What's your vision?

From Black Elk Speaks

After the long winter of waiting, it was my first duty to go out lamenting. So after the first rain storm I began to get ready.

I recently heard Kent Dobson talk about this idea of "lamenting" - what is often thought of as a "vision quest." [I apologize in advance if I get some of the details wrong in this post. A lot of what I'm writing is a combination of my memory of and the connections made during his talk. Please let me know in the comments if anything needs correcting.] 

In the Lakota tradition, when a seeker comes of age, he or she goes off with an elder to "cry for a vision." After some preparation, the seeker is left alone to lament the current state of the community and seek answers in the form of a vision. From time to time, the elder looks in on the lamenting to advise and support the seeker. When the lamenting is complete, the seeker returns to the community and shares the vision. This is an important rite for the community because without new ideas, a community withers and dies.

As I begin a new year of supervising student teaching, I want this ritual to inform my work. I hope to play the role of the elder supporting coming-of-age teachers as they experience (and lament) the current circumstances in math education and seek new answers. I will listen to those answers and take them seriously because in many ways our profession is withering. And I will help the student teachers to share their visions with the larger math education community, so that these seekers might contribute to the development of our profession.

Thank you in advance for welcoming these seekers and helping to interpret and implement their visions.

What needs changing?

The Michigan Department of Education is considering changes to Michigan teacher certification and asking for feedback. Here is more information about their proposed certification structure.

Feedback is due today. (Sorry, I'm still a bit of a procrastinator.) However, if you consider yourself as having any stake in education, I encourage you to complete the survey, as soon as possible. It'll take maybe 45 minutes to watch the video and answer some questions.

For what it's worth, here's my response. (And, yes, I know I am biased.)

My issue with the structure being proposed is that it seems to presume that new teachers ought to be completely prepared upon certification. Even if we focused on giving specific grade-level certification, there is no way to address all the issues (textbooks, culture, community, resources, …) new teachers might encounter across the state (or country). Furthermore, this structure will limit the options available to teachers and potentially restrict administrators' ability to fill positions. I can imagine school districts asking for waivers to put PK-3 certified teachers in 4th grade because of a lack of options - a teacher that may have no training at that level.

I encourage that the state look at the Grand Valley State University teacher certification program as a model. We support the development of teacher-leaders in content areas by requiring even pre-service elementary teachers to get a subject major. Local districts regularly call us looking for recent majors because they know they will be content experts who can also integrate other subjects into their practice.

Also, the MDE needs to rethink professional development for inservice teachers. Because no teacher is truly finished learning, we must return to a robust professional development process that helps teachers build on current education experiences to deepen their pedagogical content knowledge in all discipline they might be teaching. Such a system is especially important now because we have so many new, inexperienced teachers entering the profession.

This professional development ought to be in partnership with accredited bodies responsible for initial teacher certification. These institutions are aware of the training new teachers bring to the profession and can provide meaningful experiences that can expand teachers’ current abilities. The experiences would be the result of the institutions and inservice teachers collaborating on areas needing improvement.

In my discipline, we often hold up Dr. Deborah Ball (former dean of the University of Michigan’s School of Education) as the model elementary math teacher. However, we fail to recognize that Dr. Ball rose to this level after identifying this as an area of weakness in her teaching – after being a certified, inservice teacher (see p. 9). She decided to do something about it by going back to school and learning more about the subject and its teaching. We need to allow all inservice teachers to follow the path from awareness to adjustment without fear of being labeled unprepared or ineffective. 

While teacher preparation institution must make changes to better prepare teacher-leaders, teacher certification is not the primary problem. A lack of opportunities for meaningful professional development is the issue. Changing teacher certification to try to address a broken inservice support system focuses our efforts in the wrong direction and may do further damage to education in Michigan.

Thank you, in advance, for your attention to this issue.

How can we assess 3.MD.B.3?

Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. [3.MD.B.3]

Teaching-Learning Cycle

Teachers use #MTBoS as a way to find interesting and effective math lessons. Recently, some of us have noticed that assessments are often lacking as this community shares in the work of teaching. So I am proposing the #MTBoSAP (Math Teacher Blog-o-Sphere Assessment Project) as a way to pass along our wisdom and experience assessing students in our mathematics classes. I figured I could start by sharing some work I am doing in one of my courses for preservice elementary teachers.

The teachers are currently researching assessment items related to the Measurement & Data Domain (focusing on Data) from the Common Core State Standards (CCSS) in Mathematics. One of the third-grade standards in that domain is written at the beginning of this post. Because we are partnering with a school district that uses EngageNY, we looked for assessment items that already existed within that curriculum. This is an exit ticket that we found in the Grade 3, Module 6, Lesson 4. 

We thought that this item did a fair job of assessing the last part of 3.MB.B.3 but completely missed the first part, "Draw a scaled picture graph and a scaled bar graph..." So we looked through the rest of the lesson and thought this item from the problem set looked promising.

This item has students "draw a scaled bar graph" (still no scaled picture graph) and asks that they solve "two-step 'how many more' and 'how many less' problems." As we thought about it further, however, we were concerned that it was not clear whether students would use the chart or the graph to answer the questions.

Therefore, I decided to try to modify the original exit ticket in order to assess more of the standard. I got rid of two of the bars and wrote questions intended to have them "draw (a part of) a scaled bar graph" and answer multi-step questions that require some "information presented in scaled bar graphs."

Please complete the bar graph using the following information. 

• The number of books checked out on Thursday was 10 more than the number of books checked out on Tuesday. Draw the Thursday bar.
• 1,480 books were checked out Monday through Friday. Draw the Friday bar.

What are your thoughts about using this modified assessment item to gather data on students' mathematical understanding related to 3.MD.B.3? Do you have a good item for this standard that you'd be willing to share? If so, please share the item or a link to the item in the comments. Also, If you have any other effective CCSS assessment items, please share them on Twitter using #MTBoSAP. Thank you in advance for all you do to advance the profession.

Whose learning?

The following is a guest post from John Golden. Go Blue!

+++++

I was a terrible student.

I make this confession to my students all the time. When I finished undergraduate, my friends calculated whether I had attended more class, missed more class, or slept in more class. It was close. But I was a great test taker and I got good grades. I thought that class was all about what I did at home afterwards. I had no expectations of learning in class, and was delighted when I did. (From Dr. Hocking, for example.)

So I was a terrible student, but I was a good learner.

When I started teaching I focused on being entertaining and comfortable. I wanted students to enjoy being there, free to ask questions (it didn’t take too long to figure out that this was hard) and get what they wanted. But even with the ineffective assessments I was giving, it soon became clear that they weren’t understanding what I wanted them to understand.

Yesterday, I was listening to a podcast and was struck by how well the speaker captured this idea. It was Avery (@woutgeo) on Ashli’s (@Mythagon) Infinite Tangents podcast (105).

“At the beginning of my teaching I was very content focused. I need to find and create interesting content that will help the kids, that the kids will be really interested in. I had no middling success with that. I think I found some great things, and also found some other things that didn’t work and I found that teaching some of the important ideas were harder than I thought they would be.  I realized that there were plenty of ideas that I didn’t really understand myself, that I had to go back and think about a lot.  I think going back and thinking about those ideas really helped me shift a little bit to not just caring about trying to find interesting content.  But also thinking about pedagogy and how we teach things and the best ways for kids to learn.  If I had to think of an arc of my teaching career and my focus, I would say that that kind of describes my arc of going from being very content-focused to a balance between finding interesting content and also really thinking about the best way for kids to learn the content and have an experience.” – Avery

I mentioned that on Twitter and there was an interesting discussion that followed.

Shawn Urban (@stefras): Most teachers are great with content; some with relationships. The marriage of the two is often tricky.

Greg ‏(@sarcasymptote): I think the relationship piece is about trust. Young Ts fear giving up control of content to Ss

Ashli (@mythagon): I'd like to see more mentoring in teaching. Not enough time is built in for such work typically.

Shawn: I agee. Teacher prep should involve teaching & study of styles of teaching. We need models to inspire creativ

John (@mathhombre) we put a lot of effort into genuine assessment and conversation about learning. But the programming of 16 yrs...

Shawn: ... this Tweet seems incomplete. I was waiting for something mind-blowing! No pressure though.

John: we've seen so much teaching as student that we've made up what teaching is and really internalized it.

John: but it's such an incomplete picture! (And possibly not the most constructive teaching.)

Ashli:  I think it takes careful, respectful questioning to get new teachers to see beyond content.

John: @Mythagon can you say more about that questioning?

Shawn: I wonder if Ashli was refering to key questions. Why are you teaching? What do you want your students to be learning?

Ashli: I was thinking about ?s that help teachers think beyond pure delivery of content and toward formative

Ashli: + assessments, status issues. yr 1 can be a whirlwind of survival

Shawn: I thought so. Deeper, more reflective pedagogical questions.

Chris Robinson ‏(@absvalteaching): Best question any T can have while planning is "What misconceptions/mistakes will Ss have and +

Chris: + how do I plan to address these?"

Shawn: I agee. Teacher prep should involve teaching & study of styles of teaching. We need models to inspire creativ

Shawn: I think we need frequent exposure in how students learn. This is key since their learning is our job.

Shawn: When did we make teaching so complex that we forgot to teach? Even r students r distracted & forget to learn

Gary ‏(@republicofmath): IMO teaching IS very complex

Ashli: Schools can easily become a memorization gauntlet.

Great conversation about teacher learning for me.

One of the things I have grown to love about teaching is how holistic it is. When I reflect about how I learn, even learn to teach, it teaches me about how students learn. As a teacher I have to make that step to thinking about how the students learn. The kernel at the heart of this for me is a humbling one:

I can not make my students learn.

That means I can not take credit for what they understand, nor blame for what they don’t.

What can I do then? I can create the conditions of learning. I can make it as likely as possible that students will choose to learn. I can monitor what works for them and adjust. Once I care about the things I do control, I am empowered. Still frustrated at the choices some students make, but thrilled at what others do.

PS> Not that I wouldn’t be happy to have a post on the DeltaScape, but I lost a bet. I knew it was a bad bet - which of our Big Ten alma maters would go farther in the 2013 NCAA men’s basketball tournament. I didn’t think my beloved Spartans would make it past Duke. But frankly, I thought this post would be on Robert’s Casting Out Hoosiers blog. (Wait - that doesn’t sound right.) But, sadly, I haven’t learned my lesson, and would make the same bet next year.

Should I use elimination or substitution?

Be forewarn, this is probably not the post you are looking for.

A teacher I was coaching sent our this tweet after a recent observation. The class was looking at solving systems of linear equations using elimination and substitution and we had talked about how he could use data from recent Big Ten basketball games to set up scenarios that might encourage them to use the different techniques. Early in the lesson, several of the students (the class is all boys) made it clear that they were huge basketball fans.

Here are a couple of examples I came up with:

Against Wisconsin, Trey Burke scored 19 points. He made 1 free throw and 2 three pointers. What would his box score look like for scoring?

Trey Burke scored 18 points against Michigan State. He made 1 free throw and 7 shots from the field. What would his box score look like for scoring? 

Christopher Danielson has thought more and deeper about this topic than I, so I defer to his expertise (here). Sorry for the bait-and-switch, but I want to think about how this experience helped me to re-evaluate my homework policy.

About two years ago, I wrote about how overwhelmed our novice teachers are during their first teaching experience. We do this somewhat intentionally so that they can experience firsthand all that is expected of teachers, and so that they can become better at managing their time. Part of that time-management is learning to prioritize what is important and learning how to say, "No!" -- in other words, elimination. (I know that we aren't talking about mathematical elimination anymore, but this is where my mind goes. Pity my wife and my co-workers.) In that post, I went as far as to say, "It is okay to skip this assignment."

Looking back, I think that suggesting this technique, elimination, was a mistake in this situation. Simply skipping work does not match the reality of teaching. In order to prepare them for that reality, what I want them to be good at is substitution. If something else on their plate is more important than one of our assignments, then they ought to make a substitution. However, as they do their other work, I want them to recognize they are doing it on our time and look for ways to synthesize the concepts and skills we have been exploring into their "more important" work. Hopefully, this mindful effort to synthesize across classes will provide practice in attempting to meld ideas in their teaching practice.

Again, my apologies for bringing you here under potentially false pretenses. And for making you wade through the ramblings of my mind. As I tweeted Christopher recently, I need an editor. If you have your own thoughts, please add them to the comments.

Why did you do that?

We just completed another semester at Grand Valley State University, and once again I found myself asking our teachers-in-training some version of "Why did you...?" after many of the observations. I try to make it clear that this is an authentic question not some sort of accusation of wrong doing that demands an accounting. Seriously, I want to know the rationale behind some of their teacher moves. Also, I want them to be mindful of why they make certain decision during their lessons. Consequently, this is what I would call a win-win question.

For example, many student teachers struggle with time management during class, and on their Action Plan they will ask for suggestions about how to be more efficient while conducting a lesson. During the subsequent observation, I often see a typical lesson component. (Stigler and Hiebert were right about there being an often unconscious script associated with teaching math.) The teacher provides some time for individual or small group practice followed by a whole class discussion of the practice items. 

Afterwards, I ask, "Why did you go over all the items? What did you see as you walked around that made you decide that this was necessary?" Again, this is intended to be a serious question, and I sometimes get answers that teach me something about them as teachers and their students as learners. Some of the teachers say something like:

• "I noticed that most of them were struggling on [some aspect of the practice] and decided we needed to look at it as a class." or
• "There were a few different approaches and I wanted the class to see that problems can be solved in multiple ways." or
• "While they got all the practice items correct, I wanted to provide them an opportunity to communicate mathematically. They still struggle with vocabulary and precision and I thought this would be a good time to practice these given that they understood the concepts."

But occasionally, the response I get is, "What do you mean? We always go over the items after their practice. You mean I can skip this.?" The discussion that follows is nearly always goose-bump-inducing as the novice teacher begins to develop phronesis (what could I do here and what's worth doing?).

My goal as a teacher educator is not to develop a bunch of Dave Coffey clones. One is enough. Besides, my approach would not work for all the different teachers I work with over a semester. But if I become the voice in their head that asks after a lesson, "Why did you do that?" - I can live with that.

What if they ask me a question I can't answer?

I often coach student teachers assigned to teach topics they have not worked with in a long time. The last time many of them had Algebra or Trigonometry was in high school and some "rust" obscures their understanding of the subject. They are understandably concerned about making a mistake while working through examples or demonstrating solutions to homework questions. 

I can empathize having had to teach College Algebra many, many years after my last encounter with the material. Before each lesson I had to reteach myself the topics. (It didn't help that my initial interactions with the College Algebra content were what Skemp would call instrumental.) And when it came time to address homework issues, I was always concerned that I would make a mistake in front of the class. Turns out I had my own issues with math-phobia.

While I am comfortable with the idea of making mistakes in my mathematics education classes, because it represents the learning process, I have found college freshmen and sophomores taking a required math course less understanding than my preservice teachers. Students in College Algebra often have certain expectations of what mathematics teaching looks like and when they encounter something different (mistakes), they can shut down. So early on I want to establish a certain level of confidence for myself and for them.

I share this story with preservice teachers so that they can feel comfortable with the fact that any nagging doubts they have about teaching a topic they need to relearn can be expected and dealt with. There are two suggestions that I make to build their confidence. The first is to avoid answering off-the-cuff questions until they feel more comfortable with the content and the classroom. And the other is to always have the examples they want to share written out ahead of time.

When it comes to questions that arise in class, if it's not something the student teachers have prepared for I tell them it's alright to write the question down and tell the class they will get back to it. This models that all mathematical questions don't have immediate answers and hopefully helps students to become more comfortable embracing the patience required to be successful in learning relationally (see Skemp). Regarding homework questions, I suggest that the student teachers set up a procedure by which students can identify which problems they want support on. If this can be done beforehand (I used Blackboard), then the student teachers can determine which items caused the most confusion and prepare a whole-class demonstration to support student learning. Items identified by a single student can be attended to through a written example or individual conferencing. If there is no way to gather this information beforehand, then I suggest collecting it at the end of the class period when it is expected to be completed and preparing the support for the next period. In either case, have the work written out ahead of time.

When they share their examples, at least early on, I encourage the student teachers not to recopy the work on the board but to use the overhead or document camera to project their work. This addresses several issues that teachers trying to build their confidence often face. First, it tightens up time management. Writing on the board is time consuming and can allow students to get off-task. (Believe it or not, many of them are not copying it down.) I would simply project my work and add my thinking as I go over each step. Second, projecting previously written work alleviates the potential for making a mistake and eroding confidence for those already dealing with this issue. Multi-tasking when stressed is difficult, and I have found that the combination of copying and talking through my thinking can lead to errors even though I have written everything out correctly. Finally, and this is huge for new teachers in front of a class of teenagers, using written work projected on the board usually means not having to turn your back on the class. If you've ever taught in a secondary math class, you'll understand what I mean.

Let me be clear, I am not suggesting that this approach become the norm; it is just the place to start when teaching a new prep that might require some relearning of topics. Once a classroom culture of trust and respect has been developed, I have found that I can go back to responding to questions more immediately. But I still reserve the right to say, "I'll have to get back to you on that one."

How will it work?

I need your help. Due to circumstances beyond our control, the GVSU Department of Mathematics is looking at canceling two courses this fall semester. Teaching Middle Grades Mathematics [MTH 329] is required for undergraduates interested in teaching secondary mathematics (though some inservice teachers take it as part of adding an endorsement in mathematics to their certificate). Secondary Student Issues [MTH 629] is part of a College of Education's Masters of Education program. Because of the importance of these two courses to their respective planned programs, we are considering alternatives to canceling them.

One option is to combine the two courses in some way. This is of particular interest to me because of past successes with preservice and inservice teachers collaborating in MTH 329. As I said, we sometimes get inservice teachers taking this course for an endorsement, and I always try to include their perspective when discussing the realities of teaching and learning. I also connected 329 students with inservice teachers two years ago when a scheduling conflict meant that I needed to be teaching and conducting professional development at the same time. Participants report that the combined effect of preservice teachers' enthusiasm and inservice teachers' experience has been beneficial. 

All I need to do is come up with a proposal for how the combined course might work. My colleague, John Golden, and I sat down this morning to develop a draft, but I recognize that our plan would benefit from your feedback. Here's the idea:

Essentially, the content addressed will remain unchanged for MTH 329. The undergraduates in this course learn what it means to do, learn, and teach mathematics in the middle grades. In their course portfolio, they demonstrate their fluency of middle school-level mathematical content, their competencies in teaching and learning middle grade mathematics, and their ability to engage in their own learning.

Graduate students enrolled in MTH 629 will continue to focus on issues in teaching and learning secondary mathematics, but this will extend to mentoring the preservice teachers in mathematical pedagogy. The mentoring will involve supporting the undergraduates in the assessment and analysis of middle grade learners’ mathematical thinking and the design and implementation of mathematics micro-lessons. The graduate students’ portfolio will document their mentoring efforts, their results from an action research project, and their ability to engage in their own learning.

MTH 329 will meet from 6 to 7:50 on Tuesdays and Thursdays. MTH 629 will meet from 6 to 8:50 on Tuesdays. During the 6 to 7:50 overlap on Tuesdays, the class time will focus on developing a taken-as-shared understanding of pedagogical concepts through demonstrations, classroom dialogues, and collaborations. From 8 to 8:50 on Tuesdays, MTH 629 students will concentrate on aspects of effective mentoring and conducting action research that focuses on the artifacts of teaching (lessons and assessments). On Thursdays, the MTH 329 students will focus on the middle grades mathematical content typically addressed in this course.

It is my hope that this structure will help both groups to recognize that they can contribute to the positive development of the teaching profession. I want them to understand that they can improve teaching without having to wait for some outside force to tell them what that improvement would entail. In other words, I want to provide an experience that provides them with phronesis.

So what do you think? I really value your input in designing this combined course. Thank you in advance for your support.

Can you help me with subtraction?

[Note: Based on feedback from readers, it seems this post needs some context. This is part of a series on exploring a new number system in a course for preservice elementary teachers. The purpose of this unit is to provide these future educators with an experience of what it is like to learn concepts of number and operation using a conceptual approach. In this final post of the series I describe the days leading up to the unit's final assessment. You can see the entire series by clicking on the Wumania tag at the bottom of the post.]

As we near the Wumanian number system assessment, some of the preservice elementary teachers express concern about their ability to demonstrate their fluency in computing with multi-digit numbers. I suggest that they consider putting the expressions into context using one of the Cognitively Guided Instruction stories types presented in the article, Using Student Interviews to Guide Classroom Instruction: An Action Research Project (PDF). We also try connecting the stories to manipulatives (the blocks introduced at the beginning of the unit - see below) using the approaches presented in Teaching without Telling: Computational Fluency and Understanding through Invention (PDF) and Second Graders Cirumvent Addition and Subtraction Difficulties (NCTM). In the process, I explicitly point out my use of practitioner journals to inform my instruction.

Because not everyone needs support in this area, I am in the habit of posting on our class website PowerPoint Think Alouds of how I might go about solving some of the problems on this practice sheet.

Learners with access to the internet can download the Think Alouds and watch them at their convenience. This is how those first PowerPoints looked.

With current technological tools, like Jing, here is what I can post now.

These Think Alouds provide the learners who are struggling to apply the approaches they have read about a model of the approaches in action. It also models how teachers can use technology to offer different levels of support to learners. Some might call me the Wumanian Sal Khan, but I can assure you that these productions were not one-take affairs.