MeTA musings

Pi does deserve a celebration, but for reasons that are rarely mentioned. In high school, we all learned that pi is about circles. Pi is the ratio of a circle’s circumference (the distance around the circle, represented by the letter C) to its diameter (the distance across the circle at its widest point, represented by the letter d). That ratio, which is about 3.14, also appears in the formula for the area inside the circle, A = πr2, where π is the Greek letter “pi” and r is the circle’s radius (the distance from center to rim). We memorized these and similar formulas for the S.A.T.s and then never again used them, unless we happened to go into a technical field, or until our own kids took geometry. (Emphasis mine)

"Why Pi Matters" - New York Times, March 13, 2015

I'd load up with...

A friend from church tweeted a picture while he was out shopping today.  See below for the picture.

How would your students react? "Huh?" or "Are you kidding me?!"

You probably already caught it on Dan's blog, but if not the first 40 seconds are worth repeating:

"Can I ask you to please recall a time when you really loved something...a movie, an album, a song or a book and you recommended it whole heartedly to someone you really also really liked and you anticipate the reaction and you waited for it and it came back and the person hated it...so by way of introduction, that is the exact same state in which I spend every working day for the last six years.  I teach high school math.  I sell a product to a market that doesn't want it, but is forced by law to buy it.  It is just a losing proposition."  - Dan Meyer

 Check out the TEDxNYED video for the rest of the twelve minute talk on math education.



Maybe there's a reason why teaching secondary math is considered a shortage area in Iowa? (pdf)

Last week, I had an afternoon Iowa Core district leadership team meeting on Monday and had a sub again all day Wednesday to attend the Iowa 1:1 Institute.  I will be gone again for a full day to start the week to co-present with Kathy Kaldenberg, Shannon Miller, and Deron Derflinger at the IASL Spring Conference.

I typically try to schedule written assessments for days I'm out of the classroom.  It is pretty easy for a substitute teacher to hand out a test or quiz and have students hand them into the basket when they're done.  Once students work on some review problems after the assessment, it pretty much fills up an 84-minute block. 

Well, being out of the classroom this often doesn't lend itself to easy planning.  There just aren't enough written assessments I can give to my students right now.  So, I decided to try yet another experiment in virtual education, this time in my Statistics & Discrete Math course. 

First, students will be asked to watch this video explaining the day's activities- a late night clip created by me with my MacBook and its build-in iSight camera.
Second, after reading a few pages from the textbook, students will watch this video from the textbook's website to see an example of a hypothesis test being worked out. 
Last, students will complete several problems from the textbook to practice the hypothesis test algorithm.

The last day we met together, we did several activities explaining the concept behind hypothesis testing, so they should be vaguely familiar with much of the terminology presented in the examples.  This virtual setup is far from ideal, but aside from assigning more busy work, it is the only way I could think of to push forward knowing that not many substitute teachers have a deep understanding of hypothesis tests. :)

What other great ideas are out there for keeping students busy (as sad as it sounds) AND learning while you're out of the classroom?

Warning: unpolished thoughts (a.k.a. rambling) follow...

A colleague recently sent an article describing the age-old problem of math topic coverage being a mile wide and in inch deep.  A few excerpts from the article are enough to illustrate the main points math curriculum crisis transpiring in the state of Florida right now:

"The problem with the old standards, educators say, is that many topics were covered lightly and then repeated year after year."

 and

"Because the state's annual test measured performance on the 89 standards, Kuches and other teachers had no choice but to race through them in the hope that students would learn. The results were not always good, especially for slower learners."

I used to think that this problem could be quickly fixed...just teach fewer topics and explore them from many different angles so that ALL students can understand them.  After a dozen or so semesters teaching, Geometry, I still think this is a good idea in theory, but I have also realized that some young minds just aren't mature enough to comprehend certain math topics.  Think about Bloom's Taxonomy (or a more specific math example, van Hiele levels of geometric reasoning) and how some students struggle with the knowledge or comprehension level of many topics in school.  In Geometry, our curriculum (i.e. proofs, inductive and deductive reasoning) implies students should be synthesizing and evaluating geometric ideas.  From my experience, that's just not cognitively possible for some students as fifteen or sixteen year-olds.  It might not be possible until they are seventeen or eighteen.  If the topics don't repeat themselves, at least a little bit, students will be missing out on the opportunity to be exposed to these ideas later on in their "math career." 

I do like the approach of only teaching several big ideas each year during middle school math, but I'm just not sure if its reproducible at the high school level.  Some spiraling of topics seems to make sense, but how much is too much given what we know about adolescents' cognitive abilities?  I am by no means an expert in adolescent psychology, so I'm hoping y'all are able to chime in with your thoughts and experiences.

On Thursday, I, along with the rest of our district team, attended an Iowa Core Leadership Series meeting at Grant Wood AEA.  Becaue it was an all-day event, so a substitute teacher oversaw my class for the first time this semester.  Typically, I try to schedule an assessment or time for students to work on an ongoing project.  I'll be real honest in saying I don't have very high expectations for substitute teachers when it comes to teaching Geometry or Statistics.  It's a better use of my students' time if the substitute can take attendance and read a few directions before letting the students work on a task for an extended period of time rather than diving into new content.

Thursday presented a unique problem for me.  We are only a little over a week into the semester and students are not yet ready to start a new project or take a written assessment.  I decided to try "teaching" asynchronously by creating a website for my students to navigate in lieu of my usual lesson on measures of central tendency.  Here's the link to the online lesson I created in about 60 minutes late Wednesday night.  The videos, except for the first one, were taken directly from our textbook publisher's website.  I checked out some head phones and one of our laptop carts so that each student could work at their own pace.  I asked students to provide feedback at the end of the lesson as well.  Data from the google form is below.


Three take-aways from this experience are worth sharing:

1. I shared the results with students the next day and thanked them for going out on a limb and learning online in this format.  Questions such as "Why can't we find the mean of question 3?" and "Which measure of central tendency should I report for question #1?" encouraged a great discussion about measures of central tendency.
2. In the same discussion about the feedback results, it became apparent to students what quality feedback looks like.  The "keep it real dog" student response to the last question was humorous, but did not provide meaningful feedback to improve the online learning process while the "The lady that narrates the videos has a very boring voice, it made me rather read the book. Maybe if you showed us yourself, instead of having us watch the text-book ones, it would help a lot more" response was detailed and has the potential to improve the process in the future.
3. Overall, the feedback/data tells me that students are willing to engage in this type of learning again in the future with a few small modifications.  

What type of activities do your students engage in while you are out of the classroom?  Have you ever tried this type of online learning?  If so, what suggestions can you share? 

Yesterday, I observed my student teacher's statistics lesson on confidence intervals using a t-distribution.  We both agreed that the lesson went fairly well and without a doubt emphasized the main ideas needed to help students understand the day's learning objective:

Construct and interpret confidence intervals for a population mean using t-tables when n < 30 and/or the population standard deviation is unknown.

This topic is a common one taught in any introductory statistics class during the inferential statistics half of the syllabus.  In fact, just the school day before, students were exposed to the underlying ideas behind confidence intervals and the difference between descriptive statistics and inferential statistics.  After yesterday's lesson, my student teacher and I both lamented how tired and unresponsive the class as a whole was.  No matter how many jokes he cracked or the stories told about the history and relevance of confidence intervals, the students were very subdued and content with the silence.  Aside from a few strategies we discussed about alternative ways to engage the class, it inevitably seemed like one of those "it will hopefully be better tomorrow" conclusions.


Then, "after school" happened.  A student who is not currently and has never previously enrolled in Statistics strolled into our room.  For the sake of anonymity, we'll call her "Barbara."  Barbara is working on her science fair project and her teacher suggested that she pay a visit to the resident statisticians (my student teacher and me) to critique her data analysis.  (Note: I'm really excited that this type of interdisciplinary collaboration is becoming more common between Dawn and our statistics department - last week our class critiqued her class' data tables, graphs and charts for appropriate use of descriptive statistics)  As it turns out, Barbara was testing the effect of several variables on a person's ability to run.  Her averages, in seconds, were within one or two of each other.  Based on her descriptive statistics toolkit (mean, median, mode and range), it appeared as though there was a meaningful difference between her variables due to the averages being separated by as much as two seconds.  After all, to the average US citizen, a runner who has raced eight times with an average time of 23.5 seconds seems faster than another running who has also raced eight times with an average of 24.4 seconds, right?!

Enter inferential statistics and confidence intervals.

Over the next thirty minutes, my student teacher and I taught Barbara an abbreviated version of the same lesson he had taught earlier than afternoon to our statistics class.  Barbara added standard deviation to her descriptive statistics toolkit and asked questions until she not only understood the idea behind confidence intervals, but also felt like she was able to explain it to someone else, a requirement for the science fair project.  Barbara asked questions like "Would my intervals change if I increased the number of trials?" and "Why should I choose a 95% level of confidence instead of a 99% level?"  She left the room satisfied and ready to add on to the data analysis piece of her project as well add more depth to her "results, discussion and conclusion" component based on her questions. 

I am 95% confident that Barbara left the room with a deeper understanding of inferential statistics than those currently enrolled in the statistics course.  What a difference context makes...

Note: This post is an excerpt I wrote for a leadership class taken during my graduate work in curriculum and instructional technology at Iowa State University. It is also is in response to Leadership Day 2009. XYZ is a fictitious school district with some, but not all attributes similar to my current school district. The opinions stated below are solely the opinions of the author and not of my employer.


How does instructional technology fit into the mission of XYZ school district? What opportunities do students currently have to use technology that enables them to fulfill this mission statement? What changes need to be made in order for the district to best serve the students and other stakeholders? The purpose of this paper is to layout a plan introducing new policies and practices to promote more effective technology integration in the XYZ school district. The following sections will identify the current state of instructional technology in XYZ; define effective use of technology by instructional staff; suggest a plan of action to be implemented over a finite amount of time; and designate the primary role of relevant stakeholders in order to ensure the sustainability of the changes

Aside from several early adopters, the diffusion of technology at XYZ has been a relatively slow process. In general, the majority of teachers are still at the awareness and how-to knowledge stage of the innovation-decision process (Rogers, 2003, p. 172) due to their ability to simply use Keynote, Power Point, and iMovie or require students to use them for producing projects. Teachers have effectively re-invented their previous projects using student-produced technology artifacts and according to Rogers (2003), this is a generally desirable and normal function (p. 185). Cuban (2001) also supports this idea that teachers typically “maintain rather than alter existing classroom practices” (p. 71) when using technology. What, then, is the solution? The seeds of the solution are contained within the explanation of the problem (Cuban, 2001, p. 136), so let us begin by establishing effective use of instructional technology by instructional staff.

A clear separation has to be made between using computers to prepare for instruction and actually using computers for instruction (Cuban, 2001, p. 126). Effective use of technology will “transform teaching and learning into an engaging and active process connected to real life” (Cuban, 2001, p. 14). This second goal of technology in schools suggested by Cuban is the premise upon which instructional technology should be built upon. It is also important to note that technology is not appropriate for all projects, because it depends on “what the teaching and learning goals are” (Cuban, 2001, p. 70). The use of computers and other technology should be so ubiquitous in the classroom that it is not seen as an add-on but rather as a means to reach a desired outcome. The key thought behind instructional technology use should be,

“am I using this technology tool as a means of automation/communication or as a means of transformation?”

Automation and communication examples include the use of PowerPoint slides to relay information or give immediate feedback to students’ response through multiple-choice skill-based websites. These ideas in isolation are not harmful, but should not be seen as a model to follow as Cuban (2001) suggests in his book. Transformation examples include the use of software and hardware to create an environment focused on the student. This deeper learning in the form of constructivist pedagogy (Fullan, 2007, p. 266) engages the students in their own learning by using data to alter the needs and interests of the individual (Fullan, 2007, p. 180). Finally, this technology transformation involves using contextual clues to help meet educational objectives. For each educational objective that is identified as appropriate to be taught using technology, the educator would have the time and resources available to him or her to teach and re-teach in a way that has deep meaning for the students. There is not a “one size fits all” recipe for doing this, so the need for an extended amount of time and a mentality of “continual learning” to develop such ideas is pertinent. Elmore clearly supports this idea when he states,

“Improvement is more a function of learning to do the right things in the settings where you work” (Fullan, 2007, p. 153).

In summary, instructional technology involves the transformation of teaching into a dynamic process through the use of technology fueled by the needs and interests of the individual student. It is contextual. In the following sections, I will describe the details needed to “flood” the diffusion network of XYZ district with this idea.

The heart of the diffusion process consists of interpersonal network exchanges and social modeling by those individuals who have already adopted an innovation to those individuals who are influential to follow their lead” (Rogers, 2003, p. 35).

As I described in the first section of this paper, the district staff is not at principles-knowledge when it comes to instructional technology. The changes I recommend aim to accelerate the bell or S-shaped curve innovations typically follow (Rogers, 2003, p. 272). While acknowledging that change takes time, the focus will be primarily on the use of interpersonal networks to create a critical mass of instructional technology adopters (Rogers, 2003, p. 300). In the following paragraphs I will describe a multi-year, systematic process of changing the culture of XYZ district to one that favors “change” and finally a “change” towards teachers’ increased and more effective instructional technology usage.

Year one in this action plan will be called the “Year of Culture.” The focus will be on building culture throughout the district and answering the following ongoing conversation questions:

1. How do we feel students best learn?
2. How will we ensure that students are getting the best possible learning experience each and every day?

A new position will be created in the district, “Director of Pedagogy and Culture,” DOPC for short. From the day this person is hired, his/her daily assignment will be to design activities to help the entire staff of approximately 85 educators answer the two questions above. The DOPC should be an identifiable champion, someone “who throws his or her weight behind an innovation, thus overcoming indifference or resistance that the new idea may provoke in an organization” (Rogers, 2003, p. 414). The DOPC will have excellent people skills and be generally homophilous with the rest of the staff. This position, and his/her additional staff as determined on a need-basis, will be funded by virtually eliminating technology budgets. In fact, the DOPC and other district leadership will advocate for primarily planning and communication use of technology in this “Year of Culture” so that educators can focus instead on this change of culture rather than technology integration. The DOPC will focus on the “specific displacement of existing norms, structures and processes” (Rogers, 2003, p. 55) so that old behaviors and beliefs are replaced. All other “add-ons” will be eliminated such as reading in the content area, 6+1 traits of writing, special education team teaching, and literacy initiatives unless mandated by state or federal law. In essence, the district will enact a one year moratorium on doing anything “extra” in order to focus on the “Year of Culture.”

On a district-wide level, the DOPC will formalize the use of case studies and small group discussions to answer the two key questions. The DOPC and his/her staff will continually encourage small group discussions to begin and end by also answering the question, “why are we doing this?” This will create an environment in which the staff begins to agree on

“what is worth achieving…and set in motion the internal processes by which people progressively learn how to do what they need to do in order to achieve what is worthwhile” (Fullan, 2007, p. 230).

Educators themselves will be asked in these small group discussions to answer the two key questions time and time again until shared meaning is established. Constant revisions of the responses to these questions will be made available to all staff until a consensus of no more than five bullets is agreed upon over the course of this first year. Release time will be given to staff identified as opinion leaders by the DOPC in order to keep minds fresh and morale at its peak. In the midst of the small group discussions, case studies of local teachers and their most effective lessons will be shared. Because “teaching decisions often are made on pragmatic trial-and-error grounds with little success for reflection or thinking through the rationale” (Fullan, 2007, p. 24), teachers will be encouraged and allowed to take half day “leaves of renewal” to reflect and plan for upcoming lessons based on their small group discussions. The premise that “meaning fuels motivation” (Fullan, 2007, p. 39) will be constantly on the mind of the DOPC and his/her staff. In summary, the “Year of Culture” will focus on the hiring of a new position whose focus is facilitating activities designed to help the district created a shared meaning answering two questions, “How do we feel students best learn?” and “How will we ensure that students are getting the best possible learning experience each and every day?” These prompts are designed to create a change in the beliefs and understanding of the district staff so that new teaching approaches and an alteration of beliefs are the foundation of this educational change (Fullan, 2007, p. 30).

The events described in the previous section are a pre-requisite for “year two” events. If necessary, the “Year of Culture” may be extended for an additional time period until its objectives have been met. Only after the “Year of Culture” should the “Year of Technology,” year two, begin. The focus of this year is on technology. The first goal of this year will be to re-evaluate year one. What worked? What did not work? John Kotter proclaims that

“The central issue is never strategy or structure…[It] is always about changing the behavior of people” (Fullan, 2007, p. 42).

In other words, unless a change in culture has taken place, the suggestions in this section will be virtually null and void. Without truly understanding the importance of changing the way we educate students, the district will continue to focus on the innovation, technology, rather than on how technology can affect or improve the way we teach students (Fullan, 2007, p. 111).
The small group discussions in this year will now focus on two new questions that are directly related to the first two.

1. How does technology fit in with our view of educating students?
2. How will we ensure that technology will not replace “old ideas” and instead create “new opportunities” for students to learn?

The DOPC’s new responsibilities will include identifying opinion leaders to give additional responsibilities and encouragement. The strategic selection of opinion leaders should include individuals who are not too innovative themselves (Rogers, 2003, p. 318), and have a high degree of interconnectedness with a small group of individuals within the district (Rogers, 2003, p. 327). The goal of the opinion leaders will be to spread the word about instructional technology as a means of teaching students in a more effective way. Through the use of several opinion leaders, the system will be able to avoid this change in culture as being an authority innovation-decision and rather a collective-innovation decision in which the group joins together to begin the process together (Rogers, 2003, p. 403).

The underlying theme of the “Year of Technology” is obviously on technology, but more specifically the use of technology to transform teaching and learning. Individual testimonials will be encouraged to model how technology can become as routine and necessary as a read-aloud, daily oral language or mental math were once seen in an elementary classroom. The purpose of these individual testimonials is to influence teachers’ beliefs and attitudes, keep the idea of culture change in mind, and begin the process of influencing individuals one by one to match technology tools with best practices in pedagogy for their classrooms. Educators should be encouraged to ask opinion leaders through small group discussions “how, under what conditions, and to what degree” should technology be used in the classroom (Cuban, 2001, p. 192). Finally, conversations will continually connect back to the “Year of Culture” by emphasizing,

“It is not about technology; it is about learning” (Cuban, 2001, p. 184).

Because access is not the sole problem (Cuban, 2001, p. 175) and more on-demand technical support is often needed in public education (Cuban, 2001, p. 180), new money the district receives will be split between technical support staff and new technology. New technology purchases will be based on teacher-initiated projects tied specifically to strategies designed to fuel the needs and interests of individual students. Hardware and software will no longer be purchased for the sake of spending the yearly budget on new machines and the usual maintenance, but rather as a solution to implement effective teaching and learning.

Following the “Year of Technology,” the process will not come to a screeching halt. The role of the opinion leaders and DOPC will be to energize others to make good decisions in the future (Fullan, 2007, p. 300). Previous years should be evaluated continuously. The focus will continue to be on changing the beliefs of individuals one by one. Fullan (2007) boldly states,

“When enough people start doing the right thing in the setting in which they work, they end up changing their very context” (p. 302).

Every effort should be made to continue small group discussions and to make the workplace professionally rewarding so that the district will continue to “attract and retain good people” (Fullan, 2007, p. 129). In today’s competitive market for teachers, XYZ has had the advantage of being a desirable district to teach in that is geographically located in Iowa's technology corridor allowing the recruitment of quality staff. This stigma may eventually run out someday so an increased emphasis on recruiting quality staff is needed. In addition, the student as a stakeholder in education should be carefully examined. When students know what is expected of them, receive quick feedback and guidance on improving, their learning will improve (Fullan, 2007, p. 176). This should not be overlooked throughout the process. Last, parents are important stakeholders, too. XYZ educators have traditionally held positive attitudes towards parent involvement in the district and this mentality should continue in order to secure external funding for future technology purchases and increased engagement with their students’ academics.

In conclusion, the XYZ District needs to spend a significant amount of time going through a process of changing the culture of its educators. Educators’ existing beliefs about the way(s) in which students best learn need to be replaced by an abbreviated list of no more than five brief ideas. This process will no doubt take an extended amount of time. I have suggested creating a new position in order to help facilitate this process. Through a year (or more) of small group conversations, it is my hope that the staff will begin to see a “need” for change. These groups may be seen as forms of what Fullan called professional learning communities because the primary goal of the conversations was to extend the district’s commitment to continuous improvement (Fullan, 2007, p. 151). Through these groups and the ensuing culture change, instructional technology as defined early in this paper will be introduced as a means for accomplishing this change. Technology purchases transitioned from budget and maintenance-minded to being based on individual teachers’ instructional needs. I hope to find my place in this recommended change process.

“The role of leadership…concerns ‘those behaviors that enabled others to take up their role in relation to the institution’s main and defined task’” (Fullan, 2007, p. 165).

It is my hope that this plan will enable me to begin the XYZ district’s search for a defined task through a change in culture and in turn enable others to take up their role as well. We all have a stake in the whole (Fullan, 2007, p. 303).




Cuban, L. (2001). Oversold and underused: Computers in the classroom. Cambridge,
MA: Harvard University Press.

Fullan, M. (2007). The new meaning of educational change (4th Ed.). New York, NY: Teachers College Press.

Rogers, E.M. (2003). Diffusion of innovations. (5th ed.).

"Can you tell me how to do it?"

(or worse yet, "What formula should I use?")

"Can you help me figure out what I did wrong?"

Students ask these two questions of each other and of me on a daily basis. They are two totally different questions. As we reviewed for tomorrow's Geometry test, I had an opportunity to explain the difference to my students. A brief synopsis follows.

A student that asks, "Can you tell me how to do it?" most likely...

• has not thought it through first for his/herself, and/or
  
• more concerned about the "answer" than the concept/algorithm

A student that asks, "Can you help me figure out what I did wrong?" most likely...

• has already given some original thought to the problem, and/or
• has struggled through multiple ways of approaching the problem

The latter question is obviously the one I emphasized as students reviewed together.

From the Iowa Core Curriculum on "Teaching for Understanding":

"Teachers assist students in making connections between prior and new knowledge to develop deep conceptual and procedural knowledge"

When a student asks "Can you tell me how to do it?" or "What formula should I use?," I am not able to determine his/her prior knowledge. Perhaps this is why I struggle with answering whole group questions on the previous day's assignment at the beginning of class. When I spit out the solution to p. 697 #34, I am inevitably answering the question "Can you tell me how to do it?" for my students! Ideally, each student would be able to figure out their own misconceptions (a la metacognition), right?! On a more realistic note, it would involve me looking at what the student did and did not do on that particular problem to help him/her overcome the misconception.

In the future, I'm looking forward to more opportunities to model to my students the better question of the two, "Can you help me figure out what I did wrong?" To me, this is what formative assessment is all about.

Me:

"Why do I give you time during class to work on your math problems?"

Students:

"It is busy work? You want us to do something during class?"
"You're a nice teacher? You want us to have less to do outside of class?"

Wrong! Neither was the the answer I was looking for today. The point I was trying to make with my students as we end the school year is the value of collaboration. While students are in class, they have access to....

• answers posted on the board - that's right, all homework answers are "free" in this classroom
  
• other students who are generally working on the same assignment - an invaluable resource as group work is highly encouraged on a daily basis
• me, the instructor as I wander around the room and peek over their shoulders looking for common misconceptions and answering as many questions as possible in the allotted time.

Can I really blame my students for answering the "why class time?" question? Next year, I plan on focusing more on classroom culture. Emphasizing the importance of what we do and why we do it on a frequent basis is a first step in this direction. Here's a start at a few frequently asked questions and the responses I'd like my students to ideally answer after a semester in my classroom.

1. Why am I assigned homework problems each and every day?
   Homework is practice. I need it to figure out what I "know" and what I "don't know" so that I can ask questions when I don't understand something. Not doing homework is like not trying on a pair of pants in the dressing room before purchasing them - a risk!
   
   
2. Why do you get time during class to work on homework problems?
   During class, I have access to answers, friends and my teacher! At home, these resources are not readily available. It's like looking for food at the grocery store - all around me!
   
   
3. Why do we take quizzes in this class?
   Quizzes are a way to find out "what I know" and "what I don't know" in a more structured format. It's a great opportunity to get detailed feedback and learn from my mistakes.

This list is just the beginning as I continue to think about changing the culture of my math classroom. Looking ahead to next year, what questions (and ideal responses) are important to you as you think about the classroom culture?

(This post is the second in a series based on metacognition as a way of improving classroom assessment and instruction)

In response to How Students Learn, a second book was written addressing discipline-specific strategies. How Students Learn: History, Mathematics and Science in the Classroom is an outstanding resource for those who want to dig deeper in to Bransford's key principles. The emphasis on metacognition continues as well:

"Ultimately, students need to develop metacognitive abilities - the habits of mind necessary to assess their own progress - rather than relying solely on external indicators" (Donovan & Bransford, 2005, p. 17)

The application of this idea seems to be natural from a theoretical point of view - learners must be able to tell themselves whether or not they are understanding a given concept. From a practitioners perspective, I am unsure how well this idea is accepted and applied in a typical mathematics classroom. Based on my experience and observations, a secondary math class looks something like this three day cycle:

DAY #1
8:20 - Teacher reads answers aloud; students check with a red pen
8:25 - Teacher asks if there are any questions; One student raises hand to ask what answer to #6 was as s/he was too busy writing the correct answer to problems 1-5 to hear the answer to #6
8:27 - Students hand in papers to teacher to be recorded in grade book
8:30 - Lesson ABC begins; students take notes
9:00 - Students work on homework, wondering if they have the "right answers"
9:15 - Bell rings; students leave not knowing if they understand lesson ABC

DAY #2: Students come back and cycle repeats itself. Students turn in homework for lesson ABC and are assigned homework for lesson XYZ
DAY #3: Students come back and receive "graded" lesson ABC homework that they started at 9:00 two days ago.

Two questions come to mind:

1. What opportunities are we providing for our students to go back and learn from their mistakes when their mistakes are sitting in a folder on our desk "waiting to be recorded"?
2. Even if we write detailed, diagnostic feedback on each students' assignment and give it back to them the next day, how many will take the time to go back and fix their mistakes?

I believe we're amusing ourselves if we think students will take the necessary time to learn from their mistakes after 48 hours have elapsed since they first started the assignment.

"Metacognitive functioning is also facilitated by shifting from a focus on answers to just right or wrong to a more detailed focus on 'debugging' a wrong answer, that is finding out where the error is, why it is an error, and correcting it." (Donovan & Bransford, 2005, p. 239)

The teacher-as-answer-holder system as we know it seems to be missing a valuable aspect of formative assessment - providing students with an opportunity to revise and improve their thinking. In my search for literature related to metacognition and assessment during the past few days, I stumbled across an ERIC article on student self-assessment. From the article:

"...students who assessed their own work were remarkably willing to revise it."

Sounds good to me. Perhaps we've never given our students enough opportunities to assess and revise their own work!

Possible Solution: Students must be trained how to assess and revise their own work.
Notice the key characteristics...trained....assess...revise.
I started out the semester working hard to train students to "check answers with a pen as I read the answers aloud and to not copy odd answers from the back of the book." I'm finding that this is a hard habit to break. For the past three weeks, I have not read the answers aloud, but instead encouraged students to check their answers against the key both during their work time as well as the next day when they come back to class. In general, students are simply not accustomed to the idea of checking their own work - they must be re-trained. There are the few who take full advantage of this new system and check their answers regularly. Others will check their work, but are unwilling to ask questions of their peers and/or me to overcome their misconceptions. The revision never takes place. I'm looking forward to the weeks to come as I develop strategies to break this habit that's been created by me and so many others in the "teacher-as-answer-holder system."

Just as we model positive group work behavior, passing in papers and appropriate use of technology modeling self assessment must be at the core of our daily practice. The image to the left is a sketch I wrote on the board for a student today as I was attempting to help him see the value of self-assessment.

The homework checking scheme I am currently piloting clearly encourages students to become "self-assessors" by eliminating the "teacher-as-answer-holder system," but I am unsure if it lives up to the revise key characteristic made above. Language arts instructors seem to have this characteristic down through the use of multiple drafts of an essay or research paper, but this seems like a relative weakness in the math education realm.

Looking towards the final post in this series, what strategies have you found to be useful in helping students revise their work through the lens of self-assessment?

Practice with feedback matters, according to a 2007 study (pdf) published in the British Journal of Educational Technology. I've been hooked on this idea ever since I read Bransford's How People Learn two years ago. It's on my "highly recommend" list for other educators.

"The use of frequent formative assessment helps make students' thinking visible to themselves, their peers, and their teacher." (Bransford et al., 2000, p. 19)

As I mentioned in a previous post, I'm intrigued by the intersection between standards-based assessment and 21st century learning. In fact, I'm on a continual search for ways in which math education, assessment and technology might work best together. Where does "practice with feedback" fit in then? Without a doubt, formative assessment is a pretty hot topic right now at the Iowa Department of Education. In fact, I was at a district meeting of sorts last night and the "instructional decision making" model came up time and time again in conversation. IDM is the idea that instruction driven by data can help all students improve. K-6 teachers seem to "get it" through their use of DIBELS, BRI and DRA screening probes whereas us secondary folk are...well, assigning homework and giving out quizzes a few times a week! I believe that standards-based assessment/reporting has the potential to enhance my formative assessment practices, because it will better enable me to focus on what I'm teaching and how it should best be assessed. When I bring up connecting standards with assessment, the response I typically get is something like "Well, our text book matches our standards and we use the textbook tests and quizzes, so it must match up." I'm not convinced.

On the flip side, when I hear elementary teachers talk about setting up supplemental groups and choosing several different novels for their students to read based on ability, I admit that I get a bit jealous. It seems to me like the secondary folk are missing the boat somewhere. Math teachers are seldom accused of not giving our students enough practice, but I would like to propose that we are too often lacking in the area of meaningful feedback. The IDM probes referenced above are one way of enabling educators to provide appropriate instruction and in turn, meaningful feedback to those who need it, when they need it, and at a level that is appropriate for them.

Then there's the typical high school math classroom: Giving students a score out of five on a daily basis based on the neatness and accuracy of their answers, ability to show their work, and the responsibility of bringing a checking pen to class doesn't seem like meaningful feedback to me - especially when it takes 24 hours to collect papers, record scores and hand them back. A student said to me a few months ago, "I like knowing if I'm doing it right or wrong...today!" What is the "answer" to this problem? Several technology-related answers come to mind:

• Eliminate homework and make the move to more problem-based learning using appropriate technology tools
• Create some sort of electronic means (Moodle?) of providing meaningful feedback on a daily basis
• Who cares about homework?! Use student-response systems as daily probes to assess students' understanding
  

I have already decided to make the answers available to students ahead of time in order to help them get immediate feedback. I'm hoping to create a system that truly does make the students' thinking visible to both them and me. Reporting it out in a standards-based way seem like part of the picture, too. What am I missing in order to make practice with feedback a more visible part of my daily routine?

One of the most thought-provoking phrases related to education and technology that's ever crossed my RSS feed is

At a conference last week, Mark Weston from Dell computing stated that asking the question, "Does technology improve student learning?" is the wrong question....

The question should be, "Does technology support the practices that improve student learning?"

-Doug Johnson on Blue Skunk Blog

Since I started my graduate work, I've been on a rollercoaster ride of thinking about the use of techology in my classroom - both the pros and the cons. In a somewhat ironic outcome, I've started to wonder if technology itself can actually hinder learning at times rather then help. I'll come back to this thought.

Any educator whose gone through a pre-service program in the past ten years has surely heard the phrases constructivism, student-centered learning, and best practices - sometimes in the same sentence! I am no exception to this rule. As a seasoned math educator, I can attest to the fact that some learning goals are more suited for this style of learning while others are simply...not. For example, teaching Geometry students that the sum of the interior angles of any n-sided polygon is (n-2)*180 lends itself very well to a student-centered approach. On the flipside, instructing students how to solve multi-step linear equations or how to use the quadratic formula does not lend itself to a constructivist teaching style. James Cangelosi's book Teaching Mathematics in Secondary and Middle School speaks of the need to differentiate between construct-a-concept, discover-a-relationship, and algebraic skills, and developing knowledge objectives. The content/objective should determine the type of pedagogy used to teach the idea.

The same analogy applies to the use of technology in our classrooms. Are using blogs, Moodle, Google Docs, or Twitter the "best practice" for every discipline and for every standard in our curriculum? I've long been an advocate for Mishra and Koehler's TPACK framework.



With all of the talk in the education (technology) blogosphere about "best practices," I would like to propose that "better practices" is a much better term. There really is no "best" technology tool or teaching style to push unless it meets up with the corresponding content to be taught in the middle of the TPACK Venn diagram above. Wikis, for example, might have different uses in a British literature classsroom than they would in a ceramics class. At the same time, wikis may have no place in a particular P.E. unit. When we push things like "iPods" and "cooperative learning" blindly, its no wonder the masses see them as "add-ons."

Venn diagrams are hard to sell our colleagues...but so are "best practices." Are you ready for the era of "better" practices?