Upon entering the classroom this rainy morning, I was greeted with students who were smiling broadly, as they knew just how fun Polygraph is and they were eager to play it again. Immediately my mind flashed to how this scene might have looked just a few years ago without the use of Desmos. Elementary geometry historically has been a domain notoriously associated with definition memorization with little application of understanding. In my vision, I saw a teacher standing in the front of the room writing vocabulary terms and definitions while students were simultaneously recording these definitions in their notes and desperately trying not to fall asleep in class. In reality, this geometry experience couldn’t be further from that vision. After a short warm-up with Kahoot!, students eagerly awaited the teacher to provide them with the code so that they could be matched and get started. Just before she did so, she set up a few ground rules:

*Ask only yes or no questions*- This had been a struggle in the previous experience and had been the focus of their debrief. She was eager to see how well they would apply their learning

*Keep the game anonymous**Keep questions focused on the task*

As they all nodded their understanding, she unpaused the game and they jumped right in. Our goal as facilitators during this experience was the look for trends in the questions students were asking, hoping to find opportunities to provide just-in-time feedback to students that would strengthen the quality of the questions they asked. After all the students had finished at least one round, with varying levels of success, we discovered two things. The first was that students were asking questions that were either too vague or to specific. At this point, we paused the game (as the students groaned audibly because they didn’t want to stop) and I shared this question and response that we saw:

Question: “Does it have equal sides?”

Response: “Yes”

Action: The student eliminated the trapezoid, which had been the shape the student selected.

I wrote this scenario on the board and drew a picture of a trapezoid. I asked the students if the answer to the question was correct. The crowd was split. Those that diagreed with the answer shared that there are four sides and only two were equal. Those that agreed stated that there was a pair of equal sides so the answer was correct. From this evidence that they could both be correct, we decided that the question needed to be more specific. After some student discussion, these were the revisions the class suggested

Students were beginning to understand that to ask strong questions, including quantity and other descriptors was important. While not all the descriptors were specific or accurate, the concept of good question asking was beginning to emerge.

From here, we unpaused the game again and students entered back into the game, providing evidence of their ability to be more specific in their question asking.

As the game continued on, Mrs. Vallejo noticed that students questions were becoming too specific. We were regularly seeing questions like this:

Students were becoming so specific that they were asking questions that would focus more on the name of one shape, rather than the properties that defined them. Not only did this result in them having to ask more questions to identify the shape chosen by their partner, but it wasn’t pushing them to the higher level geometric skills as described by van Hiele. This was when students were asked to close their devices and a round of teacher versus student ensued. While the game was displayed for the class to see, the student selected a quadrilateral and the teacher posed the question, “Is it a square?” When the student responded with the answer, “No”, students quickly realized that they could only eliminate one option. The teacher then asked the class if there was a more specific question she could ask. The class constructed the question, “Does it have two pair of parallel sides?”, to which the answer was yes. This type of question eliminated seven options. I asked, “Can we ask a specific question about the angles?” The class then crafted the question, “Does it have four 90 degree angles?”. The answer to this, yes, eliminated six more options. With only three options left, student searched for a geometric way to describe the difference in the two remaining rectangles of different sizes. It was then that Mrs. Vallejo and I knew that the student had found the sweet spot between questions that were too specific and too general, all the while developing a real understanding of the properties of geometric shapes.

This is a far cry from the elementary geometry classrooms of years past and I am grateful to have been a small part of this authentic learning experience powered by the amazing Rosario Vallejo and Desmos!

]]>

As I consider these two concepts, they seem similar as they both help students consider whether their answer is reasonable. This is such an important ability to develop in young mathematicians! However, I think it is also important to clearly define how estimation and rounding are the same and different for both teachers and students.

It seems that many teachers focus heavily rounding, especially the “rules of rounding”, and perhaps this is because rounding is called out so directly in both the 3rd and 4th grade Operations and Algebraic thinking standards.

3.OA. 8 Solve two-step word problems using the four operations. Represent these problems using the equations with a letter standing for the unknown quantity.

Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Perhaps they focus there more as well because the rules are very clear. In rounding you must ask yourself one very important question: Is my number closer to *this* or *that*? Note my avoidance of this “5 or more, up the score” and “4 or less, clear the mess” business. While it is a catchy jingle, it has nothing to do with the place value reasoning students need to be successful in mathematics. Instead, I consider a number line (oriented horizontally or vertically) with the midpoint identified so that students can see whether it is closer to *this* or *that*.

Seeing is believing, after all.

So, if this is rounding, what is estimation? It is my opinion that rounding is a form of estimation, but it is far too rigid to only be rounding. Here is the official definition of rounding:

If I think about estimation as a rough calculation, then I have to be much more flexible in how I think about estimating a number. There is no rule for estimation. Instead, there is just the idea that by engaging on the act of estimation, I will find an “about number”. This feels like a much more rigorous skill to teach. When considering how to estimate a number you must ask yourself two questions:

- What do I know about my number(s)?
- What number(s) is close to mine that best helps me think about the reasonableness of my answer?

Let’s consider the context of division. How many of us have seen students give us unreasonable answers when dividing two numbers? I know I have! For example, what if I asked students to solve this problem?

While it is important that students understand how to find the precise answer, it is also important for students to be able to consider an “about answer”, or to estimate. Of note in this example is that rounding would make this about answer less precise (6) than estimating might (5).

Are estimation and rounding different? Yes, though they serve a similar purpose in math. Is it critical that we teach student to be flexible enough to use the strategy that best serves their needs. Absolutely! Perhaps we can consider estimation and rounding as math tools. As Standard for Math Practice 5 states, students must choose tools strategically. Let’s all be sure to put both these tools in their tool belt and help them get ever better at choosing the best tool for the situation.

]]>To this I responded:

Once I posed this idea, I began question/wonder whether it could actually work. Would people actually be interested in a show about math class? Seinfeld was a self-proclaimed show about “nothing”. It was wildly famous. If a show about nothing could be successful, why not a show about math? What would make someone watch a show like this?

Here are the reasons I watched original The Real World:

Could these same elements be found by telling the story of a math class? Could they also produce a show that would be enjoyable to watch? Let’s consider these components one by one.

I believe we can all agree that conflict can easily be found in math class in many places. The conflict of struggling through a new or difficult idea. The conflict of disagreeing about mathematical ideas. The conflict of learning in a way that is different than the way a teacher instructs. The conflict opportunities are endless! This platform could be an amazing way to redefine what conflicts in math class are and the rich learning available in well-managed conflict.

Most people enjoy watching conflict in part because we love a good success story. In each of the conflicts above exists an opportunity to show a student, a teacher, or an entire class productively struggling their way to successful learning. I currently see math as the underdog of education. “I am not a math person” is an accepted statement in our society. Wouldn’t it be great if *The Real World: Math Class* could depict the math classroom in such a way that the ultimate success story was the rise of the underdog in which all students believed they could be math people? Even more, what if they not only believed they could, but in fact proudly claimed to be math people? Math class might then come to be respected as a learnable and enjoyable subject and anything but an underdog in education.

The success stories of the characters were always most potent when the backstories of each of the characters was compelling. I believe it is safe to say that every student enters math class with their own mathematical backstory. I am reminded of Tracy Zager’s collection of mathematical autobiographies found here. Specifically, you can find my autobiography here. These autobiographies are filled with stories that will leave you cheering, but more often sobbing, as you experience the mathematical lives of these brave authors. Whether our math autobiographies are filled with heroic or horrific tales, these chapters of our lives walk into the classroom the moment we do and play a significant role in our success or lack there of.

Perhaps most importantly, the humanity of the Real World is what attracted me most to the show week after week. I recall reading that the creators of the original show grappled with whether anyone would be interested with this concept of “reality TV”. Looking through the rear view, we know now that reality TV (for all the good and bad it has to offer) changed the landscape of TV viewing. I believe this was in large part because people enjoy viewing the humanity of their fellow beings, created by the conflict, the successes, and the backstories. Seeing the real emotions, the real flaws, the real story of people is the truest of stories that can be told. If any current situation could use some humanity, it is the story of today’s math classrooms. There is a struggle in the real world learning that we are asking our students today to engage in that was not the reality of math classrooms of the past. While the math is the same, the path is different. The humanity of this struggle is exactly the element I would love to depict to connect math learners of old and new.

As a member of the MTV generation and an avid consumer of The Real World for many seasons, I remember well the introduction to each episode:

I offer this as a re-envisioned introduction to *The Real World: Math Class…*

This is a true story of a group of students enrolled in a math class who had their learning taped to find out what happens when math stops being memorized and starts getting real. The Real World: Math Class.

Would this show be successful? I don’t know. Could it be the way we educate the public about the shifts in math education? Perhaps. Would it be my dream to see math class depicted in Hollywood in a positive light? Absolutely.

]]>

So I began asking questions to determine where his understanding stopped and his confusion began. I first asked if he knew why his teacher had him decompose his thousands into hundreds. He said he wasn’t sure. I then asked if he thought four thousand was the same of forty hundreds. After some thought, he used him homework to show how each thousand had been “broken up” into ten hundreds, totaling forty hundreds. I nodded in agreement and pushed forward. Perhaps this was his point of misconception.

My next question seemed too obvious to ask, but I did it anyway. I asked him what division means. I felt like this would be a question he would have an answer to, as we have worked quite diligently on interpreting multiplication of “groups of” over the past year and I thought he would make the connection. When he looked at me with a blank stare, I realized that his misconception was at the foundation of the operation. So I offered a context I knew he would be familiar with to support his conceptual understanding of division: Money!

Here is the problem I posed:

You have $4000. You want to divide it equally among 10 of your friends. How much will each friend get?

He began looking at the representation he had drawn with his teacher. He could see the four thousands and could also identify the forty hundreds, but could not figure out how his picture gave him an answer to the question I had asked.

I, too, took another look at his drawing and realized that even though my story problem could be solved with the equation 4000 ÷ 10, just like the problem he had represented, I was asking him to make ten equal groups and his representation showed equal groups of ten. What a happy accident! Not only were we going to get to the bottom of the meaning of division, but we were going to be able compare the two types of division with the same problem.

* (If you haven’t yet seen this progression video linked above, it is worth your time!)*

So I asked him, does this picture show ten equal groups to represent the ten friends? He said no. I asked if he could create a drawing that would help him find the answer using ten equal groups. He thought about this for a while, then leaned on his understanding of “fair shares” to find an answer. His thought process was very interesting. One might call it the beginnings of partial quotients.

He began by drawing his ten circles to represent his ten friends. He then gave each friend $250. I asked him why he chose that number and he said that he felt like it was about the right amount. I was impressed by his use of reasonableness and trust in the work of estimation. Once he had given all ten friends $250, I asked how much he had given. With an answer of $2,500, I asked him what he was going to do next. He asked me to be quiet while he thought, so I knew it was time to cool it with the questions for a bit as he was now ready to take the wheel on the path to understanding. He then gave each friend $100 more dollars, realized he still had not divided all his money, and gave each friend $50 more. At this point he was satisfied that he had divided the entire $4000 into ten groups equally, with a total of $400 per person.

Wanting to push his thinking forward on these two definitions of division using the place value chart, I asked him to identify where his four hundreds would be in his previous representation.

After some serious thought, he made a vertical ring around the four dots in the left-most portion of his drawing. I then asked how many of those four hundreds did he have in his drawing. At this point he answered confidently, “Ten”, without needing his representation for support. I then circled back to the questions I asked at the very beginning, which was why the four thousands needed to be decomposed. He stated that if there were ten friends and only four thousands that there wouldn’t be enough thousands to share equally. By making hundreds, he could share them fairly. I nodded and asked one final question, “What does it mean to divide?”. He said it means to share fairly. With a smile on his face, I knew we had reached the point of understanding.

Walking away from this learning episode, I was reminded of two very important elements of teaching division:

- There are two interpretations of division, so we must be sure to model them both in our lessons.
- Context matters! Give students problems worth solving and allow the context of the problem to drive their interpretation of division.

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…

]]>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?

]]>

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.

]]>

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?

]]>

- 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.

]]>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?

]]>