days with the long weekend to finish if needed, would be adequate time for the stop motion project.  Little did I know that my district would have a bomb threat that week and that Oklahoma teachers would stage a walkout.  So, following 2.5 weeks of lost instruction, we returned to class and students resumed their stop motion project.

With such a long break, it was as if we had not done the initial paper-folding task.  Students were allowed to use their fraction strips, but they had forgotten all about their discoveries during the paper folding task.  Instead of going back and doing that activity again I encouraged students to talk in their production groups and explore their fraction kits using our routine of notice and wonder and our other regularly used problem solving questions.  It was as if they were discovering the information all over again.    This experience affirmed that students need multiple experiences and interactions with a concept to truly know and learn it.

For the stop motion piece of this activity, I anticipated that most students would create a third strip, fold that in half to create a sixth strip, and then fold that in half to create a 1/12  strip.  I also knew that students could opt to create a 1/3  strip and then fold that in half to create a sixth and then half that to create a twelfth strip.  I also knew that students could create a fourth strip, fold that into thirds to achieve their goal.  I was pleased that all three of these strategies were used.  I did have some students that created a 1/2  and took that all the way to a 1/16  thinking they were creating the correct unit strip as they had done in the initial exploration of this task.  When this occurred, I asked them how they could create a relationship with 2 and 12.  If they were not able to grapple with that and come up with an idea, I asked, what numbers are we exploring that you know that have a relationship with 12?  Some students were able to think about the factor pairs of twelve and consider the pair of two and six. They then rationalized that to create a 1/6 from a 1/2 that they would need to use the “Z”fold to fold three equal parts to create a sixth strip. Once at this point they could half that to create a 1/12 strip.  Some groups did not make this connection and chose to use the three and four factor pair and start with a 1/3 strip.

To date, students have finished their stop motion videos and submitted their scripts and plans.  An overwhelming majority of scripts and plans mirrored the movies, but there were a few that the script and plan did not depict the movie submitted.  Before students uploaded their videos to http://www.youtube.com they had to show their movie to another group and the other group had to create a  1/12   strip following the movie tutorial. All of the movies did teach someone how to create a  1/12  strip if the viewer was really watching!  The plan going forward, is for students to present their movies and entertain questions and critiques from the other students when state testing is over.  I will begin this follow-up by showing them my movie and letting them critique me!  I am looking forward to doing this kind of activity again in the future, and so are my students!

Works Cited

Sinclair, N., Pimm, D., Skelin, M., & Zhiek, R. M. (2012). Connections: lookingback and ahead in learning. In N. Sinclair, D. Pimm, M. Skelin, & R. M. Zhiek, Developing essential understanding of geometgry for teaching mathematics in grades 6-8 (pp. 69-79). Reston: National Council of Teachers of Mathematics, Inc.

Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: from research to practice. Mathematics teaching in the middle school, 268-275.

Update:

The MNmMath Nerds had wonderful couple of days watching and critiquing our 1/12 fraction strip stop motion videos!  Following each video we spent time noticing and wondering about each production.  We utilize the Notice Wonder Routine almost daily making this was a wonderful way to facilitate conversations in this situation .  Students are familiar with the process and eagerly contributed their ideas.  We critique one another’s thinking regularly as well, but we have never critiqued a final production before.  These critiques were much more personal, therefore we spent some time talking about how to provide and accept productive feedback.

 

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Following each notice and wonder the film producers answered questions from the audience.  This was an awesome learning experience for all.  To close this exploration we held a whole group discussion about what we learned about math and fractions from our experiences. We noticed that every video utilized methods that created fractions that had denominators that are factors of 12.  This was a wonderful connection to our previous Prime Time Unit that focused on factors, multiples, greatest common factors and least common multiples.  Students also noticed patterns and relationships between division and multiplication as related to fraction denominators.

This was a wonderful investigation!  All mathematicians decided we MUST do more explorations like this in the future!

Taking Order of Operations to Higher Levels of Thinking

My MNMathNerds have learned, and continue to learn that:

1. Everyone Can Learn Math to the Highest Levels. There is no such thing as a “math” person. Everyone can reach the highest levels they want to, with hard work.

2. Mistakes are Valuable Mistakes grow your brain! It is good to struggle and make mistakes.

3. Questions are Really Important Always ask questions, always answer questions. Ask yourself: why does that make sense?

4. Math is about Creativity and Making Sense Math is a very creative subject that is, at its core, about visualizing patterns and creating solution paths that others can see, discuss and critique.

5. Math is about Connections and Communicating Math is a connected subject, and a form of communication. Represent math in different forms, examples words, a picture, a graph, an equation, and link them. Color code!

6. Depth is much more Important than Speed Top mathematicians, such as Laurent Schwartz, think slowly and deeply.

7. Math Class is about Learning not Performing. Math is a growth subject, it takes time to learn and it is all about effort.

Jo Boaler

We also know that maths is so much more than memorizing facts and procedures. With all of these ideals under consideration, this week we discussed and discovered again, that there are some mathematical ideas and procedures that we just have to be told.  When we were young we could figure out that if we had 3 red marbles and 2 blue marbles we could combine the groups and count them to find out how many.  We were adding, but we did not know what the addition symbol was until someone showed us + and demonstrated symbolically 3+2=5 as our expression for our marbles.  Similarly, when we are doing multi operational maths problems we can find a multitude of solutions if we are not following an agreed upon order of operations when computing.  Mathematicians know that some numerical situations might be interpreted in more than one way. Therefore, they agreed on an order for simplifying expressions called the Order of Operations.  There are a plethora of mnemonics and tricks for Order of Operations, but after much research and readings I have settled on my favorite, GEMA as endorsed by Tina Cardone in Nix The Trix.  Once we reviewed the agreed upon Order of Operations and completed an interactive notebook entry the fun began!

Enter the high cognitive demand math problem!  I could have given my students several naked number mathematics problems to practice, but they would have just been following procedures without connections.  While sometimes these kinds of practices are good, I wanted to impress upon them the importance and affect the order of operations can play in outcomes.

So instead of giving them a problem like:  2(5×30) + 2(2×70) +10 =   I asked them to notice and wonder about the following, and then turned them loose to collaboratively strategize and solve.

• Alicia is making a reading plan for her book club selection.  She has 14 days to finish the book.  She plans to read 30 pages on weekdays and 70 pages per day on Saturday and Sunday.  Following this plan, she still has 10 more pages to read at the end of 14 days.  How many pages long is the book?

 

It was a blast to hear the conversations,thinking, and convincing in the working groups! We have been exploring the distributive property and the different ways you can notate the expressions. I loved seeing the kids explore different ways to to notate and solve and notice how they were distributing in different ways.  If I had just given my students naked number math problems, I would have deprived them of a multitude of rich experiences.  They would not have had to explain their strategies, justify their thinking, listen to and critique the reasoning of others, and they would not have had the opportunity to process the ideas, strategies and thinking in a variety of ways, thus deepening their understanding.  I was beyond thrilled with the myriad of ideas, strategies and paths to solutions explored!  I try to anticipate possible strategies and ideas, and I have to admit, my students surprised me with a few I had not considered!

I teach sixth-eighth grade students with disabilities, and I always read aloud the tasks and problems we investigate in class a multitude of times before we notice and wonder.  In addition, I always record and leave our ponderings on the whiteboard for reference during work time.  For one of my sixth-grade classes, I needed to take this wordy problem one sentence at a time so that they could notice and wonder deeply.  If I were doing this problem again, one way I might modify my delivery would be to leave off the last sentence, “How long is the book?” I realize by having this question given, I have taken the opportunity from my students to formulate a question to explore for themselves.  All in all, this exploration was a smashing success!  The affirmation came when three of my classes stood at the board, taking pictures of their amazing mathematical thinking so they could post it on social media!  YES!  I have created some awesome math nerds indeed!

 

 

 

 

 

Oklahoma Academic Standards:

6.A.2.1 Generate equivalent expressions and evaluate expressions involving positive rational numbers by applying the commutative, associative, and distributive properties and order of operations to solve real-world and mathematical problems.

7.A.4.1 Use properties of operations (limited to associative, commutative, and distributive) to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents.

7.A.4.2 Apply understanding of order of operations and grouping symbols when using calculators and other technologies.

PA.A.3.2 Justify steps in generating equivalent expressions by identifying the properties used, including the properties of operations (associative, commutative, and distributive laws) and the order of operations, including grouping symbols.

Common Core State Standards

CCSS.MATH.CONTENT.6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

CCSS.MATH.CONTENT.7.EE.B.3
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

CCSS.MATH.CONTENT.7.EE.B.4.A
Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

CCSS.MATH.CONTENT.8.EE.C.7.B
Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.–

Hidden Figures and More

Last week my class and I spent time learning about the amazing women of Hidden Figures.  We began by watching the movie, and followed up with this lesson I borrowed and adapted from the amazing Max Ray.  Hidden Figures is one of my favorite movies of all times. I have watched it countless times, and each time I laugh, I cry, and I am filled with a deep and palpable love and admiration for Katherine Johnson, Dorothy Vaughn, and Mary Jackson.

Each class period our discussions took us in a variety of directions.  Each group however, asked why did we treat people different just because of their skin color?  We talked about history, we talked about values, and we talked about how things are still not the way they should be.  Each day we start class with a spotlight on a famous failure a growth mind-set quote and a character quote.  I was proud that my students brought those quotes and ideals into our conversations and collectively decided that if the world is going to change it has to start with us.  We know that we must be the change we want to see in the world, and that a smile is the closest distance between any two people. I was heart broken when students shared stories about barriers that they face due to their disabilities, dysfunctional families and just being an adolescent.   I was encouraged that they vow that these things will not be barriers in achieving their dreams.

Following our discussions, we engaged in the Mission Control Activity and I mingled and observed.  I was pleased to hear most students using the language of mathematicians when instructing their “John Glenn” how to build their control panel.  I was also pleased when some did not use this language that others kindly reminded them of the proper way mathematicians speak. I utilized this activity as a formative assessment into my students’ spatial ability, ability to communicate mathematically, and to utilize the vocabulary and ideas of transformations we have been exploring.  I noticed that students who struggled the most are the ones who struggle with spatial sense and orientation.  I love formative assessments like these that give me usable insight into student thinking and their placement on the learning trajectory while also providing a fun and memorable experience for students!  Thursday evening, we had parent teacher conferences, and I had Mission Control and Moon Math set up in my room. It was enjoyable watching students give their parents instructions on how to draw an angle and calculate missing angle measurements.  It was also hilarious watching them teach their parents how to talk like a mathematician! 

If you are looking for other lessons to do with students using Hidden Figures, you may find some here, here, or here.

Mission Control in Action:

 

 

 

 

Quick Draw Revisited

A couple of years ago I earned my Master’s in Teaching, Learning and Leadership with an emphasis on Elementary Mathematics Specialist.  The first course in my studies was the one and only Geometry class that I have ever had the pleasure of taking.  Needless to say, I had quite a steep learning curve!  The course outline expected  the participants to become highly qualified in PK-early high school geometry content, as well as pedagogical practices focused on inquiry based, student centered, discourse rich, constructivist delivery.  I embraced and fell in love with the pedagogy, and the paradigm shift was seamless.  I also managed to learn a good portion of the content, but I knew the moment I left that course behind, that I did not learn all I needed concerning geometry. I filed away the notion that if the course was ever taught again by my advisors, that I would beg them to allow me to audit the course.  Guess what? It has been four years since that initial course, and I am currently auditing it!

While in my studies I learned about Grayson Wheatley’s Quick Draw routine for reasoning and began using it regularly with my students.  Wheatley created this routine with the goal of utilizing imagery to promote the development of spatial sense, mathematical reasoning, and recognizing mathematical patterns and relationships.

Quick Draw is a captivating routine in which students are shown an image for three seconds and then prompted to “Draw what you saw.”  Multiple quick looks can be provided until students are comfortable with their captured image.  I was following this routine, but over the course of time I managed to change the discussion portion, and instead of revealing the image and discussing what students saw, I had moved to having students instruct me so that I could draw what they saw.  I thought this was a great idea, as students loved when I drew literally each of their instructions. I had the notion that this method was developing precision and helping students communicate more effectively about the math involved.

Wow, was I ever wrong!  I revisited Quick Draw this week as originally outlined by Wheatley, and what I discovered was astonishing!  In an effort to precisely tell me how to draw what they saw I was limiting students’ ability to visually manipulate the image in their minds. They were so focused on “getting it right” that they were usually seeing  just one rendition of the image, instead of the multitued of images and interpretations possible!  This visual manipulation of images is paramount to all other maths.  The skills needed to transform visual images is foundational for mathematical reasoning, spatial sense and the ability to problem solve.

In our exit ticket for the classes, I asked my students to give me feedback on the “new” way we experienced Quick Draw.  While all of the responses were positive, some of my favorites follow:

“I noticed that when we experimented with the new way it helped us see a lot more than we did before.”

“I like how we talk about what we all saw because we can think to ourselves what other people see.”

“I loved doing this and being able to see all of the math in the shapes.”

“I like doing this new way, but I also like it when we have to tell you what to draw because we had to explain it more better.”

I am thankful that I am having the opportunity to audit this geometry class again!  Tweaking my routine to reflect the original intent ensures my students are sharing and communicating their noticings and wonderings in a non-threatening way.  They easily incorporated vocabulary such as symmetry, transformations, reflections, and more.  Had I not revisited Quick Draw I never would have heard any of the following:

“Hey do you see that the rhombus makes two triangles?”

I” wonder how many triangles we can find!”

“Look, there are really two rhombuses!”

“Where?”

“See, if you reflect that one you can see it is half of the other and then put the two halves on the ends together you get another one!”

I also had the opportunity to witness that my students who struggle the most with seeing the images more than one way are also my students who struggle most with mathematical reasoning, problem solving and flexibility in their strategies.

I believe as Darling-Hammond and Syke do when they put forth that teaching should be treated as a learning profession, I am beyond appreciative that I have the opportunity to audit this geometry course! I am looking forward to discovering all of the ways I can continue to improve my teaching practices as well as developing my content knowledge of geometry!

If you are interested in using Grayson Wheatley’s Quick Draw as a routine for reasoning you may find his resources for purchase here and here.  More from Mr. Wheatley can also be found at Learn NC until  2/1/18, but after that date these directions will need to be followed.

Acceptance and Belonging

 

I just returned from TMC17 (Twitter Math Camp) in Atlanta, Georgia where I spent five glorious days surrounded by some of the best of the best in the mathematics world.  As I sit and reflect on my time there I am consumed by a feeling of love and gratitude for the amazeballs mathematicians that have welcomed me as one of their own, and who I consider to be family!

My childhood was tumultuous to say the least.  My parents had a plethora of issues.  The most challenging being mental illness and alcoholism. These challenges created many problems that resulted in us moving so often that I went to fourteen schools.  It wasn’t uncommon to find us squatting in abandoned homes with no running water or electricity.  Many days I went to school hungry and unclean.  As hard as they tried, my parents could not handle the stress that this lifestyle brought and after seven separations and divorce fillings they stopped trying. As an adult, I have a wonderful family and career.  Even so, my husband and I have lived through oil industry busts and several rounds of bad economy. We have lived in three different states in order to go where he could find work.  Always being the new person, and as a child, going to school as the dirty little transient kid in second hand clothes,  I know a little something about not being accepted or belonging.  Even after all these years, the feelings of inadequacy and unacceptance are something I struggle with.  There are very few places I go, or people I associate with that make me feel welcome, accepted and valued.  I have to work hard not to see myself through the lens of that dirty little transient kid in the corner, doing my best to be the paint on the wall.

It is with the deepest gratitude and a huge amount of awe that I find myself not only accepted and loved by this amazing community called #MTBoS, but also respected and valued as a professional.  I graduated with a BS in Special Education in 1987 and taught for four years before deciding public school was not for me.  I took about 15 years away and worked as the director of an early childhood center.  Eleven years ago I returned to public education and realized that there was so much I needed to learn in order to be the best teacher I could be for my students.  This led me seek my Masters in Teaching, Learning and Leadership with an emphasis on Elementary Mathematics Specialist.  It was while I was seeking this degree that I was introduced to the MTBoS via Levi and TMC 14.   I have only been a math nerd for three and a half years!  I am a newbie, and while I have plenty of classroom experience, I still consider myself a novice as a mathematics teacher.  In most societies, careers, and communities, I would not be seen as a viable contributor or member.  It is with the deepest gratitude, humbleness, and awe that I can say; this is absolutely NOT the case at TMC or in the MTBoS community.  I believe unequivocally that of all the communities I have ever been a part of that THIS one has been the most open, accepting, nurturing, inspiring, challenging and downright FUN that I have ever been a part of!

Where else can you go, whether virtually or in person and have professional and personal conversations with the likes of Max, Annie, Peg, Chris, Graham,  Brian, Lisa, Bob, Andrew, Tina, John, Glenn, Hedge, Christopher, Malke, Sara, Tracy, Audrey, Llana, Steve, Edmund, Carl, Stephen, Sam, Carly, Madison, Julie, Levi, Sadie and so many more?  These conversations, acceptance and relationships do not end the last day of Twitter Math Camp, but they extend to Twitter, Facebook, NCTM and other conference events.  Every single organization, movement, company, or collective is only as awesome as its people.  I am beyond blessed and eternally thankful that this fantabulous community exists and has taken me as one of its own.  I value the learning and growth that this collective inspires me to pursue, but even more than that I cherish the acceptance, the love, the genuine sense of inclusion that I have with my fellow math nerds.  To each of you I say thank you.  I love learning, growing and becoming with you!

A Post a Day #8 – Getting Nostalgic #NoticeWonder Style

I notice I can’t think about tomorrow being our last day of school.  I notice I can’t think about not seeing my students for 2 and a half months.  If I do I get teary eyed and melancholy.  My students have stolen my heart.  I notice that they have become a part of me.  They have changed me, and I am better for having been their teacher.  I also get sad because I know that so many of them have tumultuous home lives.  I notice so many of them struggle with self confidence.  So many of them wrestle with life in general because of the disabilities that challenge them in so many areas of their lives.  I wonder  if  over the next few months they will forget that they are problem solvers.  I wonder if they will get caught up in the many conflicts that they are sure to have, and forget how it feels to be in our safe and loving community.  I wonder if  they will feel alone and helpless and forget about having a growth mindset and noticing and wondering as a way to solve any problem.  I love them.  I worry for them, and I know they will continually occupy my thoughts and prayers this summer.  As the last day with my students looms tomorrow I notice I can’t wait for next school year to begin so we can pick up where we left off and continue our #NoticeWonder journey toward becoming powerful problem solvers with growth mindsets!