What if I could take over a summer school program and put really powerful math instruction in front of the academically neediest students in the school? What if I could also leverage this time to help support teachers in making their math instruction even more powerful?

From this wondering, Summer Math Camp was born. It felt like Exteme Makeover: Summer School Edition! The format is this:

- Identify and pretest exiting 4th and 5th graders in the domain of fractions (often noted as the domain most challenging for teachers to teach and students to learn in this grade span).
- Use the pretest data (focused both on content and mindset) to determine the fractional areas of focus during our week of instruction.
- Invite teachers to watch the lessons during the morning and apply the learning from their observations in the afternoon to their own fraction lessons for the upcoming year.
- Hire a film crew to capture video of much of the learning for future use with teachers.
- Spend one full week teaching hands-on engaging lessons that support both the mindset and the learning of the students and the teachers.

No big deal, right? I’ve had some crazy ideas in my time, but this is a real doozy! Luckily, I have my math partner in crime, Chrissy Newell (@mrsnewell22) working along side me to plan, organize, implement, and revise as we make sense of this work.

As I type, it is the end of day 1 and I can tell you that the work was far from perfect. However, I had two walk-aways from the learning today that really have me thinking that we are on to something with this project. First, the students and I spent the first part of our day establishing rules and norms for math class. Initially, the students thought that rules and norms were the same thing. But when I explained that rules keep us safe and norms keep us learning, students were ready to start drafting a list. Here is what we decided as a class:

The best part about the norms were that once students had a better definition of what it meant to be a “good” math student, they were more capable in engaging in those desired behaviors. On more than one occasion today, I found myself praising a student for engaging in the lesson by using one of the norms we had set. As the day progressed, more and more student opened up and began seeing that perhaps they could be “good” at math. My plan for tomorrow is to revisit the norms with the class before we engage in our next math task to determine if we need to add or delete anything from the list.

My second reflection of the day was that in the beginning of the day, students really didn’t speak with each other when I asked them to pair-share. Many stared blankly at each other and waited for me to say something. This is when I decided to give sentence frames to support the discussion. Here’s what we started with:

When students had a frame for how to share ideas, it became easier for them to share the math mathematical ideas that were forming in their minds. It also provided accountability for each partner in the conversation. By the end of the day, students were really sharing ideas and listening to one another. My intention for tomorrow is add another sentence frame that supports the idea of adding on to other people thinking.

When it came time to reflect and plan with the teachers in the afternoon, many noted that the students who often didn’t share during the school year were engaging more in this setting. My hope is that we can build on this momentum and help these students build not only confidence in themselves, but a stronger foundation in fractions as well.

The jury still out on whether we will have a successful summer math camp program. But if today’s success is any indication of the overall success, I think we’re on the right track. Stay tuned…

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A common frustration among this group of teachers is the perception that the math they teach is “easy”. While it is true that these foundations of mathematics have gained automaticity in most adults, we must not forget the challenge of learning a new skill for the first time. Not only that, but we must realize that counting with meaning is not simply uttering the count sequence without error. Indeed, it is much, much more complex than that! In the words of David Foster, “Your child can count to 100. Great! What other songs can she sing?” While learning the count sequence is absolutely foundational to the work of counting with meaning, it simply doesn’t stop there. As children move forward in their learning, they begin to make meaning of their count sequence by corresponding one object to one number and one number to one object. This is referred to as one-to-one correspondence. Given a set of objects to count, a student would likely look at/touch/move each object as they utter the sequence of numbers to match. Students must have an accurate count sequence to be successful in this work! But they must also be building the skill of matching number to object.

Beyond this is the skill of cardinality, or the ability to state that the last number uttered in the count is the total amount of objects in the collection. A good test for determining whether a student is counting at the level of one-to-one correspondence or cardinality is to ask them at the end of their count “So…how many are there?” The student who returns to the collection to count again (“1, 2, 3, 4…” all while looking at you thinking, “Didn’t I just do this?”) is a student working in the capacity of one-to-one correspondence. The student who simply states the final number of their count as the total is working in the capacity of cardinality. Having worked with hundreds of young mathematicians over the course of my years in teaching, I have seen evidence of this sequence over and over again and believe strongly in using the progression to support students counting with meaning. Here is an image of the counting progression, as suggested by Graham’s video.

What I really grappled with after watching the video was the placement of subitizing, or the ability to immediatley recoginze the quantity of a set of objects. My original thinking was that subitizing was the end game of counting. That is to say that once I made it to the summit of my counting sequence progression, I would bury my flag in the sand of subitizing. It seemed so sensical. If I had worked and worked to learn how to count a set of objects and truly know the meaning of the numbers I was counting, I should be rewarded with “just knowing” how many were in the set. So when this video showed subitizing as the jumping point of the progression, you can imagine the cognitive dissonance this caused. Could it be possible that subitizing was not the end game, but rather the foundation for it all? How could I have been so off base?

I was fortunate enough to talk to Graham about this progression and I gained some clarity of the progression. There are two types of subitizing, *perceptual* and *conceptual*. *Perceptual subitizing* is just as I defined earlier-The immediate recognition of the quantity of a set of objects.

For example, I can look at 6 pips a die and just know that it is 6 without counting. This skill is often observed in students who have yet to acquire the learning of the counting progression and is therefore used as a foundation for this work. However, with *conceptual subitizing*, smaller sets of objects are recogized within the larger set, allowing students to use their understanding of number relationships to find the total amount of a larger set.

In this case, I might see the 6 in this image, just as it appears in the die image above, and the 3 at the top of the image. Combining the 6 and 3 would give me the total number of dots, 9. While I could not *perceive* all the dots as one collection, I could see the subsets and quickly combine them to find the total amount. My understanding of the number 9 and all the numbers contained in 9 (called Heirarchial Inclusion) supported my ability to conceptually subitize this collection.

The clarity that I walked away with from my conversation with Graham was that while perceptual subitizing serves as a foundation for counting with meaning, conceptual subitizing is the culmination of the progression. It’s like a subitizing sandwich. We start with the perceptual subitizing base, layer in counting with meaning, and end with the ability to conceptually subitize. Obviously, counting is really not as easy as 1, 2, 3 and we need to be mindful of the progression as we facilitate students’ learning in this domain.

One idea that is still swirling in my mind remains after talking with Graham. During our conversation, he shared this Number Sense Trajectory document that caused me pause.

As an A-type person (and teacher!), I would really like to believe that learning is linear. Truly, I would live and die by checking boxes off the to-do list of learning if I could. Alas, my experience has taught me that learning looks a lot more like the image on the right:

So I began to wonder what a true representation of the counting progression would look like. During a resent presentation, Tracy Zager (@tzager) spoke about mathematical intuition and shared this image:

The image really struck a chord with me as it was somewhat linear, but also represented the interconnected cycle of many mathematical ideas. My current wondering is what the counting trajectory would look like if not represented linearly, as it is above. I worry that a linear representation might lead a teacher to expect mastery of one concept before moving on to the next, as the linear model suggests. While a student may not have yet mastered the count sequence to 30, for example, I believe strongly that the same student can begin to think about one-to-one correspondence and cardinality within the set of numbers they do know how to count.

By shear virtue of models, they will all be imperfect representations of the true work. However, in the spirit of constantly pushing our thinking forward, is there a model that is closer to the true work?

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As luck would have it, I was invited to model a task lesson in a fifth grade classroom who was currently studying fraction multiplication. Jackpot!

In preparing for this task, I decided to adopt the 5 Practices model for planning. For me, this can be quite challenging, as I am a meticulous lesson planner. However, Smith and Stein (the authors of *5 Practices for Orchestrating Productive Mathematics Discussions*) state that a task can viewed as planned improvisation. *Setting a goal* and *anticipating student responses* are the first of the 5 practices, while the rest of the task planning will happen on the spot. For that reason, my lesson plan looked like this:

Game time! To get our fractional thinking started the class began with counting by fourths and noting as we made a whole. We spent some time thinking about all the patterns that existed in the recording of our count, including the diagonal pattern of the “wholes” and the difference of 5 fourths in each column.

Next, it was time for some story problem theatre. I asked the class to close their eyes while I told them the story of a school that needed help with their cornbread fundraiser. During this time, I read the introduction to the task twice, asking students to visualize as I read. After the second read, I had them discuss with their partner what they knew about the fundraiser. As a class, we came up with this collective information:

Once we agreed on what we knew, I set the students off to work independently answering the questions. The only parameter I gave was that their solutions must include pictures, numbers, and words. As they worked, I *monitored* the room (Practice Number 2), looking for evidence of student understanding. Here are some of the things I found:

This student showed evidence of partitioning a whole into fourths, but wasn’t sure how to model the 1/3 of the fourth that remained in the pan. This was a common issue among the students.

7/12 was a very common answer, as many students sensed a need for a common denominator, but weren’t sure what to do with it once they had it. I hadn’t anticipated that they would add it! I will be sure to add that to my anticipations for the next time I facilitate this task. However, a strength in the representation is that you can see the 1/3 of the 1/4 in the model. A great progression from the former representation.

As I monitored, I noticed there were a few in the room who presented the correct solution with a correct model and the *selecting* (Practice 3) and *sequencing* (Practice 4) were beginning to take shape in my mind. Under the document camera, both solutions above were presented by the respective students, as the class listened and worked to make sense of these two different ideas. After each student presented, I asked students in the class to repeat the information that had been shared with their partner. I then asked students to discuss what they noticed and wondered about these solutions. From this talk, a student raised his hand to ask why they had chosen a tape diagram rather than an area model. Neither student could articiulate the reason for their model choice, so the questioning student asked to share his model, which he had done with folded patty paper.

This student said that because the pan was square, the area model seemed the closest to the picture in his mind. He then explained how he showed a third of a fourth and how he determined that each square was worth $1. At this point, many heads began nodding and I knew we were on the right track.

Next, a student raised her hand and asked, “Wait. Isn’t this a multiplication problem? Since we need 1/3 of 1/4, doesn’t that mean we need to multiply to find our answer?” Again, nods of approval swept across the room.

As I recorded the student’s thinking, I asked students to look at the equation and notice and wonder once again. After some partner discussion, a student offered the idea that you can multiply the numerator with numerator and denominator with denominator in order to find the product. I recorded the thinking as he spoke:

At this moment, I saw the final chapter of this story nearing an end. Students had grappled through some tough application of their fraction knowledge, both finding success and struggle. Through this jouney, students pressed forward, carrying each other along to our final destination. The goal of the lesson had been to help students see that (a/b) × (c/d) = ac/bd. To see why the rule worked. I was thrilled with the sense making these students had engaged in to understand this concept.

I knew that I was going to enjoy this task, but I was thrilled to see that the students were just as excited as I was to solve the problem.

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As a math coach, I reflect back on my classroom teaching years and cringe at some of the decisions I made in math class. Some of these practices included lengthy and unneccsary quantitites of homework, teaching with abstract numbers and not attending enough to the concrete and representational work of mathematics, and engaging to “test prep”, just to name a few. However, the one decision that I made year in and year out that has been weighing heavily on my mind recently is the way in which I treated the domain of geometry in my classroom and in my mind. I perceived geometry as the domain that we did “for fun” (unlike all the other mathematics that I taught?) and therefore it didn’t seem as important to teach. In addition, because geometry was not as heavily tested on our annual assessments, it was often taught at the end of the year and rarely taught to completion.

This practice got me thinking…Can geometry be fun *and* important? How important is it to teach geometry as something more than just the “fun” end of the year activity?

While geometry is not considered major work of any grade K-7, it is very important, according to the Student Achievement Partners at achievethecore.org. (All grade level documents can be found here)

Moreover, the 6th NAEP (National Assessment of Educational Progress) indicated that students in grades 4, 8, and 12 appeared to be performing at the Visualization Level of geometric thought. This level is considered “level 0” in Van Hiele’s Level of Geometric Thought and it is considered necessary for students to have achieved a minimum of “level 2” geometric thought (Informal Deduction) by the end of middle school. (Read more about Van Hiele Levels of Geormetric Thought here)

Could these findings be in part due to the fact that I, and likely many others, did not see geometry as important? How can we begin to see the geometric connections to the major work of the grade and bring geometry back to math class?

One of the major connections I see is between the work of

- Operations & Algebraic Thinking (OA)/Numbers & Operations Fractions (NF)

and

- Geometry (G).

The major connection between these domains is the work with composing and decomposing. Just as we put together and take apart numbers (whole and otherwise), we also put together and take apart shapes. For example, consider the work we do with combining fractions. When using the concrete tools of pattern blocks, students can more easily see how the combination of these shapes can relate to the combination of fractional parts.

These representations can also apply to the inverse work of fraction subtraction and/or division in relation to shape decomposition.

In the earlier grades, when students are engaging in the beginning concepts of part-whole models, the use of geometric tools can perhaps demonstrate the equality of the parts to the whole better than using digits in a number bond can, especially for students who are still functioning in the concrete level of understanding with these part-whole relationships.

They say that hindsight is 20/20. While I can’t go back and undo these decisions I made about geometry in my past, I can move forward and support others in clearly seeing that geometry CAN be both fun and important. These examples are just the tip of a much larger iceburg that I hope we can reveal together.

What opportunities do you see for integrating geometric instruction into other mathematical domains?

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- How much do we have?
- How much more do we need?
- Will a full container be enough?

The day finally came, we decided, to determine the answers to our questions. The bottle was nearly full and it was time to begin the sorting, counting, and coin rolling process. More questions:

- How many quarters in a roll?
- Why are there more pennies in a roll than nickles if the nickels are worth more?
- What do I do with the remaining change that doesn’t fit in a roll?

The four of us sat around the shag green carpet of my childhood home and engaged in what would today be called a counting collection. (More on that topic here) With great anticipation, we watched the piles of coin rolls stack higher and higher. At the time, I was only 4 years old (the baby of my family). I was certain that by the looks of these piles, we were millionaires! Alas, I was reassured at the end of our count that, while we had not quite reached the millionaire mark, we had saved enough money to book our trip.

It wasn’t clear to me then that this experience was a mathematical one, but in hindsight I can see that it most certainly was the real world math experience that we strive to give our students day in and day out. And all of this orchestrated by my mom, who to this day will tell you she is “not a math person”. I beg to differ…

The year following, I started school and progressed through each successive year with success. Nothing spectacular or memorable. It wasn’t until the 3rd grade that I began to wonder if I, too, was “not a math person”. This was the year of memorizing multiplication facts. It all began harmlessly enough.

- Times 0 – √ (Rule: Anything times zero is zero.)
- Times 1 – √ (Rule: Anything times one is itself.)
- Times 2 – √ (I had been skip counting these since 1st grade. I was a 2’s skip counting ninja by this point. I don’t think my teacher even noticed my fingers discretely marking my skip count under my desk.)
- Times 3 – Hmmmm…no rule, no trick? I just have to remember these?
- Times 4 – Same as times 3. What is going on here? Why can’t I remember these?
- Times 5 – Oh thank goodness! I can skip count these too!
- Times 6 through 9 – These flash cards aren’t working! I don’t know what 8 x 7 is!
- Times 10 – Who cares? I’m not good at this anyway.

Needless to say, 3rd grade left me with a bad mathematical taste in my mouth. My mom isn’t good at math, so it’s okay that I am not good at math too, right?

Upon entering 6th grade, Mr. Kuhl changed my perspective on my ability in math class. He saw me as a student who was capable and put me in situations in which he demonstrated his trust in my ability to use problem solving and reason to make sense of the math problems he was posing. This was the year that, yet again, math incited wonder in me, just as it had so many years ago with a bottle full of coins. In was in large part due to this experience at this very important time in my life that I began to excel in mathematics and was placed in the advanced mathematical courses throughout the remainder of my K-12 school career. I was able to view myself as a mathematically proficient person and I thank Mr. Kuhl for believing in me in this way so that I could again believe in myself.

Jump ahead a decade and the tables had turned. I was now the teacher at the front (ok…I was rarely at the front!) of the room. I was young and didn’t know much about teaching, but I did know how I wanted my students to perceive themselves. I made it a point to avoid the “cume folder” until I had a concern about a child. While this might be a controversial decision, I felt that it was critical to allow each child to define themselves in my classroom. I had seen too many teachers determine a child’s ability before they ever entered the classroom on the first day and many of the opportunities that were afforded these children, or not, was based upon these determiniations. I was determined not to do this. Every child that entered my classroom door did so as a blank slate and, in my mind, capable of accomplishing amazing things! Getting to know the child never changed this view, nor did reading the “cume”. I wanted to be the Mr. Kuhl of my students’ lives. Especially in math, where so many students had determined they were “not math people”, I believed that I had a responsibility to show each and every one of my students that they could be, and in fact already were, “math people”.

I taught in the classroom setting for 12 years, as both a K/1 and 6th grade teacher. I enjoyed this time in my life a great deal and perhaps learned more than I ever taught. In my last year as a classroom teacher, I was honored with the experience of a lifetime. I was awarded the Presidential Award for Math and Science Teaching (PAEMST) and was flown to Washington D.C. to meet President Barack Obama. (more on this experience here) In a very surreal moment in the East Room of the White House, I had the priviledge to listen to POTUS thank *me* for my service to students. I look back on this experience and each time pinch myself to ensure that it wasn’t all just an elaborate dream.

Today, I am both a math consultant for the Stanislaus County Office of Education and a math instructor of pre-service teachers with California State University, Stanislaus. It is my hope that through these roles I can tell my story and positively influence the mathematical lives of children for years to come.

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Shortly after creating this list though, I read Sarah Caban’s blog in which she considered her own math intuition in a very raw and vulnerable way. Sarah was willing to put her imperfect and in-progress thoughts out there for others to consider. Her bravery was not lost on me.

Shortly after reading her blog and considering my own math intuition, I saw this image:

This made me reflect upon what I model for both the teachers and students I work with. Am I modeling the bravery that I expect of those that I teach? Am I putting my mathematical ideas out there, like Sarah Caben, even when I know that there is a chance that the answer is not correct?

It occured to me in that moment that I have a real #mathconfession to make to the math community and beyond…

*I am afraid of getting the wrong answer.*

*I am afraid of what others will think of me when I do.*

*I am afraid that trusting my math intuition will make others think less of my math ability, as I am sure that my intuition will sometimes lead me to make mistakes and those mistakes will be visible to others.*

There. I said it.

It is amazing to me after all the years I have shared with teachers and students that we must have a growth mindset and that mistakes are opportunities to learn in math that I still haven’t internalized this truth for myself…yet.

I felt compelled to share this not-so-pretty turth about myself as I know that acceptance of this fact will help me grow beyond it. I also thought that perhaps there are others out there who are still grappling with this fear and that we could support each other in this journey.

I recently heard Tracy Zager (@TracyZager) speak at CMC-North about mathematical intuition and the incredible need to develop it in students. I realize now that in order to help students and teachers find and develop their intuition, I must first develop my own. In order to do that, I must be willing to accept that I will make mistakes on this new path.

Looking back on my mathematical upbringing, most of my learning was achieved procedurally. As a student, I wasn’t given a chance to develop my mathematical intuition. I was taught to listen, remember, and perform. I wonder now if this was learning math at all.

Why is it that in many, if not all other curricular areas, the process of revision is emphasized and praised, yet this is not true of mathematics? In the process of writing, students are taught to create and revise drafts of work on the way to a final product. In science, the process of creation, experimentation, and revision is the way of doing business. Yet in math, performance rather than process is often what is held as the definition of success. Until we can change the script on this view of mathematical success in a major way, I fear that many will have the same view of mathematical success and mistakes that I am still struggling with today.

So here I am, with the uncomfortable knowledge that I must redefine my own definition of mistakes in math. I must be willing to make mistakes publicly in order for not only my own growth, but the growth of the mathematical community I support and from which I learn.

With that I commit to making math mistakes and learning from them in 2017 and I ask you…

What is your #mathconfession and how will you use it to propel you in the new year?

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After having attended NCTM 2016 in San Fransisco where mathematical tasks was a major theme, I have spent a great deal of time thinking about how one plans and implements a math task. One of the books that I used as a resource was this one:

I was so inspired by the ideas of this book that I took many of the concepts presented and tried them with students to see how theory would play in practice. I wrote about one of these experiences on my blog titled Fraction Math Task in Action.

One of the 5 practices is to sequence the student work to be presented to the whole class. The authors state that “the key is to order the work in such a way as to make the mathematics accessible to all students and to build a mathematically coherent story line.” (pg. 44) The authors go on to list ways in which a teacher might choose to sequence:

- Presenting a common strategy first to create more access to the task
- Sharing a common misconception first to create clarity
- Moving through the concrete to abstract progression, as these strategies will be ripe with connections to each other and will provide connections for student learning

After having an opportunity to try these ideas with students, I saw just how successful tasks organized in this way could be. Which is why the above Twitter Chat really caused me great cognitive discomfort. Is it possible that I was “taking the agency right out of the hands of students and making the processing time amost algorithmic” (pg. 140) as Humprhreys and Parker suggest in their book *Making Number Talks Matter*?

At this point, I felt compelled to go back to both these books with a new lens to determine which strategy is best when engaging students in math talk. During this rereading, I noted that Humprhreys and Parker worried that pre-selecting and sequencing can put students in the position of thinking that one strategy is better than another, especially if the most efficient strategy is always shared last in the sequence. Students are more keen observers than we often give them create for and are likely to pick up on this pattern over time. I think we can all agree that we want students to take pride in their mathematical thinking and giving more value to one strategy over another can be the kryptonite to a student’s mathematical confidence.

As I continued to grapple with these seamingly different practices, I was drawn to another of my favorite books, *Intentional Talk*. Kazemi and Hintz offer four principles for mathematical discussions and it is the first of those that shed some light on this issue for me:

Discussions should achieve a mathematical goal. The mathematical goal acts as your compass as you navigate the classroom talk. (pg. 3)

With this quote in mind, I see now that perhaps the ideas presented in both *5 Practices for Orchestrating Productive Mathematical Discussions *and* Making Number Talks Matter *are not so much different theories as they are different practices for different goals. As Principles to Actions states:

Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions. (pg. 12)

It’s really a matter of knowing what you are doing and why you are doing it. What is the learning goal for this talk and what approach will yield the greatest results for students? As we know, time is limited, so we must be intentional with all decisions we make to ensure that mathematical success can be found for all students.

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So what is a teacher to do? In the face of the dynamic tension between time and demands, how can we strive to that which we know is best for students given the limited amount of time we have to do it?

In his 2016 NCTM Ignite talk, Andrew Stadel (@mr_stadel) spoke about the #classroomclock. Since hearing him speak, I haven’t been able to get my mind to stray from this subject for long. In this talk, Stadel spoke about the 80/20 rule. He stated that 80% of what we want to do with students can be done with 20% of the tools available to us as teachers. These are mind-boggling statistics to me. After years and years of constantly adding more to my regime of teachers tools, I’m not sure I ever considered what I should have been pruning away from this same collection. The result of this was a lack of focus and extreme proximity to teacher burn out.

Recently, while reading the book *Leading with Focus *by Mike Schmoker (2016), I was taken aback by this quote:

The real path to greatness, it turns out, requires simplicity and diligence…It demands each of us to focus on what is vital — and to eliminate all of the extraneous distractions.

-Jim Collins

Having just recently worked with a group of elementary teachers over a three day period, we collected this list of routines that we had engaged in:

I began to wonder which of these tools were among the 20% that could get me the most bang for my buck? Which of these could be elimated or reduced? Which items were not on this list, but should be?

As I was considering my past practice, I thought about my beloved calendar time. There was so much good learning that happened during that time. I was able to engage students and have fun while learning. At the time, these felt like major wins. However, hindsight is 20/20. I know see that many of the components of this sacred cow in my classroom were not my standards to teach and, as such, were an example of the mathematical clutter that should have been modified to meet my standards or eliminated altogther. How many of us out there have a practice like this that we can only define as mathematical clutter? Can we be brave enough to stand up to what Michael Fullen calls “the awful inertia of past decades” and do what we know is best for students?

Because we can not create more time in our day, we have to be very strategic in how we choose to spend this valuable commodity with our students. Our #classroomclock is ticking. With each passing moment, we have the opporunity to create greatness in our students. It’s time for us all to re-evaluate our #classroomclock.

What will you add?

What will you reduce?

What will you eliminate in the name of greatness?

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- “Why does multiplication and division come before addition and subtraction?”
- “Why do parentheses come first?”
- “Why must multiplication and division be solved from left to right?”
- “Why is the same true for addition and subtraction?”
- “Does it really have to be solved from left to right????”

It was like a mathematical floodgate had opened in my mind. Once I started asking questions, I couldn’t stop. I needed to do some exploring. I needed to make some sense of these questions. I decided to solve an expression that required the Order of Operations, but pushed myself to answer the “whys” of every move.

When I thought about the expression above, I knew the following:

- If I work from left to right, the solution will be 3
- If I solve using Order of Operations, it will be -47.

Big difference, right? How could I prove to myself that I needed to multiply before I subtracted? First, I needed to know what the expression actually meant. When I put it in words, it sounded like this-

*Twenty-four groups of three, less twenty-five*

This wording was key for me to say for certain that you must work with your “groups of” (multiplication/division) first before adding or subtracting from those groups. Knowing the meaning of the operations was critical to making sense of the meaning of that expression. Alas, I finally had an answer to go with my heaping pile of questions.

Next, I aimed to determine why the parentheses must be solved first. I was struck by Wolfram Alpha’s definition:

Parentheses are used in mathematical expressions to denote modifications to normal order of operations (precedence rules). In an expression like , the part of the expression within the parentheses, , is evaluated first, and then this result is used in the rest of the expression.

In mathematics, some items must simply be agreed upon like the symbol = meaning “the same as”. Piaget calls this type of learning Social Knowledge. One learns this type of knowledge by being told or through demonstration. It is not something that can be discovered. The parentheses are grouping symbols and therefore must be evaluated first. Another question answered!

I began to feel that I was making progress. This particular layer of mathematical learning was beginning to gain momentum. I was left with my last group of questions. Must multiplication and division be solved from left to right? How about addition and subtraction? Could there be exceptions to this rule? I decided that finding a non-example would be all the justification I needed to make my decision. But where to start?

As I did some Googling (I know, I know…I was desperate!), I landed upon an enlightening article entitled 12 Rules That Expire in the Middle Grades written by Karp, Bush, and Dougherty. This article claims that the PEMDAS rule expires in 6th grade when students are given expressions that do not require this rigid sequence. This example is given to support the claim:

For example, in the expression 30 – 4(3 + 8) + 9 ÷ 3, there are options as to where to begin. Students actually have a choice and may first simplify the 3 + 8 in the parentheses, distribute the 4 to the 3 and to the 8, or perform 9 ÷ 3 before doing any other computation—all without affecting an accurate outcome.

There it was. This one non-example was all I needed to prove that solving does not always move from left to right with addition and subtraction or multiplication and division. In addition, the article goes on to state that PEMDAS limits students to thinking only about parentheses, when there are additional grouping tools like brackets and horizontal fraction bars that must also precede the other operations for the same reason.

One big question still remained. I needed to understand *why* the Order of Operations works. CCSS-M 6.EE.2c calls refers to the Order of Operations as the “conventional order”. To some extent, I believe that like many mathematical practices, we had to socially agree upon a procedure. But as I considered it further, I also now see that there is a heirarchy of groupings within the Order of Operations that is quite logical.

Socially we agree that parentheses (and other grouping symbols) take precedence over any other operation. That makes these grouping symbols top of the grouping food chain. Beyond that, exponents can be interpreted as repeated multiplication, thereby creating groupings of groupings. Logically, this is a larger grouping than making or breaking groups through multiplication and division and should preceed that work. Finally, as addition and subtraction aren’t grouping symbols, it makes sense that we would address them last.

Just like with all the other discoveries I have made on this journey from rules to true understanding, I have found that we MUST use our sense of numbers and operations when engaging in mathematics. These “rules” that we were taught all seem to have an expiration date. However, the need for mathematics never expires. Therefore, my quest for deeper mathematical understanding continues.

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“Number sense is good intuition about numbers and their relationshipsthat developsgradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Howden, 1989)

Feeling good about their working definition of number sense, I asked them to use a ten frame to build 7 using two color counters. Here is what all but one teacher in the room created:

I was intrigued by this observation. After our discussion about number sense, I anticipated that there would be many different ways of making seven in the room. Yet only one teacher made seven differently:

At this point, I had enough formative assessment data to know that these teachers were ready to build some flexibility with this tool. I first asked one of the teachers who filled the top of the ten frame first to share her strategy with the group. As she described how she built her seven she stated that she “followed the rule” of filling in the top of the ten frame first before moving to the bottom row. As many of the other teachers in the room nodded their heads in agreement, one replied in jest, “Did you get a rule book with your ten frames?” Laughter ensued…

I then asked the teacher who had filled the ten frame in differently present her solution. It was evident that “breaking the rule” made several teachers in the room nervous. We began to talk about the importance of students “seeing” the 5 in a ten frame, as that is a valuable benchmark for students in this base ten math system. I then expressed that students can use this same ten frame tool to “see” all the hidden partners within a number. In connection to this, I shared this quote from Standards for Math Practice (SMP) 5:

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Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.” (CCSS-M, 2013)

The spirit of the standard indicates that we require *mathematically proficient students* to make decisions about which tool to use and how to use it. With this new information in the forefront, I proceeded to ask the teachers to build 7 again on their ten frames, pushing them away from the 5 frame structure and into a different composition. It was amazing how quickly they became comfortable with using this tool flexibly. They then walked around the room comparing their composition to others, determining how they were the same and how they were different from each others compositions.

As we debriefed our learning from this task, we determined that there were three strategies for using the ten frame: Building 5-Wise, Pair-Wise, and Otherwise.

We also determined that their is benefit to teaching students multiple ways to use a tool in order to build greater number sense. Walking away from this day of learning with these teachers, I found myself thinking, “SMP 5 for the win!”

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