Math with Mike

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!

Mike

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.

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,

Mike

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:

Delicious.com - 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 delicious.com 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?

Scratch.com / 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.

-Mike