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.–

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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!

 

 

A Post a Day #7 – Relationships Make a Difference

Today my eighth grade boys were called to the library to get “THE Sexually Transmitted Disease Talk.”  When they walked out of class they were full of swag, and too cool for school.  I could hear them bragging that they didn’t need this talk because they already knew all of this stuff!

They were gone for about 40 minutes and then the girls were called to go hear the same talk.  When the boys returned, without exception every one of them looked like this:

Once I got them seated and calmed down they immediately began peppering me with questions that they were too afraid to ask of the school nurse.  I truthfully answered questions regarding transmission of STDs and the various protection products available.  In addition I answered questions about the way STDs can be transmitted as well as the prognosis and treatment of STDs.  I was secretly laughing on the inside about the stark difference of the boys that left my class before the “talk” and the ones that returned afterwards.  I was also thankful that they were comfortable enough and trusted me enough to boldly throw their questions at me.  I am thankful that they feel safe enough with me to do so and trust that I will answer them truthfully.  So today I am reminded that relationships really do matter with students, and if you build a safe community where everyone respects one another, that you can reach and teach students far beyond the academic requirements of state standards.  Forgive me if I giggle here and there about how quickly they lost their swagger.  Ahhh, the confidence and innocence of youth is wonderful!

A Post A Day #6 – Sometimes You Just Have to Roll With The Flow

The plan for the day was to utilize the Movie October Sky to facilitate discussions centered around growth mindset, history and social justice.  We were also going to compare and contrast the movie with Hidden Figures.  In addition I planned to discuss the struggles each of the lead characters experienced  and how those struggles were influenced by society.

Enter a snow cone vendor showing up, sports physicals, volleyball team meetings, baseball team meetings, NJHS meetings, softball meetings, band and choir practice and an impromptu locker clean out day facilitated by the administration.  There were so many interruptions today in every hour that I gave up on discussions and just let the students watch the movie.  At the end of the day we got an email stating that the snow cone vendor will be back tomorrow, the 8th grade students will be pulled out of various classes to have “THE HIV” talk, sports physicals are not done, and that locker clean outs will continue.

The lesson for today is, breathe and roll with the flow!  Tomorrow I will allow my students to watch the remainder of the movie and hopefully on Monday we can have our discussion, or not since finals are on Tuesday.  Planning and keeping students on a schedule during the last week of school is a minute by minute endeavor!

A Post A Day #5 Black Magic

The end is approaching way too quickly!  My students only have two full days left before the district mandated days of final exams on Monday and Tuesday.  Today the plan was to create flexahexagons, and if time allowed we were going to make origami cubes.  It turns out that time did not allow for the cubes like the ones in this video.

We began our activity by watching this Video, and then we colored these Patterns as per our individual likings.  We then reviewed these Directions and began the process of folding our flexes as directed.  I utilized my document camera and demonstrated the folds and then worked one on one with individuals when it became apparent that frustration levels were rising to an unproductive level.  By the end of each class period every student had created a successful flexagon, and I managed to keep from pulling every strand of hair out of my head.  This activity was one that we had been looking forward to for a while now.  We had tried the activity earlier in the year, but wound up ditching it because student’s frustration level was not productive and tears have no place in a math classroom!   Today many students still experienced high frustration levels and struggled, but there were no tears, no giving up and all were successful.  I was made keenly aware of several students who seem to have some serious spatial awareness challenges.  I am already thinking toward next school year and lessons or instructional routines that I can incorporate into our investigations to help students develop this ability.  I will also ask our Occupational Therapist about age appropriate activities to utilize in my classes.

The best moments of my day happened in last hour.  As students finished up their projects and flexed them for the first time you could hear loud gasps and excited voices exclaiming how cool their flexes were.  My very favorite was when one giddy young lady proclaimed with excitement and awe in her voice, “Mrs. Naegele knows black magic!”  I kind of like the idea of having magical powers and am blessed that I can still bring the wow factor even to the most skeptical middle schooler!  Oh my goodness, I LOVE these kids!  I don’t know how I am going to get through summer without them!

 

Update:  When my 20 year old daughter, who is home from college, came home from work tonight I showed her my flex.  She wanted to make one so I showed her how.  When she flexed her creation for the first time she declared that I am magic and wanted to know how I got it!  It’s a slam dunk day when you can impress your students AND your grown daughter!

A Post A Day #4 Keeping It Real

As the number of school days dwindles I find myself conflicted.  My body and mind need to be rejuvenated by the lazy days of summer, but my heart is sad for the many days I will not get to learn with my students!  I am fortunate as a special education teacher that I get to stay with my students for several years, and on the first day of school my students and I are ahead of the game because we already know and love one another!

So, as the year winds down we are engaging in some reflective activities and talking about how we have changed and grown over the year.  Today we revisited the Go Ahead, Break the Ice activity that we did the first week of school.  When we did the activity at the beginning of the year it took us an entire period to get through it, but today we launched, engaged and debriefed in less than 15 minutes.  The students reflected on the arguments that ensued at the beginning of the year because no one wanted to listen to or compromise with anyone else.  Today students employed a multitude of problem solving strategies and identified situations in real life where these skills are necessary, and were proud of the growth they have experienced this year.

We followed this activity by writing math acrostic poems.  Students were enthusiastic about sharing their creations and this teacher was moved to tears by positive growth mindset that these poems revealed.  Here are a few of my favorites:

(This above acrostic was written by my wonderful paraprofessional.)

To cap off the class we did a “Math Is” snowball fight inspired by this blog by Math=Love. Each of us wrote three “math is: statements on a paper, crumpled it and then we all tossed them in the air.  Once everyone got a snowball we took turns reading the statements shared by our classmates.  It was touching that the students know each other well enough that they were able to identity the author even with no name on the snowball.

Some of my favorite “math is” statements were:

• “I think math is fun now.”
• “Math is exciting.”
• “Math is making some people mad about it because they don’t know hard makes your brain grow.”
• “Math is patterns.”
• “Math is attitude.”
• “Math is art.”
• “Math is something I have problems with, but I have accepted it and learned.”

Of course I have to keep it real and honest so here are some that tell me I still have more work to do:

• “Math is boring.
• ”“Math is stupid.”

Even though these statements reflect a less than positive attitude toward math, the fact that more than one student felt safe and comfortable enough to share their negative thoughts affirm that relationships have developed and a community has been formed.  I love doing these kind of activities with my students and when so much awesome is shared it makes the nearing of the last day even more bitter-sweet!  I am going to cherish every second I have left with thESE awesome math nerds!

A Post A Day Until Summer Break #3

Today my heart is filled with pride and joy!  I have known all year long that my students were making progress.  I could see it in their faces.  I could hear it when they explained their thinking.  I could feel it in the goose bumps I regularly got when one of them would #NoticeWonder and make conjectures, connections and express their understanding.  In short, my teacher gene was telling me that all was going well and everyone was learning and progressing.

Well, today I got definitive proof that my instincts were spot on!  I conducted my last benchmark assessment for the year as well as a math attitude survey  that was developed by Donna Boucher, and the results blew me away! Not only have ALL of my student’s standard scores improved tremendously, but their positive attitude towards math has grown exponentially!  I always believe that what I am doing works and that following a pacing guide is not necessary, but it is scary when so many around you do!  I have not formally covered everything in our standards, but I have used #NoticeWonder, Estimation 180, daily instructional routines accompanied with this form and promoted think time, problem solving and productive discourse, and my students have proved to me once again that if you stay true to the pedagogy of constructivist, discourse rich, problem solving that they will learn and love doing it!   What a fantabulous Monday!  There is only one more left in this school year!  Let’s make every day count friends!