From what I understand, years 3-5 can be among the most transformative years in a teacher's career. That lands me right in the middle of that range. And despite my distaste for the New Mexico teacher evaluation system, I am thankful to my current principal, who uses it effectively as a way to reflect upon and discuss my practice. Recently, during my post-observation meeting, I sat down with my principal and we decided the next areas for me to focus are tweaking my questioning techniques and taking a deep look at methods for assessment.

With that said, I have been really focusing on low-barrier versus high-barrier questions when engaging students in a lesson. I found, that in my first two years of teaching (starting to breakaway in year 3), that I was so ready to introduce students to new content that I would introduce new procedures before I even gave students a chance to use their own reasoning and demonstrate what they already knew. I was so caught up in the idea that a diagnostic had to be a written test and I didn't want to waste a class period so why not quickly move ahead?

Now, in my fourth year, I have really made it a focus to open up new ideas by giving students an opportunity to try their own strategies for solving problems I will ultimately give them a new strategy for solving. Recently, the idea I wanted to introduce was using a line of best fit to make predictions about future data. Old me would have simply said today we are going to learn about lines of best fit and then I would move full speed ahead, using the "I do, you do, we do" method of instruction" (I shudder just thinking about my confidence in this strategy as the only logical form of instruction). Now, I present my students will the following problem:

The Google Doc version of this file can be found here.

The goal is for students to come up with their own strategy for predicting these future data points and some of the ideas are incredibly sophisticated and show a deep level of understanding. Some students said they noticed a positive correlation and estimated where the points would be at 750 and 1400 minutes. Others created an elaborate box structure to create a high and low option for each set of minutes and then put their point in the center (if I remember, I'll post a picture of this). and still others, estimated by drawing a line through the top and bottom points to estimate future values, showing they were already thinking about lines of best fit.

Year 3 me would have said, "Cool! Thanks for working through this guys (without having students show off their work). Now let me show you the correct way to do this!" Today, wiser year 4 me takes mental notes of different lines of reasoning that students came up with on their own, which they then share with the class. In this way, I am not the owner of new knowledge. Student ideas are validated and multiple lines of access to the content are opened up. Now that students have had a chance to develop their own reasoning and share it with the class, they are primed for me to demonstrate the alternative method of using a line of best fit (which some students came incredibly close to generating on their own).

In the end, opening with students presenting what they think is a powerful motivator and leads to deeper initial understanding of the underlying concepts in new material. This is similar to my opening activity for linear equations seen in an earlier post about constant difference. I find that even my lowest level learners are capable of accessing mathematics when they aren't required to "crunch numbers" right at the beginning, but can reason through the process. I am going to continue to explore these opening activities and try to generate even more. Maybe this will become a serious hobby and lead to a blog in the style of Visual Patterns produced my Fawn Nguyen.

I just want to give the MTBoS a shoutout for pushing me to be better and move beyond teaching how I was taught.

# Limits Do Not Exist

## Tuesday, February 23, 2016

## Tuesday, January 5, 2016

### Thoughts on MTBoS

I was intrigued by Brett Gilland's post on scaling the MTBoS community to include more teachers because I can tell he has a vision for what this online community can be. I also think it fits with Dan Meyer's vision of having an open-source, digital math curriculum that challenges conventional methods for teaching mathematics. With these posts in mind, I have a few thoughts:

First, having recently earned my MA in secondary education, it is clear these open-source resources, the reflections of educators, and the quality virtual discussions are more important than ever for teachers looking to improve their practice. The rigor of my MA program was offensive to me and many of my peers, resulting in limited growth for me as a professional. The worst part is that these programs are not free, though they result in modest pay bumps. What are teachers paying for if they are simply going through the motions of taking classes to get a degree because there is no challenge beyond two page reading reflections? This begs the question, what are state/federal governments hoping to accomplish by requiring teachers to earn an MA to become highly qualified? If it is simply to have nice statistics to include in reports to state departments of education, while failing to push teachers to challenge their own practice and grow as professionals, then it is a useless exercise. I know there has been increased discussion on the national stage about the quality of teacher ed programs, but I won't be surprised if that topic disappears from conversation in the next six months. MTBoS offers an alternative, a place where teachers can reflect, gain access to new lines of thought and resources, and ultimately grow into better educators. But what percentage of math teachers in this country are actually participating? Despite these benefits, the reality is that schools/districts do not recognize this informal professional community because it is not part of a formal institution. When I think about scaling this community to meet Brett's vision, maybe it involves formalizing the community in some way so that it is recognized by districts/state departments of education to incentivize trad teachers to join. National Board Certification does not necessarily mean an increase in pay, but it does garner a certain amount of respect and increases a teachers' attractiveness to schools around the country. But the National Board Certification costs a significant amount of money, yet MTBoS is free and if it remains free, many trads may be attracted to partake if they are recognized in some way by their communities for their involement.

Second, I am still a young teacher (4th year) and the MTBoS is intimidating. I do not mean this in a negative way, but it is scary to become an active participant. It has been great using resources developed by quality teachers, but giving back to the community can cause new members to tremble, fearing they will be found out as a fraud. I admire people like Sam Shah and Jonathan Claydon, who are confident writers and have incredible resources to share, but viewing them as the standard can be a deterrent to posting one's own resources (I am definitely speaking for myself in here). I think it is great that the MTBoS blogging initiative was created and pairing new bloggers with experienced bloggers is brilliant, but will this encourage more traditional teachers to join the conversation? I lean towards no as the answer to this question. So this brings us back to Brett's question. Is this community designed to be a safe space for teachers already open to progressive pedagogy to discuss teaching and share resources so that they can do more for their own students, or is the idea to transform the profession and the teaching of mathematics around the country. If the goal is transformation, how do we continue to decrease the fear associated with involvement. The blogging initiative is a great first step, but I think more has to be done.

Third, and last I think, the final point Brett made was there is an absurd amount (maybe approaching infinity) of materials out there for teachers to sort through. As a result, sometimes it feels like luck that I stumble upon something that I can use in my own classes. For those teaching the same class as Sam Shah for example, his virtual filing cabinet is like walking into Narnia. Resources from his entire curriculum can be found and so there is a foundation to build upon and tweak as one likes. But without access to teachers that have put together these filing cabinets for classes I teach, the hunt for resources is daunting and often times I feel that I am settling for tasks that I am not as excited about just for the sake of sanity. I am excited to see Dan's work on curriculum in the future, I know Brett loves CPM, and there are the virtual filing cabinets, but this is a very small amount of order in an otherwise chaotic system. There must be a way to start consolidating resources and organizing them. Maybe a MTBoS virtual filing cabinet that organizes resources from all blogs based on their tags. I'm not saying I know how to accomplish this, but too much random information can be almost as damaging as too little and for those of us early in their careers, trying to work some of the amazing tasks that are available into our curriculum can make things feel disjointed. Is there a way to create a foundation for teachers looking to change their practice, a place where the MTBoS community has organized various contributions/tasks into a skeleton curriculum for all grade levels/content areas? At this point I am simply rambling and spit balling ideas for creating a vision for the future.

In the end, I love MTBoS and it has had the greatest single impact on my teaching practice. I am excited to see what this community grows into and I have great admiration for all of its members. Thanks for reading and I apologize for any grammatical errors.

First, having recently earned my MA in secondary education, it is clear these open-source resources, the reflections of educators, and the quality virtual discussions are more important than ever for teachers looking to improve their practice. The rigor of my MA program was offensive to me and many of my peers, resulting in limited growth for me as a professional. The worst part is that these programs are not free, though they result in modest pay bumps. What are teachers paying for if they are simply going through the motions of taking classes to get a degree because there is no challenge beyond two page reading reflections? This begs the question, what are state/federal governments hoping to accomplish by requiring teachers to earn an MA to become highly qualified? If it is simply to have nice statistics to include in reports to state departments of education, while failing to push teachers to challenge their own practice and grow as professionals, then it is a useless exercise. I know there has been increased discussion on the national stage about the quality of teacher ed programs, but I won't be surprised if that topic disappears from conversation in the next six months. MTBoS offers an alternative, a place where teachers can reflect, gain access to new lines of thought and resources, and ultimately grow into better educators. But what percentage of math teachers in this country are actually participating? Despite these benefits, the reality is that schools/districts do not recognize this informal professional community because it is not part of a formal institution. When I think about scaling this community to meet Brett's vision, maybe it involves formalizing the community in some way so that it is recognized by districts/state departments of education to incentivize trad teachers to join. National Board Certification does not necessarily mean an increase in pay, but it does garner a certain amount of respect and increases a teachers' attractiveness to schools around the country. But the National Board Certification costs a significant amount of money, yet MTBoS is free and if it remains free, many trads may be attracted to partake if they are recognized in some way by their communities for their involement.

Second, I am still a young teacher (4th year) and the MTBoS is intimidating. I do not mean this in a negative way, but it is scary to become an active participant. It has been great using resources developed by quality teachers, but giving back to the community can cause new members to tremble, fearing they will be found out as a fraud. I admire people like Sam Shah and Jonathan Claydon, who are confident writers and have incredible resources to share, but viewing them as the standard can be a deterrent to posting one's own resources (I am definitely speaking for myself in here). I think it is great that the MTBoS blogging initiative was created and pairing new bloggers with experienced bloggers is brilliant, but will this encourage more traditional teachers to join the conversation? I lean towards no as the answer to this question. So this brings us back to Brett's question. Is this community designed to be a safe space for teachers already open to progressive pedagogy to discuss teaching and share resources so that they can do more for their own students, or is the idea to transform the profession and the teaching of mathematics around the country. If the goal is transformation, how do we continue to decrease the fear associated with involvement. The blogging initiative is a great first step, but I think more has to be done.

Third, and last I think, the final point Brett made was there is an absurd amount (maybe approaching infinity) of materials out there for teachers to sort through. As a result, sometimes it feels like luck that I stumble upon something that I can use in my own classes. For those teaching the same class as Sam Shah for example, his virtual filing cabinet is like walking into Narnia. Resources from his entire curriculum can be found and so there is a foundation to build upon and tweak as one likes. But without access to teachers that have put together these filing cabinets for classes I teach, the hunt for resources is daunting and often times I feel that I am settling for tasks that I am not as excited about just for the sake of sanity. I am excited to see Dan's work on curriculum in the future, I know Brett loves CPM, and there are the virtual filing cabinets, but this is a very small amount of order in an otherwise chaotic system. There must be a way to start consolidating resources and organizing them. Maybe a MTBoS virtual filing cabinet that organizes resources from all blogs based on their tags. I'm not saying I know how to accomplish this, but too much random information can be almost as damaging as too little and for those of us early in their careers, trying to work some of the amazing tasks that are available into our curriculum can make things feel disjointed. Is there a way to create a foundation for teachers looking to change their practice, a place where the MTBoS community has organized various contributions/tasks into a skeleton curriculum for all grade levels/content areas? At this point I am simply rambling and spit balling ideas for creating a vision for the future.

In the end, I love MTBoS and it has had the greatest single impact on my teaching practice. I am excited to see what this community grows into and I have great admiration for all of its members. Thanks for reading and I apologize for any grammatical errors.

## Sunday, November 22, 2015

### Constant Difference: Building Understanding

Fair warning: I typed this up quickly and posted, so there is a high probability of nonsensical statements, which you are free to comment on and ridicule me for; I can take it.

While teaching high school, I found that many students could calculate slope because they remembered the equation. Teaching juniors and seniors in Pre-Calculus and AP Calculus AB, I found that every time a student made an obvious mistake when calculating slope, it was rooted in their lack of understanding of what slope is (and as a Calculus teacher, this was a horrifying discovery). Now that I teach middle schoolers, I have the ability to try to develop that understanding so that Calculus teachers of the future do not have the same stomach sinking experience. This short story is all to set up the activity I recently used with my students to develop an understanding of slope (a word I had not yet presented them with) as the constant difference. Who says students aren't engaged by exploring patters rather than "real-world applications" (buzz word for non-math teachers or those who do not see the beauty of math for math's sake)? Below is the sheet I hand out at the start of the lesson.

The Google Doc for this activity can be viewed

After having kids read through it, I

After the 8 minutes is up, kids are then allowed to talk with their neighbors (they sit at tables of three to four students). The beauty of this activity is they become so engaged in figuring out the patterns that almost no redirection is required and I love the student reasoning.

For question 1, kids simply count three more segments/toothpicks and come up with 16.

For question 2, a few strategies are common:

At this time, I then give students an additional 3 minutes together to try to come up with a rule or equation, which could be as simple as you multiply the pattern number by 3 to get how many added toothpicks there are, then add 1 to include the extra toothpick at the beginning. Others will come up with something more complex such as 3 x P + 1 = N, where P = the pattern number and N = the number of toothpicks. Once again, when time is up, I have students present their solutions and we run a couple of samples to make sure they produce the correct values for design 8 for example, which is easily verifiable. This gives students a chance to use their reason to come up with something more abstract.

Finally, there is the challenge problem, which is much more accessible than it looks at first sight. Some of my students that struggle with mechanics were very clever and said no, 5000 could not be the number of toothpicks because they guess and checked with multiplication to find that 4999 and 5002 would work, which means 5000 wasn't possible. Students with a deeper understanding of mechanics simply did (5000 - 1)/3 and knew that because the answer was not a whole number, then it was not possible.

In the end I love this task because it builds up to abstraction after being grounded in very concrete representations. And by having students teach, while I facilitate and push them to go deeper with their explanations, they have ownership over the task and retention is increased. I follow this lesson up with a class of problem solving where students are presented with various patterns and have to work out similar problems, or I create simplistic models of diseases spreading and ask more advanced kids to make predictions about numbers of infections. In this way, the same reasoning is required of all students, but they can be pushed at appropriate levels.

While teaching high school, I found that many students could calculate slope because they remembered the equation. Teaching juniors and seniors in Pre-Calculus and AP Calculus AB, I found that every time a student made an obvious mistake when calculating slope, it was rooted in their lack of understanding of what slope is (and as a Calculus teacher, this was a horrifying discovery). Now that I teach middle schoolers, I have the ability to try to develop that understanding so that Calculus teachers of the future do not have the same stomach sinking experience. This short story is all to set up the activity I recently used with my students to develop an understanding of slope (a word I had not yet presented them with) as the constant difference. Who says students aren't engaged by exploring patters rather than "real-world applications" (buzz word for non-math teachers or those who do not see the beauty of math for math's sake)? Below is the sheet I hand out at the start of the lesson.

The Google Doc for this activity can be viewed

*and of course, feel free to use it.***here**After having kids read through it, I

__give them baggies of toothpicks they can use to construct the design. Using my document camera (I have a fairly cheap__**then***that works great with my Mac once you download the free software, thought the PC version is not as good), I then build the first two designs and ask kids how many toothpicks are in each. Kids respond 4 and 7, then I have them finish counting designs three and four, once again taking their answers of 10 and 13. At this point, I give them 8 minutes of independent time to tackle the questions where no talking is allowed. I take this time to do some formative assessment, looking at student reasoning. Kids are allowed to build with toothpicks, count in their head, or draw out the designs, giving multiple access points, which is important for multiple learning modalities.***USB camera**After the 8 minutes is up, kids are then allowed to talk with their neighbors (they sit at tables of three to four students). The beauty of this activity is they become so engaged in figuring out the patterns that almost no redirection is required and I love the student reasoning.

For question 1, kids simply count three more segments/toothpicks and come up with 16.

For question 2, a few strategies are common:

- Some kids continue to count/build by threes until they come up with the solution of 31.
- Others simply double their solution to question 1 and come up with 32 (an expected misunderstanding).
- Finally, some kids pick up on the fact that each design increases by three each time, but the extra toothpick at the beginning has to be accounted for and they also come up with 31.

- First, "Mr. D, you can't expect me to count all the way up to the 100th one?!?!" To which I reply, "can you find a faster way then?"
- Others multiply their answer to question 2 by 10 or question 1 by 20 to get 320 (once again, a predicted misunderstanding). Another common error is 31x10 = 310.
- Finally, the kids who understand the pattern can be very clever. Some will simply do 3x100 + 1 = 301. Others will cleverly use their answer to question 2 by doing 31x10 = 310, then subtract by 9 because they know that multiplying by 10 means they have counted that initial toothpick too many times.

At this time, I then give students an additional 3 minutes together to try to come up with a rule or equation, which could be as simple as you multiply the pattern number by 3 to get how many added toothpicks there are, then add 1 to include the extra toothpick at the beginning. Others will come up with something more complex such as 3 x P + 1 = N, where P = the pattern number and N = the number of toothpicks. Once again, when time is up, I have students present their solutions and we run a couple of samples to make sure they produce the correct values for design 8 for example, which is easily verifiable. This gives students a chance to use their reason to come up with something more abstract.

Finally, there is the challenge problem, which is much more accessible than it looks at first sight. Some of my students that struggle with mechanics were very clever and said no, 5000 could not be the number of toothpicks because they guess and checked with multiplication to find that 4999 and 5002 would work, which means 5000 wasn't possible. Students with a deeper understanding of mechanics simply did (5000 - 1)/3 and knew that because the answer was not a whole number, then it was not possible.

In the end I love this task because it builds up to abstraction after being grounded in very concrete representations. And by having students teach, while I facilitate and push them to go deeper with their explanations, they have ownership over the task and retention is increased. I follow this lesson up with a class of problem solving where students are presented with various patterns and have to work out similar problems, or I create simplistic models of diseases spreading and ask more advanced kids to make predictions about numbers of infections. In this way, the same reasoning is required of all students, but they can be pushed at appropriate levels.

## Tuesday, November 17, 2015

### Row Game With Proportions

Thanks to Kate Nowak, who has created numerous row games. Using her template, I created my own row game for proportions:

The beauty of this activity is it's difficult for one partner to do all of the work, so accountability is built in. It's also easier for students to act as teachers because they can share their process for solving without actually doing their partner's problem for them. Students remained engaged and got immediate feedback on their answers since their partner was supposed to reach the same conclusion. If answers disagreed, it meant the hunt was on for who made the mistake.

Also, I had to admit I made a mistake on row 4, which is always fun for my students. I changed Column B to 63/7 = 81/y. Problem solved.

This task was designed as a refresher to set my students up for some Three-Act Math problems, including the Super Bear problem, which are the result of Dan Meyer's fine work.

The beauty of this activity is it's difficult for one partner to do all of the work, so accountability is built in. It's also easier for students to act as teachers because they can share their process for solving without actually doing their partner's problem for them. Students remained engaged and got immediate feedback on their answers since their partner was supposed to reach the same conclusion. If answers disagreed, it meant the hunt was on for who made the mistake.

Also, I had to admit I made a mistake on row 4, which is always fun for my students. I changed Column B to 63/7 = 81/y. Problem solved.

This task was designed as a refresher to set my students up for some Three-Act Math problems, including the Super Bear problem, which are the result of Dan Meyer's fine work.

## Thursday, August 6, 2015

### Making Room for Ethics in Math Class

I find there is a tendency for people to present math as a value-free or unbiased subject, meaning there is no place for an ethics discussion in mathematics. When discussing the incorporation of social justice into math class, the conversation inevitably turns to examples like using statistics to expose sexism in the workplace. However, especially when teaching in low-income communities, both rural and urban, we must recognize that math has been used to ensure poverty continues to exist. Credit card companies, predatory lenders, and insurance companies have all used math to make more money, while contributing to the oppression of others. Math is used to assess the risk of insuring someone, therefore determining who can afford insurance and who can’t. Thus, the argument can be made that math is neutral in some capacities, but as soon as applications are designed, math is anything but neutral. Mathematicians have contributed to the field of economics through optimization, but that means using math to make decisions that have very real consequences for regular people. But sitting in an office making these decisions means not having to see the consequences of one’s work. Ignorance does not mean excusal from responsibility. In the end, I am saying that ethics has a place in the math curriculum and there should be discussion as to how math has consequences in the lives of our students. Instead of prepping students for industry by giving them applications without context, we should be teaching kids how to be creative problem solvers while understanding their decisions can dramatically impact the lives of others whether they immediately recognize this fact or not. I am hoping to help develop citizens of this world, not drones who can be used by industry without understanding the role of work in society. Corporations, governments, nonprofits, and any other institution all have missions, therefore work for any organization means contributing to the leader’s vision. We must therefore help students understand that they need to be sure their values align with the organization they choose to join. As I am writing this, I can only think of the scene from Goodwill Hunting where Matt Damon is in an interview with the National Security Agency (NSA). Check it out below:

This is a clear example of someone who is incredibly gifted in mathematics, yet understands his potential role in an ever more complex society. So as we continue to do as little damage as possible to this next generation, let us think long and hard about what the goals of public education are and if these goals really meet the hopes of students and educators and the needs of the planet. Personally, I see content standards and mathematical practice standards, but I do not see any discussion of ethics within these standards. I see industry working to define the desired outcomes of public education as the human aspects of education are increasingly marginalized. STEM and the humanities are not mutually exclusive, but need each other to produce humans capable of positively shaping the future. I will now step off my soap box. Thanks for reading and I am very open to comments.

This is a clear example of someone who is incredibly gifted in mathematics, yet understands his potential role in an ever more complex society. So as we continue to do as little damage as possible to this next generation, let us think long and hard about what the goals of public education are and if these goals really meet the hopes of students and educators and the needs of the planet. Personally, I see content standards and mathematical practice standards, but I do not see any discussion of ethics within these standards. I see industry working to define the desired outcomes of public education as the human aspects of education are increasingly marginalized. STEM and the humanities are not mutually exclusive, but need each other to produce humans capable of positively shaping the future. I will now step off my soap box. Thanks for reading and I am very open to comments.

## Saturday, July 26, 2014

### Organizing Blogs

This post is short and sweet. If you are trying to follow multiple blogs and struggling to keep up or organize posts you like, check out the Feedly app. Any time somebody you follow adds a new post, Feedly will put it on your homepage so you never miss out. You can also organize and share posts in a way that is useful to you. Give it a try!

### Joining the SBG Cult

For the past two years I have used a traditional grading system and wondered why grades did not motivate many students. Once I was exposed to standards-based grading (SBG), the answer seemed so obvious. There is no incentive to master skills in a traditional grading system when there is no opportunity for redemption and teachers' grade books only make sense to the individual teachers. To motivate students by showing them what they are great at and where they can improve with constant opportunities to demonstrate mastery, I hope to reinvest students in their education. I will also incorporate Dan Meyer's Three-Act Math Tasks to show students how beautiful math can be in order to challenge their assumptions that math is simply rote memorization and you are good at it or you aren't. Check out Dan's Blog, Dy/dan to the right and the link to his math tasks for some great lesson ideas.

Back to SBG, this is a big shout out to Sam Shah (Continuous Everywhere but Differentiable Nowhere) for his virtual filing cabinet. This cabinet contains countless resources for engaging math instruction in addition to general classroom resources. Specifically, his link to the SBG Beginners wiki has been helpful as I think about the implementation of SBG into my own classroom.

Back to SBG, this is a big shout out to Sam Shah (Continuous Everywhere but Differentiable Nowhere) for his virtual filing cabinet. This cabinet contains countless resources for engaging math instruction in addition to general classroom resources. Specifically, his link to the SBG Beginners wiki has been helpful as I think about the implementation of SBG into my own classroom.

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