In my classroom, I have a white board that is directly behind my desk.  A very inconvenient placing makes this board useless for instructional purposes.  Thankfully there is also a whiteboard at the front of my classroom, and it gets the job done!  I have turned the board behind my desk into a shrine of sorts dedicated to people, resources and ideas that inspire me.  I include stickers, books, photos, notes, anything and everything that reminds me that I have a village out there supporting me and my practices.  I believe these connections are paramount for educators.  We must feel we are connected to the math revolution that is so much more than ourselves.  We must surround ourselves with people and ideas that fuel our passions and push us to become the best versions of ourselves possible.  Thus I have my Board of Inspiration in my classroom as a constant reminder.

Away from the classroom, I keep connected and inspired through weekly Twitter Chats with my wonderful PLN on #ElemMathChat, and I follow some amazing blogs whose authors remind me of best practices, and inspire me to take chances in my teaching.  There are times when I feel overwhelmed by the sheer volume of amazing right at my fingertips!  I have to remind myself that Rome was not won in a day, and that the way you eat an elephant is one bite at a time!  In other words, take the resources as you can, and don’t beat yourself up mentally if you choose to take a night off to watch some television or veg with a good book!

Last week I went to see the amazing movie, Hidden Figures, and came away affected by the multiple messages the movie conveys.  I have been wondering how I can incorporate the movie into my lessons, and how I might tie the math we do in our class to the math that Katherine Johnson and the other mathematicians were doing in the movie. My thoughts have never been far from trying to find a way to explore the connections of  the racist and sexist experiences that Katherine and her fellow computers experienced in the movie to the trials and trauma that my students face because of their learning differences and disabilities.  I have been searching for a way to use the movie to inspire my students to continue in their development of growth mindsets.  Thankfully I did not have to wonder for very long as the bombdiggity Max Ray just happens to be one of my very favorite inspirations!  As I sat down to contemplate a possible lesson I discovered Max’s most recent publication,  A Hidden Figures Lesson Plan  a wonderful blog that brought to the surface all of the ideas that have been swimming in my mind ever since I saw the movie!

I cannot begin to comprehend the frustrations and failures that my students face because they were born with disabilities, nor can I begin to understand what it was like to be an educated black woman in the 1960s. I can look at both groups and admire them for their grit.  I can be inspired by their tenacity, and for being the example for us all to follow.  I am hoping that my students take the opportunity to reflect upon these ideas so that they realize the struggles they face are preparing them to be warriors in future setbacks they may encounter.  That if they are willing to take risks, make mistakes, struggle through the hard times, that they will become problem solvers and as Max says, “Find strength in the ways that adversity has shaped them, and  know those strengths serves each of us in our mathematical lives.”

We started the year off in grand style, exploring and the Weeks of Inspirational Math found there.  My students latched onto the ideals and beliefs they learned like their lives depended on it.  We have been desperate to find more stories, more examples of struggle that lead to success.  This movie and lesson are just what we have been craving!    I can’t wait to share Hidden Treasure:  Using Math to Shatter Barriers , Max’s lesson that I have transferred to Google Slides with them this week.  I am sure my students will be inspired by the mathematicians, and I am sure I will be inspired by my students!

Update 1/20/17

What a fantastic week with the @MNMathNerds!  I came away from this week as affected by our investigation into Max’s Hidden Figures lesson as I did from watching the movie.  We had a plethora of amazing discussions about history, science, math and social justice!  I am still processing all of the awesome that took place!  There were so many remarkable questions like, “Mrs. Naegele, why did they call the movie Hidden Figures?”  and,  “Why did people treat each other like that?” that inspired genuine and reflective discussions.  We spent time looking into the history that led to the setting of the movie.  We relished discussing the grit and integrity of the ladies spot lighted in Hidden Figures and wondered how we could use the barriers that each of us face to change our worlds.  Students shared personal and heart wrenching stories of times that life was not fair to them.  Through these personal interactions relationships and trust were fostered and strengthened.  It was a beautiful thing to be a part of!  The thoughts and realizations that came with the activity Mission Control were reflective as well as mathematical, and students begged to do the activity again!

Weekly engage in a routine I call Think it Through Thursdays.  I normally give students an answer and they are asked to create a question that could be asked to answer the question.  Last week I posed this question instead, “How do you use your frustrations and challenges with your learning differences to help you succeed in a moment of crisis?”  Here are a few of the responses I received:

“When I make mistakes I learn from them and then I grow.”

“I ask for help from someone I trust.”

“I remind myself that I am in charge of my happy.”

“I make up my mind to do it!”

“I decide that I am going to be bigger than my problem.”

“I keep an open heart.”

“I keep trying and never give up.”

“I believe in myself and don’t snap when things get hard.”

“I stand up to my problem and tackle it.”

“I help other people when they need help so I can be an example.”

“I use the pain my problems cause to make me stronger.  That is what life is about.                 You can’t just quit.”

These kids amaze me with their determination and veracity.  As I knew they would, they inspire me!

Learning Behind The Learning


Learning behind the learning, or soft skills, is something that I feel I am on a continual quest to improve upon.  These are the things that a teacher does to pull students into conversations, into the community, into the learning.  There are so many soft skills needed in order to create a learning environment where students feel safe, valued and comfortable enough to take risks, that it often feels overwhelming.  Add to that the fact that each group of students is different and every individual’s needs are their own, and you have a pretty heady task ahead of you!

I have found that I have to take it slow.  In my masters studies I had the fortunate opportunity to learn one piece of pedagogy at a time.  This made it possible to enact and tweak that piece for a semester before adding more skills to the mix.  This provided me time to become comfortable enough with the practice to make it my own and it allowed my students the same chance. I am in a new teaching assignment this year where I am instructing middle school students with disabilities who have never been exposed to discourse rich, problem based learning.  Therefore, I have spent a good amount of time slowly introducing them to my tools of the trade.  Teaching students with disabilities has one distinct advantage in that I tend to have the same students for the duration of the time that they are enrolled in my school!  This means that I don’t have to start over every year!

One of the first techniques I expose my students to is Talk Moves.  This is a way to get students communicating about their thinking.  A cardinal rule in my class is that every mathematician deserves think time.  This means that no one blurts out, hands do not go up, and the answer is the last thing we talk about.  If a student wants to contribute their ideas or strategies they are asked to put a thumb up on their desk.  Once a student starts speaking, it is not uncommon that they lose their train of thought or confidence.  They know that if another student has a thumb up that they may ask for their help by “phoning a friend.”  If students have an idea, but do not wish to be called upon they may lay their palm flat on the desk, and if they are still thinking they put a fist on their desk.  I have to be honest, we are half way through the school year, and students are still learning how to maneuver through our talk times.  I wish we were further along in our talk practices as we spend the majority of our class time talking about math and our ideas!

In my classroom the concrete that provides the foundation for all of my soft skills is the use of Notice and Wonder.   This technique is a wonderful tool that removes so many barriers, ensuring all have access to the math at hand.  There is not a human being alive who doesn’t notice detail and wonder about connections!  So with every problem, with every investigation, the first thing I ask my students is, “What do you notice, what do you wonder?”  Almost every problem based question starts with noticing the non-math details, and sometimes the first exposure to the ideas I present actually have no math question attached.  In these cases we wonder together what the problem may be asking us to mathematically consider.  I love posing these types of notice and wonder problems at the beginning of a unit and then returning to the scenario throughout our explorations as the math is revealed!

I have to operate in full disclosure and acknowledge that this too is a work in progress.  My students have spent so many of their learning years being told what to think, how to solve a problem, and procedures to do so, that I have to be comfortable in the silence sometimes.  I have to practice my own cardinal rule of wait time, which means I may be counting in my head at times waiting for them to think and participate!  I have noticed that my sixth grade students attack noticing and wondering without abandon.  They jump right into the talk moves and are eager to share their thinking.  This leaves me wondering if it is because they are younger and less impacted by peer pressure and adolescent hormones than my seventh and eighth grade students.  I am thankful that they will be mine for two more years and I will be able to continue to challenge them to think, notice and wonder!  These skills extend much further than my classroom walls, and if they can learn to approach life thinking, noticing and wondering, then there is no obstacle that will stand in their way for long!

One of My Favorites


This school year I am doing something I have never done in the 30 something years I  have been in education. I am teaching only one subject!  You would think that this would simplify my life and make things much easier.  In some ways that is true and in others it is not.  I find not having the same students all day makes it more difficult to create meaningful relationships with each of them.  Without meaningful relationships reaching and teaching students is difficult!    Time, that old enemy.  We can’t get enough and yet so many times we can’t wait for it to pass quickly!

With this in mind I am constantly looking for ways to stream line procedures in an effort to create more teaching and connecting time with my students.  Each of my five classes, grades 6-8, engage in daily instructional routines.  We use these routines as bell ringers as they are low threshold, high ceiling activities designed to promote critical thinking, number sense, problem solving, communication skills and the joint construction of knowledge.  At the beginning of the year I was finding that the logistics of these routines was taking up valuable time therefore, I created a Google Slide Presentation that I embed in my lesson plans on as well as a bell ringer form that students record their thinking and work on.  Since students are familiar with the routines they come into class, pick up their bell ringer form and begin thinking and working together on the routine that is projected on my interactive white board while I greet students and post attendance.  Not only has this form aided in creating more time for connecting and teaching, it has helped students stay more organized.  I am also finding that that more bell work is being turned in than before!

Time to Refocus My Efforts!

Welcome to the Explore the MTBoS 2017 Blogging Initiative! With the start of a new year, there is no better time to start a new blog! For those of you who have blogs, it is also the perfect time to get inspired to write again! Please join us to participate in this years blogging initiative! […]

via New Year, New Blog! — Exploring the MathTwitterBlogosphere

Let The Journey Begin!


 This school year I embarked on a new adventure of teaching math to sixth through eighth grade students with disabilities.  I graduated in December with a Master’s Degree in Teaching Learning and Leadership with a concentration in K-8 Elementary Mathematics Specialist.  My new teaching assignment allows me to put all of my new-found knowledge and experiences into practice.  I have to admit that while I was bubbling over with excitement to have this opportunity, I started the year with a considerable amount of trepidation.  I have worked with students with disabilities throughout my teaching career, and I know that many of these students also come to school with social and family problems that further hinder their learning.  Add to that a long history of failing and struggling in math, and you have a student who usually would much rather be anyplace else other than any math class.  I realized that if I was not able to provide experiences that would change these attitudes learning would not happen no matter what I did.

With this in mind I spent a considerable amount of time over the summer researching, reading and preparing to start this school year.  It was my goal to begin the year by establishing norms and creating a community where members trust one another and are not afraid to take risks.  One of the books I read was Mathematical Mindsets by Jo Boaler.   The ideals and message in this book were the icing on the cake that I needed.   Growth Mindsets is the plaster that hold together all of the other teaching practices and principles that have now become such a part of me.  I literally have conversations about teaching and learning math with almost every person I meet every day!  Yes, I know this makes me a math nerd, I own it humbly yet proudly!  The last week leading up to the start of school found me in a fervor putting together a plan that I hoped would reach my students and get them excited about math.  I also knew from experience that it has taken years for these kiddos to develop the mindset that they bring, and I could not expect to change it without a considerable amount of time and investment.

Here is how the first few days of my school year have gone:

The Weeks before school began:

“I am part of all that I have met.” Alfred Lord Tennyson

Start the year believing: Relationships in life are paramount.  Without them connections cannot be made.

I wonder: Will sharing all that I have met create a pathway for connections?  Am I willing to be that vulnerable?  If I show I am willing to be vulnerable, will it inspire my students to also trust and be willing to take a risk and share themselves?

I notice: Students will not trust you if they believe that you are not willing to see past their struggles or life situations.  They crave to be seen for who they really are as a person.

I notice: It still takes more.  Students who struggle academically often bring a defeatist attitude with them to class.  They have developed a closed mindset towards math.


I wonder:  Can I convince them that: Anyone can learn to do math to high levels; Speed doesn’t matter; Mistakes Grow your brain; We are in this together; I’ve got you and you’ve got me.

I resolve:  I am going to see my students as the beautiful people they are.  I am going to look past all of the baggage they bring due to their disabilities and other life struggles.  I am going to introduce them to what I believe is the most promising and beautiful kind of learning I have known.  I am going to facilitate activities and learning experiences with them that will turn them into powerful problem solvers who notice and wonder at every opportunity.  I am going to continually remind them that they can grow their brain and change their mindset if they are willing to look for patterns, take their time, be willing to struggle and make mistakes.


First day of school –  “I am part of all that I have met.” Alfred Lord Tennyson

My story: I laid a lot out there –  my childhood, my fears, my likes, my dislikes and, my struggles.  I kept it age appropriate and left out some things, but I was honest and vulnerable.  I also shared my life now –  12 Things You May or May Not Know About Mrs. Naegele

Students also shared  things about themselves with the class:  Getting to Know You

Math Attitude Survey  This was eye-opening even though not surprising


School Required Review of Student Handbook (time-consuming, boring and necessary)

Math Is Snowball Activity

Math Is Wordle – our graphics were not very positive!


Go Ahead Break the Ice  – This activity was designed to introduce students to the problem solving process in a non-mathematical/non-threatening experience.  You may read about how it went here: Melting Ice on Day Three.


The GOAT video –  What did you notice and wonder about Michael Phelps?

Talk about what we do when things get hard – reference the video games discussed yesterday.

Spend more time on Four 4’s

Day Two: Brain Crossing Video – What did you notice and wonder? Number Visuals with Animation – What do you notice and wonder?


Day Three: Which One Doesn’t Belong Wednesday – We had some great justifications for our choices!

Speed Video followed by Modified Paper Folding Activity – What do you notice and wonder?  We wondered and then noticed that the denominator doubles when the paper is folded in half!


Have a reminder talk about what we do when things get hard – reference weight lifters and athletes.  Do they get better by lifting ten pound weights or only working out for a short time each day?  What does that tell us about the math we need to do to grow our minds?

Day Four: Patterns Video followed by – What do you notice and wonder about the patterns found here?

I honestly didn’t think life could get better after Thursday!  Here is my Facebook post: Today my class explored 2015’s day four of Week of Inspirational Math. ( ) My students LOVED it! They were noticing and wondering about patterns left and right. They made conjectures and we tested them. They struggled and got frustrated. They agreed and disagreed with each other, and they completed the activity. I had students thanking me for pushing them to struggle. Some said, “This is the funnest and hardest math I have ever done, and I love it!”  I lost count of how many students requested to take a copy of what we did today home so they could do it again with their families! How stinking cool is that?

And then came Friday.


Day 5: Mistakes Video    Growing Shapes Lesson Plan   How Do You See The Shape Growing?

On Friday we explored 2015’s Day Five Week of Inspirational Math.  My students came in still excited about Thursday.  Many asked if we were going to watch another “Jo” video, and were happy to hear that we were.  Our bell-ringer was a problem that gave mathematical clues that the students were supposed to use to find each place value in a five digit number.  I admitted to them that I had to read the clues more than one time to be able to find a solution.  I asked if they were ready to notice and wonder and beat that problem.  They were ready!  The last clue was; the sum of the ones, tens and hundreds place is fifteen, and the tens place is four times the hundreds place.  They were stumped.  I asked them what strategy they could use to figure it out.  They were not sure.  I said, “Well can you guess and check and come up with something?”  I kid you not when I tell you they were stunned to be given permission to guess!  I reminded them of the check part and they were off!  When they collectively found the solution I asked what they noticed and wondered about the process of finding the solution.  They still wanted to talk about being given permission to guess and check.  We talked about it and decided if you have no idea where to start in a problem, guessing and checking is a pretty good place to start.  I followed that with this question: “If you guess and check, what is the worst thing that can happen?”  I then showed the Mistakes video for day five.   As usual we followed the video with a notice and wonder talk.  Everyone was amazed to learn that mistakes make your brain grow.  I have a Thomas Edison poster hanging on my wall with the quote, “I have not failed, I have just found 10,000 ways that won’t work.”  We talked about what this meant and how it applied to what is shared about Michael Jordan in the video.  In one of my classes a sixth grade boy felt safe enough to share how promising the message in the video was to him.  He was moved to tears by the message that mistakes are not bad, and it is through them you learn.  He expressed how happy that made him because he has always felt like a failure because things are hard for him.  Every student respected that moment, and this teacher’s eyes were brimming with tears as well!   Every class on Friday had a plethora of noticings and wonderings about how the shape grows.  They noticed physical patterns as well as numerical patterns.  They connected this day’s patterns to the patterns explored in Pascal’s Triangle and commented how doing that activity helped them understand this one even more. I asked them to work in small groups to find the fourth, fifth and sixth pattern.  We moved fluidly from whole group to small group back to whole group and each class was able to take the task to their highest level of understanding.

how do you see the pattern growing

In each final large group discussion I recorded the class observations into an empty table.  Every class observed all of the information in the table below.  I was thrilled that they had gotten to this point!  I challenged each group to see if they could look at the table to come up with a rule that would tell what number of squares was in any case. They worked together thinking and drawing, talking and comparing ideas, but the bell rang before they could solidify a rule.  But wait, there is more!


Case # or c # of Squares Increase Relationship


1 4


2 9 5



3 16 7



4 25 9



5 36 11


+1 6 49 13


At the end of the day, I challenged my eighth grade students to look at the table and notice and wonder about what they observed.  I was beyond ecstatic that they not only noticed the increases, and relationships, but also that “If you multiply 3×3 you can see the number of squares in the case above that case, and it is the same for the others.”   They then wondered about that relationship and came up with the plus one.  I challenged them to think about that plus one.  I also asked them to see if they could come up with a rule for the number in any case.  They talked among themselves, and one student stood up excitedly and said, “I don’t know if this is right, but I think if you add one to the case number and then multiply that number by itself you will get the number of squares.”  I asked the group to talk to their shoulder partner to see what they thought.  They worked together, tried the idea, drew some more cases, and came to the consensus that the rule: (c+1)∙(c+1) would give you the number of squares to any case. I wrote their rule on the board and showed them you could also write it (c+1)².  When I turned around to ask them to use their rule to tell me what the hundredth case would be they were sitting in stunned silence, and a few mouths were hanging open.  I couldn’t resist, I said, “And that my friends is how you do Algebra.”  I mic dropped my dry erase marker and walked away from the board.  There were a few whispers about my  mic drop, and then an eruption of excited chatter about how smart they were filled the room.  I let them live in the moment, and truth be told, I don’t know if I could have talked without crying anyway!

These kids, now my kids, in a weeks’ time went from believing that they were failures who hated math to excited mathematicians celebrating their victory!  I don’t think I have ever witnessed a more beautiful educational moment in my life!

I know we still have an uphill battle to climb.  I realize that those nagging negative thoughts may return when we encounter problems that cause them to struggle a little too hard.  For now though, I too am going to live in this moment.  I am going to cherish it as the first step, just as I cherished my own children’s milestones.  I have more resolves now.  I promise that I am going to continue to provide positive experiences for my students that challenge and stretch them.  I am going to continue to teach them about developing a growth mindset.  I am going to strive to make every week better than the last, and not let this be the only mountain top experience we have this year.  Because I saw so much happiness, so much confidence, so much love of math on that mountain top that I cannot bear the thought of it being their last.  They deserve the chance to climb those mountains and reach the top, and I resolve I will bring the mountains to them.


Melting Ice on Day Three!


The first full week of school began today, and thanks to Amy Zimmer, @zimmerdiamonds, we are off to a fantastic start!  Amy shared one of her favorite back to school activities, Break the Ice, at Twitter Math Camp, #TMC16,  this last summer, and I fell in love with it!  I brought it home and added my name to her Google Slideshow and I was ready to go (Mrs. Naegele’s Awesome Math Nerds) .  I launched the activity by introducing students to our group creator cards that I placed on each desk and had students find their group according to the graphic in the one frame.  Then we worked through the slide show that Amy created.  The students loved having the opportunity to talk in their small groups about the games, movies and books that they like the most.  In our follow up conversation about the strategies employed to reach a consensus, every class was able to identify several strategies including; strong arming, majority rules, listing, throwing out ideas until one resonates with the group, pickiest gets the choice, most strong willed gets the choice.  We then thought about and discussed the pros and cons of each strategy if we were engaged in mathematical problem solving.  I had recorded each group’s responses on the whiteboard, and I then challenged the class to come to a consensus for one book, one movie and one game that we would agree upon as the class favorite.  I read the lists and asked who would like to promote and defend a particular choice.  The speaker was taxed with convincing the class to vote in favor of whatever book, movie or game they were promoting.  The students were eager, and passionately described why they loved what they were promoting.   When the first few students had shared I drew attention to myself and let students know I would be modeling restating in my own words what the presenter was promoting.  When a student was finding it difficult to enunciate their feelings I explicitly modeled adding on.  It wasn’t long before several students were asking to restate or add on to each other’s justifications.  When a student was defending a movie, book or game that I found particularly unappealing I modeled respectfully disagreeing and providing counter arguments.  You guessed it, the students quickly picked up on this and they too started respectfully disagreeing and providing counter arguments!  When the class had reached a consensus on each category we then talked about how we can take these techniques into problem solving in math.  I followed this activity with Jo Boaler’s video, Mindset, found on,  and we discussed what we noticed and wondered about the video.  Right away my students picked up on the fact that the brain can be changed, they called it plastic.  They were impressed that the brain can grow with the right kind of activities.  They noticed that people have to continue to engage in activities to maintain the expanded brain capacity.  I closed by sharing with them why this excites me so much.  I highlighted the fact that each one of my students has one kind of disability or another that has resulted in them being in special education classes for math.  I reminded them of the research that proves when students are engaged in the right kind of activities that the brain can and does change and grow.  I talked about how I have been using these activities and methods with students for a while now and I have seen first-hand how life changing these activities have been for my students.  I told them success stories about former students and how excited I am that none of the students I had at my previous school who are now in 6th and 7th grade are in my class now.  My previous students have changed their mindset, and they no longer need intervention in math.  I challenged my new students to come to class every day ready to engage in activities in which they may struggle. I asked them to keep an open mind and be ready to strategize and problem solve with one another.  I told them if they are willing to put themselves out there and take a risk that together we will change their thinking, change their mindsets, and they will become mathematicians.  I hooked them and delivered the close.  Almost every one of them cannot wait to explore the math problem that I promised to bring tomorrow!  Thank you Amy for such an amazing opener!  The ice is not only broken, but I can already see it is beginning to melt!

I’m Not Procedural, I’m Divergent


Recently I began reading Jo Boaler’s amazing book Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching (Boaler, 2016).  At this point I have read about half of the book and even when I have set the book aside for a while, my mind keeps contemplating the ideas shared.  The research and ideals are fueling my teacher’s soul.  Each page and new concept serves to reaffirm my own pedagogical practices and beliefs. I can’t contain myself, and find I am yelling out “Yes, yes, yes!” with every encouraging page.

As a teacher for students with disabilities, I have been educated to teach with a closed mindset, and to educate my students likewise.  For example, Saunders, Bethune, Spooner, and Browder (2013), contend children with moderate to severe developmental disabilities are frequently deficient in their rudimentary mathematical abilities.  These researchers assert that students are capable of learning content required of grade level standards if educators utilize a systematic, procedural approach in their teaching.  At the core of these procedures is the use of prompting.  Prompting begins with ensuring the student gets the right answer followed by a series of prompts that are scaffolded to reduce support.  The student is trained to follow procedures that lead to the correct answer while eliminating instructor support.   This approach is supported by the final report of the National Mathematics Advisory Panel.  Educators are advised that students with learning disabilities should receive some, but not all, systematic instruction that has teachers lecturing, exhibiting exact strategies, or using prescriptive teaching methods (2008).  According to Witzel and Riccomini (2007), once teachers optimize their mathematics curriculum, they must model and guide students through the educational resources they have chosen.  While the National Mathematics Advisory Panel supports that these strategies are successful with students with moderate to severe learning disabilities, the practice of rote, procedural teaching is in direct conflict with the philosophies of other great mathematical philosophers such as Vygotsky, Bruner, and now, Jo Boaler.

As I am reading Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages and Innovative Teaching (Boaler, 2016), I keep wondering if Boaler would consider creating a student friendly version of this book.  I believe my students would find the information liberating and empowering!  I know I can continue to hone my open mindset teaching skills while encouraging students to develop their open mindset, but I want more.  Just as we strive for students to conceptually understand mathematics, I also want them to have a conceptual understanding of the teaching approaches utilized and the practices I ask them to employ in my mathematics classroom.  In the whisperings of my mind an idea has emerged, but right now it has no backbone, it has no form, only a name; I Am Not Procedural, I Am Divergent.   I find myself wondering, how to take the ideas from the Divergent series and utilize them in a math classroom.   My mind is whirling with ways to compare and contrast the factions to the practice of rote, procedural learning. The possibilities are endless: “Faction vs. Algorithm, What’s the Connection?”  “Tradition vs. Innovation, What’s the Risk?”  “Conformist vs Divergent, How do I Break the Mold?”

At this point these are only ideas in my mind.  However, these ideas keep tickling my brain, and I can’t let them go.   Therefore, I feel the need to put my thoughts in writing for others to consider.  I have discovered in order to have a true mathematical mindset, joint construction of knowledge is paramount.  Operating in a vacuum, believing that what we have going on is the end all, is a death sentence to innovation and growth.  We need to share notions with one another, collaborate and challenge each other’s thinking.  Just as our students need to engage in constructive dialog, we also do.  It is with this belief that I am leaving my pondering here, hoping that they will spark a conversation, generate ideas, and challenge my thinking, so that I may continue on my journey to cultivate a mathematical mindset.

Works Cited

Boaler, J. (2016). Mathematical mindsets: unleashing students’ potential through creative math, inspiring messages and innovative teaching. San Francisco: Jossey-Bass.

National Mathematics Advisory Panel. (2008). The final report of the national mathematics advisory panel. Washington DC: US Department of Education.

Saunders, A. F., Bethune, K. S., Spooner, F., & Browder, D. (2013). Solving the common core equation teaching mathematics CCSS to students with moderate and severe disabilities. Teaching Exceptional Children, 24-33.

Witzel, B. S., & Riccomini, P. J. (2007). Optimizing math curriculum to meet the learning needs of students. Preventing school failure, 13-18.



The Moments That Fuel Us

I find that this time of year, when I realize our school year is half over, I can begin to panic about all that is left to teach.   If I am not careful I can get stressed and I start to doubt myself.  If I am not diligent I forget to stop and smell the roses and to reflect on the difference I am making.  Teaching special education is rewarding and challenging.  My students do make progress, but often times it is not at the same pace as their non-disabled peers.

At the beginning of this school year I began utilizing Quick Draw and Quick Look explorations designed to build number sense.  I have purposefully and diligently engaged students in activities to develop subitizing skills, “the ability to quickly identify the number of items in a small set without counting. Researchers have demonstrated a strong relationship between subitizing skill and math achievement” ( 2016) .  My students have risen to the challenge and are making excellent progress.  They are becoming master subatizers and are beginning to carry this over to  composing and decomposing numbers to twenty.  Furthermore, they are beginning to think of themselves as mathematicians.

The second week back from winter break we were engaged in a writing activity.  Students were challenged to think about the thing that makes them special.  The one thing that they are proud to call their own.   This image captures perfectly the moments that fuel us as teachers.  It is important to remember that yes, we do make a difference.  Yes, our students are learning.  Don’t be hard on yourself if some kiddos learn more slowly than others.  If you are developing relationships with your students and  using sound pedagogical practices they all can and do learn!





Building Number Sense Through Quick Draw Explorations

quick drawThis school year I have embarked on a new career experience and am teaching 1st-4th grade students with moderate to severe learning disabilities.  Almost all of my students are with me all day except for when they go to specials with a general education class.

I am a firm believer in discourse rich, inquiry based learning and am committed to bringing this experience to my students.  This poses a unique challenge as my students are a  very diverse group of learners.  I have tried a few estimation stations with them and have facilitated some sessions of, “The Answer is ____.  What Could the Question Be?”  These have proven to be too challenging for most of my kiddos  as they struggle with subitizing and basic number sense.   With this in mind I have established the routine of  using Quick Draws and Quick Looks.

Prior to launching a Quick Draw I choose an image and think of all of the possible questions I can ask, and the possible challenges students will have when deciphering an image.  I try to think of follow up questions that will facilitate deeper conceptual understanding.  I also try to see the image in as many different ways as I can as well.  Almost all of the images can be seen in 2-D or 3-D, and these present different questions and explorations.


Two of the Quick Draws we explored this week are above.  I have told my students that it is not important that they are able to draw the images exactly as they see them.  This is necessary as I have discovered that my students who receive Occupational and Physical Therapy get very discouraged and upset by their inability to draw what they perceive.   We use the quote in our room, “If you can’t make a mistake, you can’t make anything.”  I assure my students that what is important is their ability to communicate to one another what they see and how they see it.  This has freed their anxiety and everyone is eager and willing to participate.

We begin the activity in listening, learning position and I make sure all eyes are on the board before I reveal an image.  Many of my students also have Attention Deficit Disorder and if they miss an image because they have been distracted they become very upset.  My students are wonderful and cheer each other to success and encourage each other to pay attention.  I also assure them that we will take quick looks, draw, and discuss until they are comfortable with their drawings and ideas.  Once we have taken our first look and students are drawing I walk around observing.  I ask students what they are thinking and this allows me to scaffold the discussion.  I always allow my students to give a thumbs up if they wish to share and allow them to opt out if they choose not to share their ideas.  I do ask the ones who opt out to restate what others have said to promote good communication and listening skills as well as to allow productive discourse to take place.  I usually start by calling on the student who only has a vague idea of what was displayed and once we discuss and validate the piece that they bring to the discussion I ask who would like to add on.  As we go along I draw exactly what the students are explaining and I encourage them to correct me if I make a mistake.  If they are not using mathematical vocabulary I will restate what they are saying with the desired vocabulary.  As I do this students are picking up the vocabulary and utilizing it in their descriptions as well.  On Friday one of the students said that there was a diamond in the middle of the square and another student kindly reminded her that mathematicians use the word rhombus, and that Naegele Navigators are Mathematicians.


My drawings as the students share their perceptions.

 I have noticed that as we progress in our year students are quickly picking up mathematical language.  I routinely hear them use correct terminology such as rhombus, vertical, horizontal, triangle, vertices.  They notice that others are drawing the same thing as they are but  the image may be rotated or with different dimensions.  We are comparing and contrasting shapes and wondering what would happen if we removed pieces of the drawings and put the remaining pieces together.  On Friday we actually created a square piece of paper and cut out the middle triangle.  I asked the students to conjecture what shape would be created with the remaining pieces.  Half of the students said we would have a square and the other half said we would have a rhombus.  We then discussed if we should just move the pieces together to check our conjectures or if we should rotate a piece to join them.  Unanimously we agreed to do both!  The students were thrilled to discover a kite, “that is almost like a rhombus,” and a rectangle that is similar to a square.  When we first began these explorations several of my students would get upset if they didn’t get the “right” answer, but I am finding that they are more likely to take risks and make conjectures and are learning from mistakes rather than being disappointed! As the teacher of these amazing mathematicians I notice that my students who struggle the most with fine motor skills and drawing the images are the ones who can communicate what they observe the best.  I find myself wondering if continued experiences with these Quick Draws will enhance spatial awareness and if this awareness will lead to better number sense skills.

I find the use of Quick Draws to be a dynamic mathematical experience for my students.  They feel proud and accomplished and through these experiences are gaining confidence in their mathematical thinking and are considering themselves to be mathematicians.  We are exploring and being introduced to a plethora of geometric concepts while developing spatial awareness and it is my hope that as spatial sense develops so too will number sense.

IMG_20150917_121421694 IMG_20150917_121428505IMG_20150917_121433468IMG_20150917_121442392IMG_20150917_121450502IMG_20150917_121500982IMG_20150917_160326691IMG_20150917_160333088IMG_20150917_160340070IMG_20150917_160346327IMG_20150917_160354671IMG_20150917_160402604IMG_20150918_134459641IMG_20150918_134525155IMG_20150918_134537641IMG_20150918_134645416IMG_20150918_134652543IMG_20150918_134701209