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. 

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.

Followup

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! 

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?

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?

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.

Next I read "The 12 Habits of Highly Collaborative Organizations."

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?

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.