Friday, September 25, 2015

Common Core - #2 of Studying Structure - More on the Practices

In the first post of my Common Core series, I discussed the importance of really looking at the Standards of Mathematical Practices (SMP), because the title alone is not enough. You cannot assume that each title of the 8 SMPs gives enough information to truly know what students should be saying and doing to enhance their mathematical understanding. The descriptive narratives for each practice are necessary reading to clearly make good instructional decisions that will truly support student understanding of mathematics

My last post focused on SMP #4, Model with Mathematics. In this post, I'd like to choose another of the SMPs, and be a little more explicit in breaking down the structure to show how both the title and the narrative help inform instruction. Then, hopefully, you can do your own investigation of the remaining SMPs.  This is a great exercise to do with collaborative groups of teachers - maybe your next team/faculty meeting?! All of this is from work I have done being part of the International Fellows working with The Charles A. Dana Center, UT Austin.

I am choosing SMP #1, Make Sense of Problems and Persevere In Solving Them. Every teachers dream that students approach problems and understand them and not give up, right?!  But what does that look like in class? How do teachers support students so that they reach this point of being able to take a problem, make sense of it, and continue to try to solve it even when the going gets tough?  Here's where the narrative of the SMP comes into play - title is not enough, remember?!!

Narrative of SMP #1Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
That's a lot of stuff going on there!  I have underlined some key components related to STUDENT actions.  Notice the narrative is about the students - what they should be doing, saying, considering. Taking this to the next step, reflect on students actions and write down (T-chart works) what students might be saying and doing if they are demonstrating this practice. Create a list, like the one below (this is NOT an exhaustive list, just some examples.  See posted pictures for more suggestions).

Students should saying/doing
Teacher should be saying/doing
-asking questions to their peers

-highlight key words or numbers to understand the problem

-making decisions about a plan to solve

-trying out solutions and CHANGING their plan if it is not working

-continually ask themselves “does this make sense”

-drawing pictures, getting tools (i.e. rulers, calculators, manipulatives) to help make sense and plan a solution

-checking their solution using a different method

Once you've generated several things students are saying and doing, THEN look at teachers actions. What you want students to do informs the actions of the teacher, so this step is done after getting a clearer understanding of the student actions.  Again, here's a list of teachers actions/behaviors, but it is NOT an exhaustive list.

Students should saying/doing
Teacher should be saying/doing
-asking questions to their peers
-provide a classroom environment that encourages conversation and student dialogue
-highlight key words or numbers to understand the problem
 -teach strategies to help students identify key words, numbers and questions in problems
-making decisions about a plan to solve
 -model and demonstrate various problem solving strategies
-trying out solutions and CHANGING their plan if it is not working
 -Allow for productive struggle and act as a facilitator
-continually ask themselves “does this make sense”
 -Asking students questions like "is there another way?" "Can you explain how you did this?"
-drawing pictures, getting tools (i.e. rulers, calculators, manipulatives) to help make sense and plan a solution
 -Provide multiple resources 
-checking their solution using a different method
 -Encourage collaboration and sharing of solutions; require students to explain their thinking

This process, of really studying the structure of the SMPs, the title AND the narrative, and then focusing on student actions first allows for a clearer path of what teacher actions are necessary to help students achieve the SMP skills.

I encourage you to do this yourself, but better yet, do some collaborative learning with other teachers around the structure of the SMPs and you will get a better understanding of what students need in order to become the mathematicians and problem solvers that the Common Core Standards strive to achieve.

Next in my continuing series will be looking at the structure of the content standards - a much more complex structure than the SMPs.

Monday, September 21, 2015

Common Core - Studying Structure & A Personal Fight Against Politicking

I am tired of the #CommonCore bashing, so I am just going to start writing some posts about things I have learned that have helped me realize how powerful and SUPPORTIVE of student learning these standards and practices can be.

Don't get me wrong - I never needed to be convinced that the Common Core State Standards for Math (CCSSM) were a good thing for math education.  I cheered their adoption because to me, these standards supported what I had been practicing and preaching, both as a teacher and a professional development provider. But - even though I have spent the last several years studying and coaching others in implementing the Common Core in a variety of ways, I just recently learned more to solidify, for me, their power. It was sort of an ah-ha moment: if I, who consider myself very familiar with the CCSSM, just learned something new, imagine how those who have spent little, if any, time with these standards (i.e. parents, news reporters, politicians) could benefit from looking a little more deliberately at the standards instead of relying on the rhetoric or a cursory glance to make their decisions?!

I've been working with The Charles A. Dana Center at UT Austin supporting math teacher training. As part of this endeavor, I participated and delivered some training that focused on looking at and understanding the structure of the CCSSM and alignment of the standards (within grade and vertical). My huge take-away was how important this process is for any educator who is implementing the CCSSM. This in depth look is something I think is missing in much of the PD that surrounds these standards, which might explain a lot of the confusion about how to teach and help students understand the standards and practices, which in turn leads to the confusion among parents, the media and politicians.

My plan here is to write a couple posts about some of the ways looking deeply at the standards can bring clarity and understanding to how they work. More importantly, it emphasizes the importance of seeing the Math Content Standards and the Standards of Mathematical Practice as equally important in helping students learn, understand, and apply mathematics. You cannot look at isolated standards or practices or problems and hope to know how the standards and practices work to BUILD mathematical competence over time - (which, unfortunately, seems to be what is happening - looking at isolated components versus the overall picture).

This post is going to focus on the structure of the Standards of Mathematical Practice. Pretty simple really - there are 8 practices, and their structure contains the title and then the narrative description. The problem here being the title  is what most educators focus on, and the title DOES NOT give you enough information to really know what students need to do and what educators need to do to support students.  You really need to read the description to get a better sense of what students, as mathematicians SHOULD BE SAYING AND DOING, which then informs what resources the teacher needs to provide and what the teacher should be saying and doing.

I am guilty of assuming the title of the practice was enough. I will use Standard of Mathematical Practice #4 as my example. The title of this practice states that students should be able to:
Model with mathematics.
Seems pretty simple. On its surface it seems to imply using models - i.e. manipulatives or physical objects to help students understand math - which is wrong, or rather very incomplete thinking. Here's the narrative description of this practice:
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
I've highlighted just a few key phrases that show what this standard really means is students can USE mathematics to model and explain problems, which is NOT the same as using manipulatives to learn mathematics, which in my experience is what many educators think this practice means. What the practice says is students can approach a real-world problem and APPLY and use some type of mathematical model (i.e. equation, function, geometric figure) as a way to explain the problem and and come up with a solution. If you continue reading the narrative, you will see that it references students ability to make assumptions, approximate solutions, interpret their results and create mathematical models that help them solve the problem.  It's a lot more descriptive than just the titl, and something all educators need study and understand so that they are not misinterpreting what the titles alone might be saying.

The pictures to the right are of some of the work I did with teachers that shows their end-result of studying structure of the practices. These t-charts show an understanding of how the narrative informs both students and teacher actions (sorry, they are a bit out of focus!)

The point I am making here is that the title is not enough - the narrative is more important than the title because it gives expectations for what students should be saying and doing, which in turn informs what the teacher should be saying and doing. The structure of the practices, if looked at as a whole, provides information to help teachers create a classroom that supports student learning - you need to go beyond the title.

Friday, September 11, 2015

Global Math Challenge September 27 - CCSS anyone?

In perusing the internet for edtech and math news of interest, I came across Sony Global Education Inc.'s Global Math Challenge.  On September 27 you can get online, take a brainteaser math test, competing against mathematicians (young and old) all over the world.  Wow!

According the the description, "GMC is full of beautiful illustrations, helping to put math puzzles into a real-world context". This is true if you look at the sample.  You can sign up for the free-Trial Plan, where you will just get your score, or the Standard Plan, in which you actually get to review your results and compare your results with all the other competitors. There are some sample questions for you to 'try it out' before signing up.  Which of course I did....

I was much relieved to find I could actually answer them, though I did get #3 wrong because I misinterpreted the question (I assumed we started from 2, vs. using the given five already assembled). What I loved was the logic needed to get to a solution.  I had to analyze, make a guess, justify, readjust - I used pictures, numbers - lots of ways to make sense of the problem and looking for patterns....i.e the Common Core!

My thoughts are teachers might try some of these problems with their students - it really gets the conversation going.  And - encourage them to sign up for the Global Math Challenge. What a great opportunity for doing challenging logic math problems and then sharing the results after and comparing to others around the world - a way to connect globally and mathematically!

Wednesday, September 9, 2015

Technology for Learning - We Might Want to Ask the Students

I have been doing some independent research on technology use in classrooms.  With all the news about 1:1 classrooms and students using technology apps to learn, I was curious about what is actually happening.  Some of my findings have made me really question whether some of our approaches - i.e. 1:1, computers vs. tablets, are really supporting the way students are learning with technology.

I found this interesting report from Project Tomorrow called "From Chalkboards to Tablets: The Emergence of the K-12 Digital Learner"  with some fascinating statistics regarding the technology students are using to help them learn.  You should download the full report (available as a PDF), but I will share a few of the findings here.

One thing that stood out for me is the prevalence of smartphone use by students, especially as they get into the upper grades.  Table 1 below shows the use of both personal and school provided devices.  Very apparent that 1:1 and school provided technology is NOT as dominant as news reports would make us believe.

Smartphones are definitely the preferred student  technology choice. I should't be surprised, since my two daughters seemed to do most of their homework using their phones. I asked my daughters why the phone, and they said it's just easier, quicker, and you can get everything you need (i.e. videos, research, answers/help). The tablet/laptops are only needed if it is necessary to write something that must be turned in or create a project. Though they did say they used the calculator for math, mainly because it was the only thing allowed in the math class and on tests.

This of course begs the question, what are students doing with these smartphones and other devices to support their learning? I know my own daughter was a huge fan of the Khan Academy for helping her learn calculus. The findings from the Project Tomorrow report showed the following results, most of which are not surprising, except perhaps the texting with teachers (at least for me). The video statistic definitely was supported by what I saw my own daughters doing.  I was a bit surprised to not see researching information via the internet.

There are several other charts/tables in the report, but the last one that really struck me because of the disparity between students and teachers, was the chart depicting assigning/using the internet for homework/classwork. As you can see, using the internet definitely increases as students age, with daily/once a week use by students in grade 9-12 at 61% or higher, and yet teachers assigning homework that utilizes the internet decreases.

I've only pulled out a few tables/charts from this repor - the ones that struck me. Students are clearly using the internet, using their smartphones, and making choices about technology that helps them learn, regardless of what the schools provide or don't provide. It seems, especially from this last chart, that we are not supporting how students use technology to learn and therefore missing a huge opportunity to create learning experiences that fit students use of technology. It seems we need to talk to students about what they need, rather than just buying technology we think is going to support their learning. Right now, it seems there is a clear mismatch.

Thursday, September 3, 2015

QR Codes: Implications in the Classroom

I hadn't really thought about QR code use in educational settings until my recent exposure and personal experience with a new app. EDU +, released by Casio to support their Classwiz scientific calculator. It got me interested in how QR codes are being utilized in classroom settings.

In my research I have found some interesting suggestions for using QR codes in education. Steven.Anderson has some great postings and links about QR coding ,which led me to several articles and resources. One in particular, Tom Barrett's slide show on "40 Interesting Ways to use QR Codes in the Classroom" gave me insight into the possibilities of QR codes for teaching and learning. I particularly liked idea #7 of adding QR codes to word documents for students to check their answers. He has 40, so it's worth a look!

Basically, QR codes are information - up to 4000 characters. QR codes can be used to link students to more information (reviews, pictures, graphs). Embedding QR codes into web pages, documents, and other resources allows students to access additional facts about what they are seeing/reading.  For example, placing QR codes on books in the library can allow students to link to reviews of books. There are multiple uses simply by scanning a QR code. I think of it as a short cut to more information, allowing students to quickly expand on their learning.

I can give an example of this 'more information'  from my brief, personal experience last week in Japan at the Global Teachers Meeting at Casio R&D working with the Classwiz scientific calculator. This calculator has the ability to create a QR code directly within the calculator function (example, a QR code of a table of values). Using the EDU+ app from either an iPhone or iPad, the QR code is scanned directly from the calculator and creates the link, which can then be embedded on webpages, documents, emails and other places for students to access. Clicking this link brings up a graph of the data, allowing students to see and change graphical display of their data. (Here is an example of the QR code link I created while I was hands-on learning with the calculator and my iPhone).
I realize this begs the question, why not just use a graphing calculator? I think accessibility to graphing calculators is one reason - scientific calculators are more prevalent in elementary and middle grades, even lower grades in high school.  I think another reason  is diversity - a QR code can be used in many places so students can visually see data graphically, whether they have the calculator or not, The teacher can use QR codes to show things on tests, websites, emails, allowing access to information/pictures that  may be limited normally. Lots of possibilities.

QR Codes on books
My example is just one specific use of QR codes, in a math setting. There are many examples of using QR codes for different subjects and purposes,  such as enhancing class projects or communicating with parents. If you are interested in exploring QR codes further, here are some suggested links:
LiveBinder/Steve.Anderson , 12 Ideas for Teaching with QR Codes, QR Codes in the Classroom (this one by Kathy Schrock is great - lists QR readers, and has tons of links), 50 QR Codes for the Classroom,

For me, what I have gleaned from my hands-on experience and research on QR codes, is you do not have to be limited by your space or devices. QR codes provide an ability to post information in multiple ways and places, allowing additional information to reach students in a variety of ways. Obviously, there is the limitation of needing a device that can read QR codes, but with the availability and prevalence of smartphones and apps, I think QR codes might show up much more frequently in educational settings.