Wednesday, December 31, 2014

A Reflection

It's the time of year when many of us do some reflecting of one sort or another. For many, the winter break allows us a time to catch our breath and think about the changes we would like to make for the second part of the school year. Although my job as a secondary math specialists does not provide an official break from the job, it is a bit of a chance to prepare for the training that will be upcoming and to plan for some improvements.

The information that is gathered from session evaluations is helpful for looking critically at what I do and how effective it is in the eye of the teachers. One area of concern for me recently has been my ability to convey the connection between the various workshops that I have led and the AZCCRS (that is Arizona College and Career Ready Standards for those outside of AZ). 

In short, this post is more for me than any audience but perhaps it will inspire at least a few people to set some goals based on data. 

The last two days have been spent preparing for an upcoming session focused on increasing student engagement at a higher cognitive demand with a colleague. At the beginning of the process we wrote the goals for the session and discussed the idea that we wanted to stress the connection between the strategies we will be modeling and the AZCCRS.

Having the data has been helpful in helping me set a personal goal. However, the most important factor in me taking the first step toward reaching this goal is the collaborative process that has occurred over the last two days. Having another person to reflect and brainstorm with has been invaluable. Soon enough I will be able to see if this first step has been successful.

Happy new year!

Monday, December 15, 2014

Biggest Bang for your PD Buck

At our most recent staff meeting we were presented with some information regarding "effects" and their impact on student learning. The information was based on John Hattie's "Visible Learning for Teachers." This report is a synthesis of 1000 + meta-analyses which included 240 + million students and reported by Stephen Kendall-Jones. The basic idea is to see what types of things (teaching styles, teacher actions, school environment, etc.) are the most and least effective in terms of student growth. I have no specifics about the measurement of student growth or how the data was analysed. Some of the findings were interesting to me. Below I have listed the 10 least and the 10 most effective "effects."

Least Effective Influences

  1. Whole language
  2. Perceptual-Motor programs
  3. Out of school curricula experiences
  4. Distance Education
  5. Teacher subject matter knowledge
  6. Diet
  7. Gender
  8. Ability grouping
  9. Teacher education
  10. Mentoring
Most Effective Influences
  1. Self-reported grades
  2. Piagetian programs
  3. Providing formative evaluation
  4. Micro teaching
  5. Acceleration
  6. Classroom behavioral
  7. Comprehensive interventions for learning disable students
  8. Teacher clarity
  9. Reciprocal teaching
  10. Feedback
I must admit that I don't know what some of these are. I do have some observations. 
  • Using evaluation to provide feedback is effective - I do not have the specifics and am unsure if this is regarding students or teachers. However, I do feel that it is important for both teachers and students to have formative evaluations. At times it is difficult for us as teachers to make changes unless we are given a nudge in the direction of improvement. Often times these changes are unnatural and uncomfortable for us. Focusing on these changes through the lens of a formative evaluation can be helpful if done in a supportive, non-punitive setting. I can see the same being true for students. Evaluations are an indicator of how a student is performing. We need to help students use evaluations to determine both there strengths and their weaknesses. We also need to help them improve in their weak areas through the development of a plan.
  • Feedback is important - Feedback does help us grow. Others see what we have done and how we perform differently than we do. Through feedback we change our perceptions about ourselves and learn how we can improve our performance. This feedback must be specific and immediate (or as immediate as possible).
  • High expectations - Most of us have no idea what we can really accomplish. We like to live comfortably and rarely reach beyond what is comfortable. Setting high expectations can be an effective influence if done so in a supportive environment.
  • Self-reported grades - This allows students to take ownership for their learning. Students learn to be self-reflecting which teaches them meta-cognitive skills. Through development of these skills students will recognize where they need to improve and what skills they have to help them through learning difficulties. 

Although it didn't make the top 10 list, another influence that has a high effective is teacher-student relationships. If every teacher could make the extra effort to connect with all students (especially those that have not experienced success in education) we could see huge gains. It is not always easy to do this with our busy schedules. However, a concerted effort is worth every second of time spent. Doing this right away and staying with it can have a great impact on students who otherwise may not experience their true potential. Often times we get caught up in the logistics of teaching. We forget that it really is about kids and helping them reach their potential. They are people and often need our support outside of content. Life is tough for way too many kids. Letting them know that we do care about them not just as a math student can go a long way to helping them do things they never thought possible. 

Thursday, November 6, 2014

Invert-and-Multiply...But Why?

If you teach division of fractions (the other "F" word) using the standard algorithm of invert and multiply, then you may have wondered, as I have, why it works. As a self proclaimed "math nerd," I have been thinking about this off and on for about a year now. As I started to work deeply with the AZCCRS (that's "Arizona College and Career Ready Standards" for those of you who do not live in AZ), I was really impressed at how the standards require the use of drawing and modeling with concrete objects to help students understand and make connections to procedures. I believe this is a key piece math education has been missing for far too long.

However, there was one common procedure (algorithm) that I could never connect to a model or drawing for myself - that is the standard algorithm for dividing by a fractions. The "Invert-and-Multiply" procedure is a mysterious procedure to most (if not all) students and I would guess to most adults, including teachers, as well. No matter how I tried, I was not able to connect a drawing to the standard algorithm of flipping and multiplying that you are probably so familiar with. Three education specialists (two math and one STEM) in my office have tried for months to find a way to demonstrate the connection for students. We were able to use drawings to answer division of fraction questions, but the elusive connection was always absent.

That all changed recently. As part of an effort to roll-out a math program to schools in Pinal County I have been participating in training offered by the Rodel Foundation of Arizona. The training is targeted at 3rd grade teachers. As part of the training we were asked to read selected chapters from "Teaching Student-Centered Mathematics - Developmentally Appropriate Instruction for Grades 3-5" by John A. Van de Walle, Karen S. Karp, LouAnn H. Lovin, and Jennifer M. Bay-Williams. It is a wonderful resource and provides insight into helping students make connections so that they understand mathematics at the deep level required by the AZCCRS. 
I started reading the chapter on fraction operations as required. I expected to read another technical proof on why the invert and multiply routine works. I had seen multiple explanation of this and understand and agree with them. However they always fall short of making the connection using a drawing or model. As I read the section I was encouraged and finally it happened. As I worked through the suggested questions to ask of students, I began to use drawings to help me make sense of the problems I suddenly realized each time I was dividing my drawing by the denominator and then multiplying by the numerator. Although I know I was doing this with every problem, the secret for me to unlocking the connection between the drawing and the algorithm was the context that was given to the problem. It finally made sense! And I understood it! 

I worked several more problems just to make sure I understood it. Then the other two specialists and I practiced at the circular table in the middle of our office space. This is where a lot of great discoveries take place (or a lot of junk piles up - depending on the work load of any given week). We tried what ifs and asked each other to explain it again. After an hour or so, feeling accomplished, we went back to our individual tasks for the day.

A few weeks later, while I was working with a group of middle school teachers, who are working on studying the standards and writing effective lesson objectives in preparation for writing curriculum for their district, the question arose. How can we help students understand the process of dividing fractions. I pumped up my chest, and smiled and stated, "I can show you." Feeling pretty smart, I walked to the white board and asked for them to give me a question. They asked the question (of course it was a word problem) and I started. I drew some rectangles, divided them accordingly and then completely forgot everything that I thought I knew about making connections. (This ever happen to anyone besides me?) I tried another approach. Maybe a different context. Cookies! Yeah, there was something about cookies that helped me understand it last time.

I drew some cookies. No luck.
I drew some ribbon - you know for making bows. No soup.
Rates! - That was it. If it takes 2 and 1/2 hours to travel 3 and 1/8 miles how far can you travel in one hour. Nope - still couldn't put it together.

Every adult (7 middle school and high school math teachers and two math specialists) tried for 45 minutes to make the connections. Finally our time ran out. I left for 10 days of vacation and the teachers moved on to greener pastures. The first thing I did when I got back to work, was grab "Teaching Student-Centered Mathematics" and make the connection for myself again.

So now, after learning and forgetting and relearning, I decided it would be a good idea to have it where I can always access it. Isn't the Internet great! I can post it here, share it, get feedback, and find it when I need it.

First - start by asking students questions where the divisor is a unit fraction (i.e. 3 divided by 1/2, 5 divided by 1/4, 3 and 3/4 divided by 1/8). Also, to help students visualize, give them a context. (How many servings of 1/2 are in 3 containers?) This will help them create pictures to help them with the math.

This is how I approached the question "How many servings of 1/8 are in 3 3/4 containers?"

I first drew a picture of 3 3/4. I divided it into 1/4s. I kind of did this intuitively. I'm not sure if students would work with way at this point, but as they work and get stuck I may direct them in this direction later.

I then needed to decide how many 1/8 fit into the 15/4. I see that if I divide each section into 2 parts I will then have 1/8s and I can simply count.

Simply by counting I see that 15/4 divided by 1/8 is 30. Thus, there are 30 1/8 servings in 3 3/4. The standard algorithm would result in my multiplying 15/4 times 8/1 or (15*8)/(4*1). At this point I was still not able to make the connection. But I could see that in effect when I counted 2 1/8s for each quarter which means I multiplied by 2. 15 * 2 is 30. I can also see that if I had first reduced the fractions when multiplying by the inverse I would have multiplied 15 times 2 since the 8 would reduce to 2 and 4 would reduce to 1.

The next question really got me on the way. "You have 1 1/2 oranges, which is 3/5 of an adult serving. How many oranges (and parts of oranges) make up 1 adult serving?"

More drawing. I started by drawing a representation for the oranges. 
I also knew the drawing would represent 3/5 of an adult serving. I thought to myself, if I could figure out what 1/5 of an adult serving is, then I could simply multiply by 5 so that I would have 5/5 or 1 adult serving. (Here's where I started to put this together.) If 3/2 of oranges is 3/5 of a serving then I divide the 3/2 into 3 equal parts (dividing by 3 - which is the numerator of the divisor) I will know 1/5 of an adult serving. 

Back to the drawing.

I know this mean 1/2 of an orange is 1/5 of an adult serving. Therefor multiplying 1/2 times 5 gives me 5/2 oranges or 2 1/2 oranges. I can now see with the picture that I divided by 3 and although I didn't multiply by 5 using the picture I can see the connection to the standard algorithm. I can see the light!

Now to try something a little more difficult, where the fractions aren't as friendly. "Aidan found out that is she walks quickly during her morning exercise, she can cover 2 3/5 miles in 4/7 or an hour. How fast is she walking in miles per hour?

I start by drawing 2 3/5 miles. I am also thinking that this represents 4/7 of an hour. Again, if I can determine how far in 1/7 of an hour, I will be able to determine how far you can travel in an hour, or mph.
To do this I need to split the 1/5 sections into 4 equal groups, or divide by 4. Since 13 doesn't divide into 4 even groups I will need to rename the sections. I know that if I split each 1/5 section into 4 equal parts, then I will be able to split the 52 sections in 4 even groups.

This means that she travels 13/20 of a mile in 1/7 of an hour. To determine how far she can travel in 1 hour, I multiply 13/20 times 7. This means that she can travel 91/20 miles in one hour, or approximately 4 1/2 mph or exactly 4 11/20 mph. So in summary, I divided 13/5 by 4. In order to do so I changed 13/5 into 52/20. When I divided I dot 13/20 miles for 1/7 hours and then multiplied by 7 to get 91/20 or 4 11/20 mph. I divided by the numerator and then multiplied by the denominator. I have made the connection using the algorithm to what I have done with the drawing.

I still think I can make this connection more explicitly, but I am much closer than I was and think I could help students connect these ideas. In some instances I think that students find this easier they have no knowledge of the standard algorithm and do not enter the conversation with misconceptions that we as adults sometimes bring to the situation.

Let me know if this makes sense to you and/or if you have another way to demonstrate this connection to help students make sense of and understand this often poorly understood algorithm.

Monday, October 13, 2014


I really enjoy my job as a math education specialist. However, I have had a lingering concern about the lack of follow-up that my job allows. Most of the time I work with teachers for very short periods of time. Sometimes as short as 1 hour. I then may not see those teachers for several weeks, months, or longer. This always makes me wonder how much of the information that teachers are able to effectively integrate into their practice.

How can I effectively follow up with teachers to support effective change?

This question has been following me since I began delivering PD to teachers in Pinal County over a year ago. Today I think I came up with an idea. Although I have not ironed out the details, I feel that asking teachers to report back electronically after a specified period of time will allow them to feel supported and to ask questions that come up as they attempt to change the way they operate within the classroom. Below are some ideas that I will think more about and use as follow up activities.
  1. Using a forum on my website - I would ask participants to report what they have implemented into their classes after a given time (1-2 weeks) and their reflection on how it went. Also, they could ask any questions that may have arisen or ask for advice from myself and other participants.
  2. On a PD blog - similar to using the above method however this would be more public. Perhaps a benefit of this format would be the reach. The questions, advice and conversation could potentially reach more people therefor allowing for collaboration on an even greater scope.
  3. A scheduled Google Hangout - have the participants "meet" at a predetermined time to hold a video conference concerning what they have incorporated and the success and challenges they have faced. I like this idea the most, but scheduling will be a challenge.
  4. Set up courses in Blackboard or Moodle - This would have the added benefit of having all of the materials available for participants in one location. As in using a forum on my website I would use the forum feature to facilitate discussions.
I'm sure there are other ways to do this and I would be super happy to have you share those with me. This is an endeavor that I feel is important and I have no idea of whether or not it will have any positive effect but as my Mom always told me, "You never know unless you try."

Happy mathing! 

Tuesday, August 19, 2014

To 3-Act Math or to not 3-Act Math?

Recently, I was tutoring and a student asked me for help with what I found to be an interesting question.

"There are 2 stools, one 3-legged and one 4-legged. One of the stools wobbles slightly.
  • Which one is it? How do you know?
  • Suppose that each stool is placed on a flat surface that is slightly sloped. Do you expect either of the stools to rock side to side? Explain why or why not. 
 Life experience led me to the correct answer yet I had to think for a moment to consider how to help the student understand the underlying concept that any 3 points make a plane.

Concurrently, I was thinking that a much better way of presenting this question to the students would be during an inquiry activity. It would allow students to discuss their ideas with one another and collaborate in order to make a precise statement defending their position. It would allow students to make a connection between "real-life" and mathematics - an opportunity to understand that there is a mathematical explanation for an what many students may have already observed during their lives. It demonstrates that people use math to explain the world around them. This idea also has applications such as the use of tripods.

I thought more about the question as I drove home and considered how I might have introduced this to students and what tools I may provide to them in order to experiment and get them interested in the task at hand. I turned to my PLN for ideas and posted the question and asked "Could this be improved? How could tech be integrated?"

A conversation followed between myself, +Jeremy Bell, and +Kyle Pearce. They both offered suggestions. The conversation then turned to whether or not the concept warranted the creation of a 3 Act Math Task or not. I would argue that this is an opportunity for students to have a math talk about a concept that is pretty simple, yet difficult to explain without mathematics vocabulary. Perhaps I view this activity as one that could be done at the beginning of the semester in Geometry. It could be used as an opportunity for the teacher to demonstrate to the students how to engage in math talks and share ideas. Also of emphasis could be that learning will take place through inquiry and discovery activities. This is a significant change for many students as they have been taught in a more traditional direct instruction model.

I also understand the other side of the argument. The same idea may be conveyed through a quick demonstration or even an explanation. The concept is not one that is of great focus in Geometry. It may be wasted time to create a 3 Act Math Task for this small concept and it may be a waste of time to have students work through a process.

So the question stands. Do you think that a 3 Act Math Task is an effective way to help student uncover the idea that 3 points always make a plan, or is it wasted time and resources for this particular topic?

Wednesday, August 6, 2014

Mini Goals

Lately I have been working hard to try to stick to a workout schedule. With the start of the school year (yes in AZ it starts at the end of July or beginning of August) it means my wife will be going back to work and I will once again have the pleasure (or duty - depending on the day) of transporting our three year old to daycare a preschool. Sticking to a workout schedule becomes more difficult. I also tutor at Grande Sports Academy, a residential soccer program, three nights per week. So the schedule gets more hectic and getting up with the alarm at 5 gets tougher.

While running earlier this week I found myself wanting to take a walking break after running just over a mile. With another 1.5 miles to go, I was having a sort of internal struggle over whether or not to take a short walking break. My hip was hurting, I was out of breath, and my allergies were causing issues beyond the normal. The struggle led me to keep setting mini-goals. Keep running to the next intersection, then the stop-light, then catch that guy that is walking, then to the corner in front our house. Before I knew it, the run was over and I never did take a walking break. I find myself doing this when I work out. One more set of 10 burpees, 15 more seconds planking, 3 more sprints, etc. I break down a workout that seems overwhelming into a series of mini-goals. This helps me keep from giving in to the aches, pains, or lack of motivation.

Amid the argument in my head, I began to realize that this strategy can apply to other areas of life. I specifically thought about using this strategy in education. Instead of being overwhelmed by a larger goal you may have (integrate technology, implement problem-based learning, etc.) break a larger, long-term goal into one mini-step at a time.

As a trainer this is something that I sometimes fail to do. I fail to help participants break an overwhelming task into smaller more attainable mini-goals. This is one great weakness that I see in the model of PD that I am sometimes asked to deliver. One-time, all-day PD sessions often have excellent content, strategies, resources etc. However, they truly lack in follow-through and support. Fortunately many schools are beginning to ask for follow up support as their teachers implement the new _______________ that has been presented. It is more effective and it seems that teachers are much more willing and able to truly change their teaching when this support is provided.

My goal is to be more cognizant of what my long-term goals are and work to create mini-goals that lead me to the finish line. Although, if you are a runner you know that one finish line leads to another start.

What are your long-term goals and what are some mini-goals that will help you get there?

Tuesday, July 29, 2014

Collaboration vs. Working in Groups

Yesterday I lead PD with a wonderful group of teachers who are preparing for the coming school year. The topic of the day was writing effective, standards based learning objectives and assessments. Although the day was filled with great conversations, questions, ideas, and just overall good fun, one particular conversation really caught my attention. We discussed the question... 

How is “collaboration” different than “working in groups? 

A rich discussion ensued in which it became quite apparent that no one in the room could clearly define collaboration or explain exactly what it would look like with a group of students. Further, it was stated that this is why students fail to meet our expectations of working "collaboratively" and ultimately may end up simply "working in groups." After all, if the teacher does not have a clear understanding of what collaboration is and what it looks like, how can one expect the students to "collaborate effectively."

We did however reach consensus on a few points.

  • Collaborative work requires that all members contribute in a meaningful way
  • All members must improve their understanding of the topic/task through the work
  • Clear communication is a must
Some other ideas were tossed around but were not agreed upon.
  • All members have an established role
  • Divide and conquer is a form of collaboration (this was quite controversial)
  • All members do an equal part
The conversation lasted at least 15 minutes and could have carried on much longer but we decided that our collaboration on the topic must cease so that we could try to get through the remained of the scheduled activities. Another great point came about was the impact of status in the classroom. We all know those students that are looked at as experts in the classroom and others depend on them for the answers. Status can have a great impact on how students collaborate (or fail to collaborate) with one another.

As I reflect, I was still struggling to identify a set of characteristics that could be used to identify effective student collaboration. I did a little Googling this morning and read some information about effective collaboration. Although some of what I read was focused on education, the majority was about collaboration in the business world. Much of it can be applied to classroom settings. Especially since one of the major goals of education should be to prepare students for life after school. Many students will at some point need collaboration skills in the pursuit of higher education and/or in their chosen career.

I first read "What is Collaboration?" The article defines collaboration as; a working practice whereby individuals work together to a common purpose to achieve business benefit. It also lists two keys features 1) synchronous collaboration such as online meetings and instant messaging 2) asynchronous collaboration such as shared work spaces and annotations. The author points out that collaboration depends on openness and knowledge sharing but also some level of focus and accountability.

Clearly the "math guy" I am was going to need a bit more investigating to figure this out. Synchronous and asynchronous are not words used in my vocabulary with any regularity (or ever if I am being honest). But thanks to the good ole interweb, my suspicions were confirmed. Now time to apply this to teaching and learning. It appears that our discussion was on track if we follow this definition. Synchronous collaboration takes place while individuals are working on something at the same time, or at different times. The key is that they are all contributing to one product and all portions of the product. Communication, openness, and knowledge sharing are important to successful collaboration. This confirms our thoughts that all members must contribute their knowledge and understanding and be willing to provide and receive feedback in an open manner. 

Now we are getting somewhere. In the haze, a shape is starting to take form.

I like the visual representation, but at first was unsure how many of these relate to school and students. After reading the descriptions, a few fit really well. 

Lead by example - Some teachers tend to work in isolation. How can we expect students to collaborate or see the value in collaborating when we don't collaborate with our peers. It's also a good idea to collaborate with your students. Let them see that they do have value and that their ideas can help improve the class. Letting go means you don't have to have all the answers and models how to discover, learn, and adapt through a collaborative process.

Create a supportive environment - This is vital. In order for students to collaborate effectively they need to feel comfortable taking chances and sharing their thoughts and opinions. This is especially true when there are students of differing status within a group. 

Persistence - Collaboration is not innate to students. We need to have persistence in helping them collaborate with one another. The article speaks of following through with creating a collaborative atmosphere even when initial attempts fail. Persisting, even in the face of failure, is another opportunity to model attributes we would like our students to develop. 

Collaboration can make the world a better place - Students are social beings. Why not use this to our advantage. Collaborating brings energy, excitement, and new ideas to a group. Not to mention a sense of responsibility. If I know my team is counting on me, I am more likely to do my part. Especially if I know that I will be supported and appreciated. 

So back to the initial question, how are collaborative groups different than groups that work together. Collaborative groups are interdependent and interactive. The groups strength is the group itself. Without the group the individual parts will experience a lower level of growth. Groups that are simply working together are not dependent on one another. One member of the group may simply do most or all of the work and tell others what they have done with no critique of input from the others. Unfortunately I see this happening in classrooms (including my own). It is a difficult task to create a truly collaborative experience for students, but one that holds great value. 

I have now outlined what I feel differentiates "working in groups" from working collaboratively. What are your thoughts?

Tuesday, July 22, 2014

Goals for a New Year

Many teachers and students are already headed back to school. Depending on where you live you may have a few weeks left to enjoy before you head back to school, excited, inspired, and rejuvenated for a new year with new students. Since my job is a 12 month position I am not headed back to the grind after a summer off, but I am however reflecting on the past year and creating some goals for the 2014-2015 school year.

We all naturally head into the new year thinking about what it is we want to change, things to do better, things to experiment with, and certainly those few things that were such a catastrophe that they will never again see the light of day. I thought I would take this opportunity to write about some of my goals and put them out there for the entire world to see - or at least the one person who is now following my blog. Thanks Heather! Congratulations! You are my first "follower." There are no special prizes or gifts, but know that you have made me feel important. Pretty exciting for me. And really the inspiration for me to write this today.

Here are some of my professional goals for the following year.

  1. Blog more - Although this is only the 3rd blog post that I have added to this blog, I really do see the importance of it. According the stats, there are at least a few people reading what I write. Ideally they are reading and thinking about their practice - whether they agree or disagree with me is really unimportant. I also find blogging gives me a chance to reflect on my own beliefs and practices. I find myself thinking about possible blog topics while I am working out in the morning. This time allows me to collect and work through my thoughts. Most of the time these inner "conversations" never make it to the blog, but still hold value. 
  2. Get more people using Twitter as a part of their PLN - After working with many of the schools in the county where I work, very few are using Twitter as a professional learning tool. I sort of stumbled on Twitter as a PLN myself this year and am amazed at the interactions I have had and the information that is constantly being shared. I am often intimidated and feel that I am not an "expert" like those that I follow, but I am discovering that I do have valuable information to share. I would really like to have some teachers from every district following me on Twitter. Last week I gained my first followers from Pinal County. Very exciting. I think that a virtual PLN is vital to the success of our rural districts due to their isolation and their size. It's pretty difficult to collaborate and gain momentum for change when your whole department consists of one teacher. Pretty lonely conversation.
  3. Be part of at least one grant opportunity - This is definitely one of my weakest areas. I know next to nothing about grants, so I can not even make this goal more specific.
  4. Conduct a webinar - Our office has discussed several times of how to reach more teachers, especially those in outlining areas. Webinars are one solution that has been talked about, but no one has jumped on the train. Sad, but unfortunately it is reality. My goal is to regularly conduct webinars.
If you feel so inclined, comment and share ideas for my goals (how I can accomplish, if you think they are worthy, etc.) and/or share your goals for the year. I know that every teacher has great ideas and "big thoughts" for the new school year. Share some of these great ideas and goals and maybe you will inspire someone else.

Oh one more. I need to start wearing more shirts like this.

Thursday, May 15, 2014

Unproductive Beliefs

I recently attended "Mathematics in the Mountains" in Flagstaff Arizona. The conference was being held for the first time and organized by Arizona Association of Teachers of Mathematics. It was held on the campus of Northern Arizona University. It was also a great time to be away from the valley as it was the first 100+ degree day of the season. Many more to come very soon.

One of the sessions I attended was "Principles to Action: Ensuring Mathematical Success for All, NCTM's New Blueprint for School Mathematics" led by Jane Gaun of the Flagstaff Unified School District. All day I was looking forward to attending this session. I had previously watched the video of Steven Leinwand talk on the "Principles to Actions" at NCTM. When I saw that there was going to be a session based on the "Principles to Action" I was excited. Especially after seeing the passion with which Mr. Leinwand spoke of the book.

During the session, the presenter asked us to complete an activity. We were given a set of cards with a short phrase on it. We were then asked to sort the cards into two categories - those that are productive beliefs and those that are unproductive. We were to complete the task as a group and discuss any we disagreed on. It led to some rich conversation. It caused me to think about some of the common beliefs that are roadblocks to allowing all students access to high quality, math education. I hope that by writing some of these statements below a conversation can begin about these beliefs and how we can start hearing and seeing the productive beliefs more and the unproductive beliefs less. The statements are below and are in no particular order. Please comment on which ones you think are productive and which are unproductive with a rationale.

Let the conversation begin! Consider putting your response on Twitter using #PrinciplestoAction.

  1. The role of the student is to be actively involved in making sense of mathematics tasks by using varied strategies and representations, justifying solutions, making connections to prior knowledge or familiar contexts an experiences, and considering the reasoning of others.
  2. Students can learn to apply mathematics only after they have mastered the basic skills.
  3. Mathematics learning should focus on developing understanding of concepts and procedures through problem solving, reasoning, and discourse.
  4. The role of the teacher is to tell students exactly what definitions, formulas, and rules they should know and demonstrate how to use this information to solve mathematics problems.
  5. All students need to have a range of strategies and approaches from which to choose in solving problems, including, but not limited to, general methods, standard algorithms, and procedures.
  6. Students can learn mathematics through exploring and solving contextual and mathematical problems.
  7. Mathematics learning should focus on practicing procedures and memorizing basic number combinations.
  8. The role of the teacher is to engage students in tasks that promote reasoning and problem solving and facilitate discourse that moves students toward shared understanding of mathematics.
  9. An effective teacher provides students with appropriate challenge, encourages perseverance in solving problems, and supports productive struggle in learning mathematics.

Friday, April 25, 2014

The Common Core Standards as I See It.

The other day I had one of those "Aha" moments. Seemingly from nowhere, a random, moderately intelligent thought popped into my head. Perhaps this rare moment of clarity was due to the work that I have been doing with Disciplinary Literacy and preparing for presenting a Socratic Seminar to college students at the local community college. The latter is a pretty scary and daunting task for a "math specialists." However all survived and the students even seemed to understand the take-away that I was hoping for.

The moment occurred when I was driving home. For some unknown reason I was thinking about my days as a student in middle school woods class. I wasn't a very handy youngster. Breaking things and losing them were more my specialty than building and fixing. However, I was always very curious about how things worked. I saved up all my pennies from allowance to buy a Fast Traxx remote control car and took it apart countless times.
 I studied the gears and all of the internal parts to figure out just how everything worked. Like most people, unless I had the opportunity to get my hands on something and see and feel how it was put together I didn't understand it. Unfortunately my middle school woods teacher's teaching style didn't match my learning style. He was more of  a "show you how to do it once and you better get it" kind of teacher. I don't think he had any ill will toward me or had cooked up a scheme to keep me from ever having the ability to build something on my own. However, I was very frustrated in the class and remember crying with my mom about how difficult it was and I "Just didn't get it." The class was split into two portions. Book-work, where we learned the names of the tools, safety, different styles of cuts, joints, etc, and measuring, and the project, which required building a clock.

I excelled at the book work portion, acing assignments, quizzes, and tests. I bombed the clock. Several times the pieces I built were rejected by the teacher's quality control team that consisted of himself. Rather than giving me a second opportunity to learn what I did wrong and rebuild the piece, he would select the proper piece from a surplus of clock pieces that were collected over the years from student projects that were never completed. Half of "my clock" is really due to the work of other students. I feel like there should be a list of credits on the back. "Thanks to Billy for the face-plate since mine sucked!"

The point is, I knew how to use the tools, understood the safety and could measure like an expert. The area where I lacked, was using the tools to create a product. I'm not blaming the teacher. I'm sure that he was doing the best for me that he knew how and probably thought I had no interest in ever building anything after his 1 semester class. In fact, I should probably thank him. Perhaps this event is partially responsible for my view on teaching. If a student doesn't understand something and be able to apply it when appropriate, I take it very personally. It becomes my mission to help these students understand and conquer!

In essence, I am no different than many students in math class. Many students can repeat the steps that are taught to them by a teacher. Isolate the x, keep flip and divide, use the quadratic formula, etc but are completely lost when asked to apply their math skills to solve a "real-life" problem (how long will it take to fill your pool?). I knew how to use the tools that were available but was unable to use them in an application (build a clock).

This is exactly what we have been doing in math classes for a long time. We have been teaching kids a set of procedures that are seemingly unrelated and are then surprised that even though they may score well on our assessments they fail miserably when asked to apply these skills in novel situations. This is the concern that Common Core addresses. Not only in math, but across all subjects. Common Core asks us to help students really understand the concepts that we are teaching them and give them opportunities to apply what they know. The inquiry portion of Common Core math lessons are great! These answer the question that students have been asking for decades...I'm sure I don't have to write it but I will. "When will I ever use this?"

 I know this change in teaching is challenging and scary and tons and tons of work. But what a great opportunity. I have been talking with teachers recently and one of the things I emphasize is when students are in the inquiry phase we shouldn't care about the answer. In fact if we don't know the answer, or how to get the answer, or scariest of all, if there is an answer, it's ok. Life will continue. The students will be uncomfortable, frustrated, whinny, and even obstinate. But, they will be thinking and learning math as long as we foster the conversations and allow students to discover, experiment, fail, and try again.We need to put our focus on their thinking. Concentrate on what they are saying and how they are supporting their claims. This is the mathematical thinking that we have been missing. We need to focus on teaching students both the skills (procedures) and how they can use these skills to build a product (use their understanding of math to solve a problem). Just as in woods class it is important to be able to use the tools and build a product.


Monday, March 24, 2014

EdTech Team Arizona Summit - Featuring Google for Education - Part II

Day 2

Keynote - Monica Martinez
Day 2 started off with a keynote by Monica Martinez. She discussed how the development of the web has changed the world around us and how we access information. Here is a link to the presentation. She also talked about the amount of information that is available today via the Internet and how one of the most important skills we can teach students is how to access, analyze, and process this information. There has also been a change since the early days of the Internet. Today most information is generated by individuals who are sharing content. 

In school we use the Internet for completing reports which requires being able to learn efficient search techniques and the knowledge of databases and online encyclopedias. She led us in a competition using TodaysMeet by posting a question and asking us to answer the question by posting our answer as quickly as possible. I admit that I was really bad at this even though I think of myself as being fairly adept at locating information online. One tip she gave was using "*" when there is an unknown word in your search. For example if we wanted to know what Magellan spent more money on than weapons during his circumnavigation of the world we could type [Magellan spent more on * than weapons] to complete the search. Try it to find the answer.

Another tip is to complete simple calculations by simply typing them into the search bar on Google. Google will automatically bring up the calculator once you have entered the search. Simple, yet brilliant. For a complete training on becoming an expert searcher click the link.

Next she discussed how students are learning. It turns out that students are supplementing what they are learning in schools by using the Internet. We are no longer tasked with holding all information. It is my belief that to really help students we need to teach them how to collaborate, communicate, locate, synthesize and apply information. We need to help them "connect dots" rather than "collect dots." 

Next Monica reviewed the SMAR model and gave examples of what each part looks like in practice. Check out this video of a young man who is trying to acquire new information about a skill that he has tried to learn and failed repeatedly. This is a great example of how technology can allow students to learn in ways that were never possible before. Students can create a product and connect with experts in a particular field. This is also a great skill for teachers to have. We all create or own materials, lessons, activities, etc. We can now easily connect with experts (not only other educators, but those in other fields) to collaborate with on our creations. 

Finally, she shared this video about the power of technology. Really all we have to do is get out of the way of our students and facilitate their learning. 

After another wonderful keynote, it was time for more sessions and great information.

Session 1 - What is the next stop on your Google Apps Journey? Round table / Q&A Session - Peter Henrie (,
This was a bit out of my comfort zone as it was more technical than where my knowledge level is. I wanted to attend this session because I wanted to see if I could find out how to best advertise and make educators more aware of all the tools that Google Education can provide. Most of the attendees to this session were IT people and admin. The biggest take away for me was that going Google provides a lot of benefits for a lower cost. Every IT director in the room spoke to the advantages of Chromebooks and the ease of management compared to IPads. Peter was very helpful by allowing the participants to direct the session. He is very knowledgeable and was able to explain things so that even I could understand what he was talking about. 

Session 2 - Close Reading with Google Docs - Chris Bell (
Really pushing myself today. Math guy at a Close reading session. I learned a lot. Chris shared with us the process of using Google Docs and some of the ad-ons for completing Close Reading. He demonstrated how to find an article ( and save it as a Google Doc, showed how to use the research tool and dictionary tool and, best of all, TextHelp add on. This add-on allows you to collect highlighted words into a new Google Doc. This is a great tool for creating student vocabulary list, creating consensus maps, and a whole bunch of ELA concepts that are beyond my mathematical knowledge at this point. I can really see the value of doing this and wish that this had been around when I was in college. Really!!! Google Docs will create my bibliography as I complete my research within the document itself. Crazy Awesomeness! The ELA teachers in the room were super impressed and so was I.

Session 3 - Creating Formative Assessments with GAFE - Cherie Stafford (,
After a much needed brain break and lunch, Cherie presented creating formative assessments using Google for Education. This session was focused on making formative assessments through the use of Google. Cherie also stressed how the use of available tools can make formative assessment more authentic. One easy way to do this is to publish the completed project to a class web page. Many of the tools available from Google allow this to be done easily. Also, collaborating can be done and tracked easily. One idea that I like is having students create a list of resources on the class site for sharing. Another that I like is using Google drawing and have students label and explain parts of something, although I'm not sure this activity in itself is enough to make it authentic. To do this, it would have to be a part of some bigger project, but could be authentic if used as a part of some student project. 

Session 4 - Gearing Up: Project Based Learning to the Core with Google Apps for Education - Cori Araza (,
This was possibly my favorite session. least in the top 5. They were all awesome in their own way! Anyway - Mrs. Araza presented how she uses Google Docs in her classroom through project based learning. Here is a link to the presentation.  She leads a group called GenYES in which students become the IT experts for the district and provide technology pd to the teachers. The class is heavily dependent on the use of Google docs. One of her former students was also in attendance to share his insight into the use of Google as a tool for researching, collaborating, presenting, documenting and completing the assignments within the course. I was amazed at what the students are able to accomplish and take away from the class. This really is pbl in action. The other big take away I had is the process that she uses. She shared with us one of the assignments that students complete. It involves the student writing a professional email. The assignment begins by asking students to write a professional email. This is really all the guidance they receive. After they have done so, they are asked to complete a self-reflection based on a Rubric that she has created. The student completes the rubric along with their justification for grading. After the self-reflection the student is asked to write a second professional email which is submitted and put through the same rubric by the instructor. I thought that this was a very good match for what the Common Core standards ask students to be able to do and could be applied to a variety of content and topics. 

At this point my mind was saturated and it was off to the final key note speaker. 

Final key note - Jim Sill
Jim's final message was very entertaining. He formerly worked in the video and television industry and his message was that teachers have the power to change the lives of their students and we have opportunities to do so as never before with the availability of technology. He spoke of some of the great successes in the classroom as well as some of the even greater failures that lead to learning opportunities. It was an awesome wrap-up to a great conference. I will definitely be back in 2015!

Friday, March 21, 2014

EdTech Team Arizona Summit - Featuring Google for Education - Part I

This is a summary of my experience at the GAFE Summit held this week at Grand Canyon University in Phoenix. First off I want to emphasize how great this conference was. I knew that Google was a powerful tool, but had no idea to what extent these tools can impact how teaching and learning could be impacted in the right hands. Now the challenge is to get this information out to anyone willing to listen - especially teachers in Pinal County where I work. This is more an attempt to collect and sift through the information for myself in hopes of making some cohesive and useful information for others, so forgive me if I wander around a bit. Here is a link to the schedule of events which has links to lots of information provided in the sessions. 
Day 1
Day 1 started off with keynote speaker Jaime Casap (, I honestly don't remember all, or much of, what he said, only that his message was to not be afraid of failing. This also seemed to be a message from several of the presenters at the conference. More about this later. He also talked about the need to adapt and accept that things are different now and so should our classrooms. He spoke about the power teachers have to affect change in the lives of our students saying, "Teachers are the most important people in a kids life." Scary - yet errrr encouraging? After all, who better to effect a student in a positive way than a teacher that is passionate about making change in the world. He also challenged the audience to think about the technology that we had used just that morning. This got me thinking that it is rather hypocritical to not allow students to use the technology that they use in their "real-lives" inside the classroom yet we would not do the same. Technology is a door-way to student motivation and inspiration. The possibilities truly are endless. Unfortunately educators take a hangar-door size opening and restrict it to doggy-door size. In doing-so we inadvertently crush the desire of the students. 
Session 1 - Connecting the More in Google Drive
Cherie Stafford (, presented how to connect apps through Google Drive and then highlighted some apps. - Tutorial on how to connect apps through Google Drive - Handout on the apps presented along with my notes and links to tutorials for some of the apps. 
I personally like WeVideo and PowToon. I can see the application in the classroom for both of these as ways to 1) present/introduce/hook students 2) have students use to create product as an assessment. There is more information within my notes section and on the tutorials. It was also during this session that I started to see how awesome Google Apps could be for teachers. Through add-ons teachers can push out documents, collect and organize student documents, give feedback to students, and on and on and on....Great stuff already and it's only 11 am on day 1. 

Session 2 - A Beginner's Guide to Google+
I will start this off by saying that 1 month ago if you asked me about Google Apps, Google+, Google Docs or Google __________(fill in the blank) I would have told you that I had heard of it but not much else. I am also pretty new to the Twitter and have started to rely on in it more and more for information, feedback, and idea sharing. I'm starting to become just a bit addicted. I sometimes dream of having a smart phone (yes, I know you are laughing. My wife and I may be the only two people left in America that don't have smart phones. She teaches 2nd grade and most if not all of her students have their own) so that I can keep better track of my Twitter feed. So I am a bit behind but am gaining speed as I roll down the mountain. I even have started to use HooteSuite so I can post to all of my new social networks and stay in contact with my growing PLN.

Anyway, the presenter for this session was Michelle Armstrong (, I felt right at home as her Canadian accent sang to me and reminded of how some people talk in my hometown in northern Minnesota. She first introduced us to what Google+ is all about and how to use it most efficiently to grow your PLN. Visit my Google+ page to see my progress. Maybe you can give me some pointers (for instance how do I get I nice handy url like Michelle instead of this Like I said still a rookie but learning none the less. My take away from this session was this is another great way to connect with others that are interested in similar stuff (teaching math, tech integration, outdoorsy stuff, etc.) Here are the links to info. provided by Michelle.

We learned about Circles in Google+ and most awesomely, Hangouts. Certainly will be testing the waters here in the future.
Session 3 - Get More from Your Google Drive with Apps
Monica Martinez (, presented more apps to use in the classroom.
Above is the link to the website where you can find highlighted apps. My favorite from this session is , an app that allows you to take notes on a YouTube video while you view it. Every time you enter a new note, a time stamp is recorded. To return to that point in the video later, all you have to do is click the comment. I can see using this with students in a flipped setting or even putting in questions when having students watch a video. I'm sure there are many other uses that people much smarter than me can think of so please share your ideas too.

Session 4 - Chromebook 101

Chris Bell ( provided information on how to operate Chromebooks. This was the first time I took the opportunity to use our office Chromebook extensively. I am really impressed with it. We have the Samsung version which by my estimates seemed to be the most popular by far at the conference. I am really impressed. It did everything I needed it to and was very easy to use after learning a couple of little tricks from Chris. I really want one for myself. The battery lasted most of the day and I was using in constantly from 8:30 in the am until 5:30. I did have to find a place to plug in at about 4. A big step up from our office lap tops which last 2 hours max. I also liked that as soon as I opened it, it was ready to go. No waiting for boot-up and forgetting what note I was going to write or site I was going to visit. Lots of information can be found on Chris's website about Chromebooks and other Google related stuff.

Demo Slam -
A plethoro of information was thrown at us in 4 minute chunks. My favorite was Jim Sills presentation about some of the tools available in Gmail to help avoid getting fired. He included how to enable the "undo" button to take back a sent email and how to create auto reply messages. Very entertaining and a great way to end day 1.
Wow. Only halfway home. I think this will be a 2 part blog.
Enjoy! Connect with me via Twitter or Google+ and make sure to drop some comments.