Inspiration for a lifetime

“So what does matter? How will the value of your days be measured?

What will matter is not what you bought, but what you built.

Not what you got, but what you gave.

What will matter is not your success, but your significance.

What will matter is not what you learned, but what you taught.

What will matter is every act of integrity, compassion, courage or sacrifice that enriched, empowered, or encouraged to emulate your example.

What will matter is not your competence, but your character.

What will matter is not how many people you knew, but how many will feel a lasting lost when you’re gone.

What will matter is not your memories, but the memories in those that love you.

A life lived that matters, is not of circumstance, but of choice.”


-A Beautiful Truth 


So what matters? Everything!

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An Honest Debate about Standardized Tests

John Lockhart’s A mathematician’s lament is an inspiring piece of work. Wow. everyone should read it. Everyone. Please read this (it’s on the web) and continue to read.

Let’s assume that we all agree that he is right. Further, let’s assume that we all want to create a math education environment fitting to Lockhart’s ideas…What do we do?!

Those two huge assumptions are so unrealistic that it would be time wasted to try to answer that question. A better question is, for those who buy into Lockhart’s ideas, how do we create math learning like Lockhart envisions with our current system? What can we actually do to truly change the system? 

Well, it I think I know what the first major step would be and this is truly radical: Remove all standardized testing.

Now, I’m not saying that my idea is novel or that this idea obviously hasn’t been discussed before. However, the idea of removing standardized tests is truly radical.

I would love to see/have an honest debate about this. Why do we need these tests? What would schooling look like without them? What forms of informal assessment should we adopt? Can we create a system of accountability without testing?

I take the position that in today’s socially networked and open access world, that we (educators/teachers/administrators) can truly communicate and cooperate with manageable expectations about how our students learn mathematics without any formal testing in schools. Period.

Well, I could keep going, but hopefully this sparks some conversations (either online or in person). 

To change!


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My First Proposal

Well, I am in the middle of my first year and feel good about submitting a proposal for a poster presentation for the 2013 PME-NA Annual Conference in Chicago!

My first email confirmation of proposal submitted.

My first email confirmation of proposal submitted.

Anybody know if they’re going? I’m not sure if I’m even going, but if I do I’ll be very happy. It’ll be my first time ever presenting on research that I have done. Although the quality of my work will be in question, at least I’m swingin’ right? Also, I think a lot more work can be done with what I’m presenting. My research question is: How do students’ experience and skill-level with Geometer Sketchpad influence their visual conceptualizations of math topics?

The work basically advocates for more software instruction with dynamic geometry as it affords students a different experience from the usual reification of math topics.

Well, that’s it for now.



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Minus Sign and Negative Sign in Harmony

I’m being overly ideal, but in an effort to unify the cultural context by which our society views the minus sign, I propose a national initiative to have all creators of math symbols (publishers, teachers, etc.) implement a distinction between the minus sign and its close relative, the negative sign. The minus sign is commonly known, but the negative sign isn’t far behind in terms of exposure in math classes around the country. The negative sign is a smaller minus sign that is positioned higher and close to an integer indicating a negative value.

To be clear, never should we see this,

Clumsy subtraction symbolsbut rather,

Ideal subtraction sentence

As difficult or impossible it might be to mandate a distinction between the negative sign from the minus sign, if ever there were a moment in time to call for some unified math notation, it would be in today’s socially networked world.

In terms of culture unification, the possibility of developing sociomathematical norms within a classroom has been documented. We currently have such norms from a societal perspective as well. For example, when a number does not have a negative sign, we automatically assume it is positive. That is a societal sociomathematical norm. So, we can potentially add to the current set of national notational norms by establishing a readily acceptable way to format negative numbers.

Additionally, within the majority of today’s elementary schools, we deliver a fairly linear approach to teaching the two operations of addition and subtraction, so within that framework, we should recommend all schools to implement negative numbers with a symbol separate from subtraction. Why would we allow negative numbers to be mis-represented by a symbol that already means something else to children? I think our children deserve better.

As a more important reason for such an initiative, students have been shown to think logically about operating on negative numbers via addition and subtraction and this notation would provide students the affordance to think in this manner. By explicitly distinguishing the minus sign from the negative sign, understanding the two signs will help students logically construct the movement of integers Fostering such a logic within the addition and subtraction of integers provides foundational thinking for children learning formal mathematics.

From a more practical viewpoint, one could argue that by isolating the binary operative meaning for the minus sign from its other meanings could prove beneficial. It is documented that the minus sign has other meanings: a sign for negative numbers and the unary operator as opposite of. We should apply the unary operation to the negative number symbol and keep the minus sign solely for what it was originally meant to be: a binary operation.

As for criticism about this initiative, one could argue that students will be exposed to the minus sign as representing a negative number and won’t be prepared to handle such a situation. This can be easily remedied. For example, students can spend some time in class to learn that whenever they see two minus signs together followed by a number, the second sign can always be changed or viewed as the negative symbol for that number. By distinguishing a negative symbol from the minus sign, students will already have an image in their minds to alter the meaning of the clumsy –(–5) or – –5.

Overall, as a researcher, I am interested in improving math learning for students. Symbols are a crucial component to the epistemology of mathematics, and by avoiding unnecessary pitfalls, like multiple meanings for one symbol, we allow our children the affordance to develop reasoning skills within reason. I would interested in modifying some current published studies that provide students with number sentences that apply the minus sign in the negative context by altering those problems to include the two different symbols discussed here. Will students perform better on the same questions with these two different symbols? Will they exhibit a higher percentage of logical reasoning during clinical interviews? Also, I would also be inclined to develop a teaching experiment that provides such a model for students to see if keeping the minus sign solely as a binary operator facilitates operating on integers.

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Harlem Shake in a Science Class

Many thanks to Mr. Black’s science class for gifting me with a great surprise. Their test average was over 90%, so they were rewarded with the opportunity to do a harlem shake while I was recording the classroom.

Not only is it great that Mr. Black has allowed me to video-record his classroom, his students are awesome enough to put this together:


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Relational Thinking for Algebraic Reasoning

In the article,

Jacobs, V.R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children’s algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38(3), 258-288.

the authors led a professional development that had an algebraic focus. Rather than educate teachers about higher-level math or provide new lessons that might be a refresher for the same content covered, this study basically changed how teachers should view and teach the current math curriculum. They did this by embedding algebraic reasoning within the arithmetic, and conceptualized algebraic thinking as a form of reasoning and justification. More specifically, they introduced the study of relational thinking in 3 ways:

  • Applying fundamental properties of numbers
  • Using number relations for calculations
  • Modifying how students view the equal sign.

In terms of how students see the equal sign, most think of the sign as an operator for the numbers that come before it. For example,

2+5 =

makes students think that the equal makes 7. So, this professional development focused on viewing the equation sign as a relational object (which really an equal is stating two expressions are equivalent). They accomplished this by providing teachers (and ultimately students) with equations like (p. 262)

57+36= ∎+34

which changes how students view the equal sign. With the above symbols, student would reasonably see that the equal sign indicates that both sides are equal and that the black box has a number in it that makes the equation true. It’s easy to see the value in this perspective in terms of its application to learning algebra. By structuring number sentences like this, the teacher educators were able to introduce number relations that ultimately could be leveraged in the classroom. The above example can ultimately allow students to see that 36 is 2 more than 34, and in order for the equation to be valid, the number in the box must be 2 more than 57. With minor calculations, this strategy for solving the question is more efficient than the standard way (combining terms on the left and then subtracting 34). It’s important to note, as Jacobs, et al. point out in the article, thinking relationally “is different from applying a collection of tricks or memorizing a set of mathematical properties” (p. 263). Relational thinking is a form of reasoning that exploits the fundamental properties of numbers, and the authors make this explicit in the professional development as well.

The teacher educators also worked with the teachers to illustrate how students can understand explicitly the foundational principles of numbers (i.e. commutative, associative, and distributive properties). Research has shown that students are able to understand and explore these generalities. As the authors learned from their previous development work, they “found that a focus on students’ mathematical thinking provided opportunities for teacher learning that led to changes in classroom practice and improvement in student achievement” (Jacobs, et al, p. 264). Ultimately, it was the goal of the professional development to promote such thinking in the teachers through relational thinking.

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A new semester

Well, I’m hoping I can be more active with my blog. I have had some illusions about what it means for me to be a blogger and consequently I’ve stifled creativity and production in the process. My goal is to be myself, say what it is I want to say without worrying if what I’m saying is something worth reading. I am going to speak my mind, share my thoughts and provide information that may be or may not be valuable to a particular reader.

So, I have finished my first semester as a phd student, and I feel good and bad about where I stand. For some reason, knowing that Ill be a student for the next 3 years makes the process feel very slow, but on a day-to-day basis feels very fast. I feel bad because the future feels unstructured and I’m not used to that, but I feel good because I am a part of a program that will support me towards success. I am grateful for that.

My goals are to continue to connect with my surrounding community here in San Diego and meet teachers, educators, and students. My hope is to learn more about how people are feeling about technology in the classroom. I think this is a good first step: how do people like tablets/technology in the classroom?

From there, I hope to dive much deeper into the various components of teaching with the perspective of 1:1 or BYOD classroom environment. There are a lot resources for teachers in learning how to implement pedagogy within this environment, but what does it mean to research it?

My goal is to figure that out.

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Dinner with Dr. Micki Chi

Italian dinner in La Jolla. With (from left) C. David, John, Dr. Chi, Me, and Jamie. Good times!

A couple weeks ago I had the opportunity to share a dinner with my fellow graduate students and Dr. Micki Chi. We had a great conversation over dinner and addressed several topics relating to math and science education.

The most interesting portion of our conversation was about bridging the gap between education research and the implications for teaching from said research. Dr. Chi is developing a project that looks to address this issue, but from our conversation it sounds like this topic is a legitimate concern that requires more attention. Here are a few thoughts/suggestions that came up:

▪   Professional Development for teachers could include research components that help them build their craft. One could argue the best researchers are right in the classroom.

▪   How much research is being done on the implications for teaching that education research brings? A lot of times articles and studies mention implications towards the end of their research papers, but what is being done to actually test out their theories?

▪   Can we develop an institution that works on bridging this gap?

Well, plenty to write and talk about, but hopefully this is a conversation that can continue beyond this post.

To Progress,


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Summary of Technology tools for Teachers

The last few months, I have embarked on learning the technology tools that teachers are using in the classroom, or at least are being taught in professional development and credentialing programs. This post is a place to explore the list that I have accumulated from my experiences. I participated in a Stanford Workshop that took place over the summer that was conducted by Dan Meyers, and I enrolled in a course at UCSD that consisted of pre-service elementary teachers taught by Dr. Chris Halters.

Now, I’m not sure how useful this is going to be for anybody, because teachers are already diving into tools and even being forced to take on new tools. From my experience teaching, there’s plenty to do with the tools already in place. But, I do think that productivity is the name of the game, and if you take your profession seriously, these tools can help in providing support in improving skills, staying informed, and developing professionally as an educator/researcher.

With every item, I will also take on the task of including potential research questions and/or implications for teaching that may provoke some level of insight into how one may analyze these tools within the context of math education and reform. Your input and/or comments are highly encouraged. Please feel free to add to the list.

Without further ado and in no particular order: – Are you an active peruser of the web? Are you bookmarking every link you like? Well, if you would like a more effective way to classify and search your sites, then look no further. This website is great for managing your sites, but its “x-factor” is the social media component an develop.  Register and check it out.

How can we observe teacher practices from a research standpoint? Aside from all the current work relating to learning, can we improve on what teachers do besides the curriculum?

I am always interested in how technology can actually add value for teachers, and I would seriously consider this tool for professional development. With all the information on the web, it would be wise to follow blogs and sites that are modifying and adding new lessons. We should educate teachers how to utilize this source for doing professional development on their terms. With an appropriate system in place, it may be valuable to have teachers stay active in adding/modifying their account and providing some form of evidence of executing new ideas from their account in order to show some form of professional growth.

Google Reader/Drive – This one is self-explanatory, but if you’re not sure what this is, well, google it. The suggestion I make with applies also to Google Reader. Again, having teachers staying informed about their profession is vital for furthering their teaching skills. I treat my role as a teacher as a professional one. As such, it is my responsibility to grow as an educator, and there should be a system in place that readily allows teacher to participate in this important behavior. With technology, it’s never been this easy.

Google Drive provides essentially everything you would need for teaching math concepts that Microsoft office has to offer, but there’s a plethora of ways of utilizing The Drive. How does the social component change learning? Google Forms provides a easy way to poll students and gather information. How can we research this? Are there ways students are learning by sharing documents/spreadsheets? How does this affect socio-cultural learning?

Edmodo – Teachers should definitely consider managing their classroom with edmodo. Again, social media is changing the classroom. How can we design instruction fully optimize these tech tools? How is this affecting learning? Are we increasing understanding? / Alice – Students need to learn computer programming. What’s interesting is how we have introduced new Common Core Standards, and nothing is mentioned for computer science. We’re falling behind yet again on what we need students to learn. But, if you really believe students should learn programming, then here are two great sites for doing that. What if we taught teachers how to develop programs? Is there value there for teachers too? I think we’ll find that out as new programs and languages are made available for people without any programming experience. Either way, learning logic and programming can be beneficial for anybody.

Well, there are a ton of more sites and softwares we can talk about, but I think some interesting questions reside with how teachers learn and development their skills. One major concern is time. Teachers need more time to develop as a professional in the field of education. One or two hour prep periods is not enough and having PD once a month is also a failure. Secondly, how PD is actually implemented is another issue altogether.

Would love to hear your thoughts.


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EDS 203 – Day 1

Today I went to my first class at UCSD. The majority of the students in this class are preparing to become elementary teachers, but the purpose of the course is to learn ways to integrate technology in the classroom. Since I’m in the beginning of my doctorate program, taking this class is a great way to jump start the year. My head will be in the game with what is happening out there and how teachers are using the current technology. Knowing how to actually use some of the programs is a great way to start advancing my skills.

Professional development for teachers seems to be happening everywhere with the technology theme ever-growing. As our baby boomers retire and more new teachers enter the workforce, the attractiveness of knowing technology tools for teaching increases. I wonder how much research is being done on the actual effectiveness of all the tools? Are teachers actually using them? From personal experience, without real hands-on training and continued support, I have easily discarded tools I learned during a workshop. I just didn’t have the time to really work on them. I was too busy grading and lesson planning. But, this class is showing me that pre-service teachers are getting the necessary training to truly integrate new technology.

I am creating a website for the class that will include some of my past curriculum. Thank you EDS 203 for jump starting my first website. Hopefully it will be useful for teachers. I will be sure use this website to post material moving forward. I’ll use it to also post materials from my graduate program. It will continually be a work in progress.

It’s here.

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